From 7af7383d52199fcd8bb0a3ec3aa55d0f30dca924 Mon Sep 17 00:00:00 2001 From: KellyJDavis Date: Thu, 8 Oct 2026 07:58:08 +0200 Subject: [PATCH 1/6] Blueprint: state Minkowski Axiom 3 in a common containing diamond Restate local commutativity through the isotony maps into any diamond containing both regions, parallel to the curved Axiom 3, and record the quasilocal form as an equivalent lemma. Move the canonical embeddings into the quasilocal-algebra definition, and correct Chapter 5's argument that isotony cannot place two spacelike algebras in a common algebra. Co-Authored-By: Claude Opus 5.5 --- .../sections/sec10/haag-kastler-axioms.tex | 73 +++-- .../src/sections/sec5/quasilocal-algebra.tex | 25 +- .../sec9/axiom-3-local-commutativity.tex | 2 +- .../plan-axiom3-common-diamond-2026-10-08.md | 283 ++++++++++++++++++ 4 files changed, 350 insertions(+), 33 deletions(-) create mode 100644 notes/agents/plan-axiom3-common-diamond-2026-10-08.md diff --git a/blueprint/src/sections/sec10/haag-kastler-axioms.tex b/blueprint/src/sections/sec10/haag-kastler-axioms.tex index 8b4dd67..04e2f41 100644 --- a/blueprint/src/sections/sec10/haag-kastler-axioms.tex +++ b/blueprint/src/sections/sec10/haag-kastler-axioms.tex @@ -47,7 +47,7 @@ \section{Haag Kastler Axioms}\label{sctn:haag-kastler-axioms} \subsection{The Quasilocal Colimit} -The quasilocal algebra of \ref{def:quasilocal-algebra} below rests on a chain of algebraic facts about the directed system of Axiom 2, which we record in this subsection. They fall into two groups: those putting a normed $*$-algebra structure on the colimit, assembled in \ref{lmm:quasilocal-union-normed-star-algebra}, and those passing from that to its completion, assembled in \ref{lmm:quasilocal-completion-cstar}. Axioms 3--5 depend only on \ref{def:quasilocal-algebra} itself and not on how it is built, so a reader after the axiom presentation alone may skip this subsection. +The quasilocal algebra of \ref{def:quasilocal-algebra} below rests on a chain of algebraic facts about the directed system of Axiom 2, which we record in this subsection. They fall into two groups: those putting a normed $*$-algebra structure on the colimit, assembled in \ref{lmm:quasilocal-union-normed-star-algebra}, and those passing from that to its completion, assembled in \ref{lmm:quasilocal-completion-cstar}. Axioms 4 and 5 depend only on \ref{def:quasilocal-algebra} itself and not on how it is built (Axiom 3 does not use it at all), so a reader after the axiom presentation alone may skip this subsection. \begin{lemma}[Alexandrov Diamonds are Directed under Inclusion] \label{lmm:alexandrov-diamonds-isDirected} @@ -360,69 +360,86 @@ \subsection{The Quasilocal Colimit} \emph{The construction route is settled: colimit, then completion.} The project constructs the quasilocal algebra \emph{from the net alone} --- the directed colimit of the local C*-algebras along the isotony family, followed by completion --- which is exactly the chain running from \ref{def:completion-standing-hypotheses} through \ref{lmm:quasilocal-completion-cstar}. This is a decision that has been taken, not an option still under consideration, and the reason is physical. The alternative below is rejected. -\emph{The rejected alternative, and why it is rejected.} The alternative was to realise the quasilocal algebra as a \emph{closed} $*$-subalgebra of an ambient C*-algebra: take the $*$-subalgebra generated by the images of the canonical embeddings $\iota_{\mathbf{B}}$ --- data that Axiom 3 (\ref{def:local-commutativity}) already assumes --- and close it off with \texttt{StarSubalgebra.topologicalClosure}. That description is accurate and the saving would have been real: \texttt{StarSubalgebra.cstarAlgebra} supplies the \emph{entire} C*-structure as an instance --- involution, norm, completeness, the C*-identity, the algebra structure and the \texttt{StarModule} field all at once --- so \ref{def:completion-standing-hypotheses}, \ref{def:completion-star}, \ref{lmm:star-extends-to-completion}, \ref{lmm:completion-cstar-identity}, \ref{lmm:completion-normed-algebra} and \ref{lmm:completion-of-cstar-normed-star-algebra} would all have become unnecessary, with none of the density arguments above written by hand. +\emph{The rejected alternative, and why it is rejected.} The alternative was to realise the quasilocal algebra as a \emph{closed} $*$-subalgebra of an ambient C*-algebra: take the $*$-subalgebra generated by the images of the canonical embeddings $\iota_{\mathbf{B}}$ --- data that the quasilocal algebra structure (\ref{def:quasilocal-algebra}) already assumes --- and close it off with \texttt{StarSubalgebra.topologicalClosure}. That description is accurate and the saving would have been real: \texttt{StarSubalgebra.cstarAlgebra} supplies the \emph{entire} C*-structure as an instance --- involution, norm, completeness, the C*-identity, the algebra structure and the \texttt{StarModule} field all at once --- so \ref{def:completion-standing-hypotheses}, \ref{def:completion-star}, \ref{lmm:star-extends-to-completion}, \ref{lmm:completion-cstar-identity}, \ref{lmm:completion-normed-algebra} and \ref{lmm:completion-of-cstar-normed-star-algebra} would all have become unnecessary, with none of the density arguments above written by hand. It is rejected because of what it presupposes. \texttt{StarSubalgebra.cstarAlgebra} takes an \emph{ambient} \texttt{CStarAlgebra A} as a hypothesis and equips a closed $*$-subalgebra of it with the induced structure; it does not produce a C*-algebra out of nothing. So the shortcut requires an ambient C*-algebra containing isomorphic copies of every local algebra $\mathfrak{U}(\mathbf{B})$ to be given in advance --- and there is no physical justification for such an algebra. Nothing in the physics supplies one, nothing says where it would come from, so it cannot be assumed. That is the whole of the argument: the shortcut would have the blueprint assume $\mathfrak{U}$ rather than construct it from the net, and assuming it has no basis in the physics. The completion chain above starts from the net alone and \emph{builds} a C*-algebra, discharging the existence claim that \ref{def:quasilocal-algebra} makes. \emph{The existing route-2 material in this repository is not invalidated.} \texttt{Physicslib4.AQFT.HaagKastler.dense\_adjoin\_iUnion\_range\_\ensuremath{\iota}} (in \texttt{QuasilocalIntertwiner.lean}) and \texttt{Physicslib4.exists\_starAlgHom\_extend\_of\_dense} (in \texttt{Physicslib4/Analysis/CStarDenseExtend.lean}) implement the dense-extension technique and both take the quasilocal algebra as given. Taking it as given is perfectly legitimate for a \emph{consumer} of the construction: these are downstream results about a quasilocal algebra, not competing constructions of one, and they remain valid verbatim. What the decision changes is only that the blueprint must now \emph{also} construct such an algebra, which is what the chain above does. Indeed \texttt{CStarDenseExtend.lean} remains the closest worked example in this repository of transporting star-structure across a dense embedding, so it is the reference to read before formalizing the completion nodes above. -Before introducing the next axiom, we must introduce the definition: +We next introduce the quasilocal algebra. Axiom 3 below does not need it, but has an equivalent form in it (\ref{lmm:local-commutativity-iff-quasilocal}), and Axiom 4 is stated through it. \begin{definition}[Quasilocal Algebra] \label{def:quasilocal-algebra} \lean{Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra} - \leanfile{Physicslib4/AQFT/HaagKastler/QuasilocalAlgebra.lean} \uses{def:local-algebras, def:isotony, lmm:minkowski-diamonds-upward-directed, lmm:quasilocal-union-normed-star-algebra, lmm:quasilocal-completion-cstar} \leanok Consider the union of all $\mathfrak{U}(\mathbf{B})$, taken along the isotony family of Axiom 2 (\ref{def:isotony}). This union is a normed *-algebra; taking its completion one obtains a C*-algebra denoted $\mathfrak{U}$, called the \textit{quasilocal algebra}. + A quasilocal algebra comes with \emph{canonical embeddings} $\iota_{\mathbf{B}} : \mathfrak{U}(\mathbf{B}) \to \mathfrak{U}$, one for each basis set $\mathbf{B}$: injective unital $*$-homomorphisms, each the map of a local algebra into the completion of the union of the local algebras. The map $\iota_{\mathbf{B}}$ is \emph{not} the unital $*$-monomorphism $i_{\mathbf{B}_1\mathbf{B}_2} : \mathfrak{U}(\mathbf{B}_1) \hookrightarrow \mathfrak{U}(\mathbf{B}_2)$ of Axiom 2 (Isotony), which goes from one local algebra to another and not into $\mathfrak{U}$. + + The two families are required to be compatible: for all basis sets $\mathbf{B}_1 \subseteq \mathbf{B}_2$, + \begin{align} + \iota_{\mathbf{B}_2} \circ i_{\mathbf{B}_1\mathbf{B}_2} = \iota_{\mathbf{B}_1}. + \end{align} + This is the cocone condition making the $\iota_{\mathbf{B}}$ a compatible family on the directed system of Axiom 2 (\ref{def:isotony}), and it is what makes $\iota_{\mathbf{B}}$ well defined on the colimit: an element of $\mathfrak{U}(\mathbf{B}_1)$ may be regarded as an element of any larger $\mathfrak{U}(\mathbf{B}_2)$, and all such readings must have the same image in $\mathfrak{U}$. Without it, a product $\iota_{\mathbf{B}_1}(a_1)\, \iota_{\mathbf{B}_2}(a_2)$ in $\mathfrak{U}$ would depend on which local algebra $a_1$ and $a_2$ were viewed in. + The union is to be read as a \emph{directed colimit}, not as a set-theoretic union. This is what Axiom 2's identity and composition laws buy: with $\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$ a functor on the inclusion order, and the Alexandrov diamonds upward directed, the local algebras form a directed system and the colimit carries a well-defined $*$-algebra structure --- the product of $a_1 \in \mathfrak{U}(\mathbf{B}_1)$ and $a_2 \in \mathfrak{U}(\mathbf{B}_2)$ is computed in any $\mathfrak{U}(\mathbf{B})$ containing both, and functoriality is exactly what makes the answer independent of that choice. Without the two laws there is no such structure: a bare set-theoretic union of the $\mathfrak{U}(\mathbf{B})$ has no multiplication at all, since elements of different local algebras live in unrelated carriers. The three supporting results this definition rests on are now declarations of this blueprint rather than appeals to the informal discussion of Chapters 6.1 and 6.2: (i) upward directedness of the Alexandrov diamonds (\ref{lmm:minkowski-diamonds-upward-directed}), which supplies the containing diamond $\mathbf{B}$ in the description above; (ii) that the colimit of the directed system is a normed $*$-algebra (\ref{lmm:quasilocal-union-normed-star-algebra}); and (iii) that its completion is a C*-algebra (\ref{lmm:quasilocal-completion-cstar}). All three are formalized. What would pin $\mathfrak{U}$ down up to isomorphism rather than merely produce it is uniqueness of the complete C*-norm; that is discussed as prose in the passage following \ref{lmm:quasilocal-completion-cstar} and is deliberately not claimed by any declaration here, for the reasons given there. - The colimit-and-completion construction of $\mathfrak{U}$ is therefore formalized, while the Lean structure continues to take $\mathfrak{U}$ as \emph{given} --- an ambient C*-algebra together with the canonical embeddings $\iota_{\mathbf{B}}$ and their cocone condition, as recorded in \ref{def:local-commutativity}. There is no longer any discrepancy between this blueprint's formalization markings and the Lean, and the construction and the structure are now connected: that connection is made by \ref{thrm:quasilocal-algebra-exists}, which produces a quasilocal algebra for the net out of the colimit-and-completion construction, and it is from that theorem that a Haag--Kastler net (\ref{def:haag-kastler-net}) obtains its canonical $\mathfrak{U}$. See the formalization note following \ref{lmm:quasilocal-completion-cstar} on the two available construction routes. + The colimit-and-completion construction of $\mathfrak{U}$ is therefore formalized, while the Lean structure continues to take $\mathfrak{U}$ as \emph{given} --- an ambient C*-algebra together with the canonical embeddings $\iota_{\mathbf{B}}$ and their cocone condition, as recorded above. There is no longer any discrepancy between this blueprint's formalization markings and the Lean, and the construction and the structure are now connected: that connection is made by \ref{thrm:quasilocal-algebra-exists}, which produces a quasilocal algebra for the net out of the colimit-and-completion construction, and it is from that theorem that a Haag--Kastler net (\ref{def:haag-kastler-net}) obtains its canonical $\mathfrak{U}$. See the formalization note following \ref{lmm:quasilocal-completion-cstar} on the two available construction routes. \end{definition} \begin{definition}[Axiom 3: Local Commutativity] \label{def:local-commutativity} \lean{Physicslib4.AQFT.HaagKastler.LocalCommutativity} - \leanfile{Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean} - \uses{def:alexandrov-topology, def:minkowski-spacetime, def:chronological-future-and-chronological-past, def:local-algebras, def:isotony, def:completely-spacelike, def:quasilocal-algebra} - \leanok - Let $\mathbf{B}_1$ and $\mathbf{B}_2$ be any two basis elements of the Alexandrov topology on Minkowski spacetime, i.e. any two sets of the form $I^+(p_1) \cap I^-(q_1)$ and $I^+(p_2) \cap I^-(q_2)$. + \uses{def:alexandrov-topology, def:minkowski-spacetime, def:chronological-future-and-chronological-past, def:local-algebras, def:isotony, def:completely-spacelike, lmm:minkowski-diamonds-upward-directed} + Let $\mathbf{B}_1$ and $\mathbf{B}_2$ be any two basis elements of the Alexandrov topology on Minkowski spacetime, i.e. any two sets of the form $I^+(p_1) \cap I^-(q_1)$ and $I^+(p_2) \cap I^-(q_2)$. Assume Axiom 2 (\ref{def:isotony}), and let $i_{\mathbf{B}_1\mathbf{B}_2}$ denote its chosen isotony family. - If $\mathbf{B_1}$ and $\mathbf{B_2}$ are completely spacelike, then $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute in the quasilocal algebra $\mathfrak{U}$, i.e. for any $a_1$ in $\mathfrak{U}(\mathbf{B_1})$ and $a_2$ in $\mathfrak{U}(\mathbf{B_2})$ it follows that + If $\mathbf{B_1}$ and $\mathbf{B_2}$ are completely spacelike, then for any Alexandrov topology basis element $\mathbf{B}$ such that $\mathbf{B_1}, \mathbf{B_2} \subseteq \mathbf{B}$ the algebras $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute in the C*-algebra $\mathfrak{U}(\mathbf{B})$: for any $a_1$ in $\mathfrak{U}(\mathbf{B_1})$ and $a_2$ in $\mathfrak{U}(\mathbf{B_2})$, \begin{align} - \iota_{\mathbf{B}_1}(a_1)\, \iota_{\mathbf{B}_2}(a_2) - \iota_{\mathbf{B}_2}(a_2)\, \iota_{\mathbf{B}_1}(a_1) = 0 + i_{\mathbf{B}_1\mathbf{B}}(a_1)\, i_{\mathbf{B}_2\mathbf{B}}(a_2) - i_{\mathbf{B}_2\mathbf{B}}(a_2)\, i_{\mathbf{B}_1\mathbf{B}}(a_1) = 0 \end{align} - in the quasilocal algebra $\mathfrak{U}$. Here $\iota_{\mathbf{B}} : \mathfrak{U}(\mathbf{B}) \to \mathfrak{U}$ is the \emph{canonical embedding} of a local algebra into the quasilocal algebra of \ref{def:quasilocal-algebra} --- the map into the completion of the union of the local algebras. It is \emph{not} the unital $*$-monomorphism $i_{\mathbf{B}_1\mathbf{B}_2} : \mathfrak{U}(\mathbf{B}_1) \hookrightarrow \mathfrak{U}(\mathbf{B}_2)$ of Axiom 2 (Isotony), which goes from one local algebra to another and not into $\mathfrak{U}$. + in the C*-algebra $\mathfrak{U}(\mathbf{B})$. - The two families are required to be compatible: for all basis sets $\mathbf{B}_1 \subseteq \mathbf{B}_2$, - \begin{align} - \iota_{\mathbf{B}_2} \circ i_{\mathbf{B}_1\mathbf{B}_2} = \iota_{\mathbf{B}_1}. - \end{align} - This is the cocone condition making the $\iota_{\mathbf{B}}$ a compatible family on the directed system of Axiom 2 (\ref{def:isotony}), and it is what makes $\iota_{\mathbf{B}}$ well defined on the colimit: an element of $\mathfrak{U}(\mathbf{B}_1)$ may be regarded as an element of any larger $\mathfrak{U}(\mathbf{B}_2)$, and all such readings must have the same image in $\mathfrak{U}$. Without it the displayed commutator would depend on which local algebra $a_1$ and $a_2$ were viewed in. + Such a $\mathbf{B}$ always exists, by upward directedness of the Minkowski diamonds (\ref{lmm:minkowski-diamonds-upward-directed}), so the condition is never vacuous. - Note that the curved counterpart (\ref{def:local-commutativity-in-curved-spacetime}) has a different shape: there is no quasilocal algebra in curved spacetime, so commutation is stated inside a common containing algebra $\mathfrak{U}(\mathbf{B})$ using the Axiom 2 embeddings directly, and no $\iota$ appears. + This axiom has the same shape as its curved counterpart (\ref{def:local-commutativity-in-curved-spacetime}): in both, commutation is stated inside a common containing algebra $\mathfrak{U}(\mathbf{B})$ using the Axiom 2 embeddings directly. The one point where they differ is the existence of $\mathbf{B}$: in Minkowski spacetime a containing basis element always exists, as just noted, whereas in a curved spacetime it need not, and there the condition can be vacuous for a given pair $\mathbf{B}_1$, $\mathbf{B}_2$. - By \ref{lmm:local-commutativity-any-quasilocal} the choice of quasilocal algebra in this axiom is immaterial, so the canonical $\mathfrak{U}$ of a Haag--Kastler net can be used throughout. + In the quasilocal algebra (\ref{def:quasilocal-algebra}), this axiom is equivalent to commutation of the images $\iota_{\mathbf{B}_1}(a_1)$ and $\iota_{\mathbf{B}_2}(a_2)$ (\ref{lmm:local-commutativity-iff-quasilocal}), and it then holds in every quasilocal algebra of the net (\ref{lmm:local-commutativity-any-quasilocal}). \end{definition} \begin{lemma}[Local Commutativity Holds in Every Quasilocal Algebra] \label{lmm:local-commutativity-any-quasilocal} \lean{Physicslib4.AQFT.HaagKastler.LocalCommutativity.commute_ι, Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra.commute_ι_iff_commute_map} \uses{def:local-commutativity, def:quasilocal-algebra, def:completely-spacelike, lmm:minkowski-diamonds-upward-directed} - \leanok - Suppose the net satisfies Axiom 3 (\ref{def:local-commutativity}) with respect to some quasilocal algebra $\mathfrak{U}$, i.e. an ambient C*-algebra with injective unital $*$-homomorphisms $\iota_{\mathbf{B}} : \mathfrak{U}(\mathbf{B}) \to \mathfrak{U}$ satisfying the cocone condition $\iota_{\mathbf{B}_2} \circ i_{\mathbf{B}_1\mathbf{B}_2} = \iota_{\mathbf{B}_1}$ for all basis sets $\mathbf{B}_1 \subseteq \mathbf{B}_2$. Let $\mathfrak{U}'$ be any quasilocal algebra of the net (\ref{def:quasilocal-algebra}), with canonical embeddings $\iota'_{\mathbf{B}}$. Then for any completely spacelike basis sets $\mathbf{B}_1$, $\mathbf{B}_2$ and any $a_1 \in \mathfrak{U}(\mathbf{B}_1)$, $a_2 \in \mathfrak{U}(\mathbf{B}_2)$, + Suppose the net satisfies Axiom 3 (\ref{def:local-commutativity}). Let $\mathfrak{U}'$ be any quasilocal algebra of the net (\ref{def:quasilocal-algebra}), with canonical embeddings $\iota'_{\mathbf{B}}$. Then for any completely spacelike basis sets $\mathbf{B}_1$, $\mathbf{B}_2$ and any $a_1 \in \mathfrak{U}(\mathbf{B}_1)$, $a_2 \in \mathfrak{U}(\mathbf{B}_2)$, \begin{align} \iota'_{\mathbf{B}_1}(a_1)\, \iota'_{\mathbf{B}_2}(a_2) = \iota'_{\mathbf{B}_2}(a_2)\, \iota'_{\mathbf{B}_1}(a_1) \end{align} - in $\mathfrak{U}'$. In particular Axiom 3 holds in the canonical quasilocal algebra of \ref{thrm:quasilocal-algebra-exists}, and does not depend on which quasilocal algebra is used to state it. + in $\mathfrak{U}'$. In particular these images commute in the canonical quasilocal algebra of \ref{thrm:quasilocal-algebra-exists}. \end{lemma} \begin{proof} - \leanok - \uses{def:local-commutativity, def:isotony, lmm:minkowski-diamonds-upward-directed} - By upward directedness of the Alexandrov diamonds (\ref{lmm:minkowski-diamonds-upward-directed}) pick a basis set $\mathbf{B}$ with $\mathbf{B}_1 \cup \mathbf{B}_2 \subseteq \mathbf{B}$. By the cocone condition, in any quasilocal algebra $\iota_{\mathbf{B}_k}(a_k) = \iota_{\mathbf{B}}(i_{\mathbf{B}_k\mathbf{B}}(a_k))$ for $k = 1, 2$, where $i_{\mathbf{B}_k\mathbf{B}}$ are the isotony embeddings of Axiom 2 (\ref{def:isotony}). Since $\iota_{\mathbf{B}}$ is an injective $*$-homomorphism, $\iota_{\mathbf{B}_1}(a_1)$ and $\iota_{\mathbf{B}_2}(a_2)$ commute if and only if $i_{\mathbf{B}_1\mathbf{B}}(a_1)$ and $i_{\mathbf{B}_2\mathbf{B}}(a_2)$ commute in $\mathfrak{U}(\mathbf{B})$. That condition does not mention the quasilocal algebra, so commutation in the algebra for which Axiom 3 is assumed transfers to $\mathfrak{U}'$ with its embeddings $\iota'_{\mathbf{B}}$. + \uses{def:local-commutativity, def:quasilocal-algebra, def:isotony, lmm:minkowski-diamonds-upward-directed} + Let $\mathbf{B}_1$, $\mathbf{B}_2$ be completely spacelike. By upward directedness of the Alexandrov diamonds (\ref{lmm:minkowski-diamonds-upward-directed}) pick a basis set $\mathbf{B}$ with $\mathbf{B}_1 \cup \mathbf{B}_2 \subseteq \mathbf{B}$. By the cocone condition of \ref{def:quasilocal-algebra}, $\iota'_{\mathbf{B}_k}(a_k) = \iota'_{\mathbf{B}}(i_{\mathbf{B}_k\mathbf{B}}(a_k))$ for $k = 1, 2$, where $i_{\mathbf{B}_k\mathbf{B}}$ are the isotony embeddings of Axiom 2 (\ref{def:isotony}). Since $\iota'_{\mathbf{B}}$ is an injective $*$-homomorphism, $\iota'_{\mathbf{B}_1}(a_1)$ and $\iota'_{\mathbf{B}_2}(a_2)$ commute if and only if $i_{\mathbf{B}_1\mathbf{B}}(a_1)$ and $i_{\mathbf{B}_2\mathbf{B}}(a_2)$ commute in $\mathfrak{U}(\mathbf{B})$, and that is what Axiom 3 (\ref{def:local-commutativity}) gives for this $\mathbf{B}$. +\end{proof} + +\begin{lemma}[Local Commutativity in the Quasilocal Algebra] + \label{lmm:local-commutativity-iff-quasilocal} + \uses{def:local-commutativity, def:quasilocal-algebra, def:completely-spacelike} + A net satisfies Axiom 3 (\ref{def:local-commutativity}) if and only if there is a quasilocal algebra $\mathfrak{U}$ (\ref{def:quasilocal-algebra}), with canonical embeddings $\iota_{\mathbf{B}}$, in which, for all completely spacelike basis sets $\mathbf{B}_1$, $\mathbf{B}_2$ and all $a_1 \in \mathfrak{U}(\mathbf{B}_1)$, $a_2 \in \mathfrak{U}(\mathbf{B}_2)$, + \begin{align} + \iota_{\mathbf{B}_1}(a_1)\, \iota_{\mathbf{B}_2}(a_2) = \iota_{\mathbf{B}_2}(a_2)\, \iota_{\mathbf{B}_1}(a_1); + \end{align} + and in that case this holds in every quasilocal algebra of the net. +\end{lemma} +\begin{proof} + \uses{lmm:local-commutativity-any-quasilocal, thrm:quasilocal-algebra-exists, lmm:minkowski-diamonds-upward-directed, def:isotony, def:quasilocal-algebra} + ($\Rightarrow$) A quasilocal algebra exists by \ref{thrm:quasilocal-algebra-exists}, and \ref{lmm:local-commutativity-any-quasilocal} gives the commutation in it, and indeed in every quasilocal algebra. + + ($\Leftarrow$) Let $\mathbf{B}_1$, $\mathbf{B}_2$ be completely spacelike and let $\mathbf{B}$ be any basis set with $\mathbf{B}_1 \cup \mathbf{B}_2 \subseteq \mathbf{B}$ (one exists by \ref{lmm:minkowski-diamonds-upward-directed}). By the cocone condition, $\iota_{\mathbf{B}_k}(a_k) = \iota_{\mathbf{B}}(i_{\mathbf{B}_k\mathbf{B}}(a_k))$ for $k = 1, 2$, with $i_{\mathbf{B}_k\mathbf{B}}$ the isotony embeddings of Axiom 2 (\ref{def:isotony}). Since $\iota_{\mathbf{B}}$ is an injective $*$-homomorphism, commutation of the images in $\mathfrak{U}$ is equivalent to commutation of $i_{\mathbf{B}_1\mathbf{B}}(a_1)$ and $i_{\mathbf{B}_2\mathbf{B}}(a_2)$ in $\mathfrak{U}(\mathbf{B})$, which is Axiom 3 for this $\mathbf{B}$. + + The right-hand side of the equivalence was the form of Axiom 3 in earlier versions of this blueprint. \end{proof} The next axiom makes use of the following new definition: @@ -764,7 +781,7 @@ \subsection{Lorentz Covariance and the Haag--Kastler Net} \subsection{Einstein Causality} -Local commutativity (\ref{def:local-commutativity}) is an algebraic statement about the quasilocal algebra. Its physical content - that spacelike-separated measurements do not interfere - is its operator form: in any representation of the quasilocal algebra, spacelike-separated local observables commute as bounded operators on the Hilbert space. +Local commutativity (\ref{def:local-commutativity}) is an algebraic statement about the local algebras, which in the quasilocal algebra takes the form of \ref{lmm:local-commutativity-any-quasilocal}. Its physical content - that spacelike-separated measurements do not interfere - is its operator form: in any representation of the quasilocal algebra, spacelike-separated local observables commute as bounded operators on the Hilbert space. \begin{theorem}[Einstein Causality in a Representation] \label{thrm:einstein-causality} @@ -776,8 +793,8 @@ \subsection{Einstein Causality} \end{theorem} \begin{proof} \leanok - \uses{def:local-commutativity} - This is the image under $\pi$ of the local commutativity relation (\ref{def:local-commutativity}); since the GNS representation of any state is such a $\pi$, spacelike-separated local observables commute on every GNS Hilbert space. + \uses{def:local-commutativity, lmm:local-commutativity-any-quasilocal} + This is the image under $\pi$ of the commutation relation of \ref{lmm:local-commutativity-any-quasilocal} in the quasilocal algebra, which holds by local commutativity (\ref{def:local-commutativity}); since the GNS representation of any state is such a $\pi$, spacelike-separated local observables commute on every GNS Hilbert space. \end{proof} \subsection{Local von Neumann Algebras} diff --git a/blueprint/src/sections/sec5/quasilocal-algebra.tex b/blueprint/src/sections/sec5/quasilocal-algebra.tex index c43aa8b..960ed9d 100644 --- a/blueprint/src/sections/sec5/quasilocal-algebra.tex +++ b/blueprint/src/sections/sec5/quasilocal-algebra.tex @@ -3,9 +3,17 @@ \section{Quasilocal Algebra}\label{sctn:quasilocal-algebra} When one requires that $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute, this tacitly implies that $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ are subalgebras of a single, larger algebra, an algebra in which products and sums of their elements are defined. -Naively one might hope that isotony would come to the rescue. However, this isn't the case. If, say, $\mathbf{B_1} \subset \mathbf{B_2}$, isotony would imply that $\mathfrak{U}(\mathbf{B_1}) \subset \mathfrak{U}(\mathbf{B_2})$. This would place $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ in a single, larger algebra $\mathfrak{U}(\mathbf{B_2})$ in which they could commute. However, the assumption that $\mathbf{B_1} \subset \mathbf{B_2}$ would imply that $\mathbf{B_1}$ and $\mathbf{B_2}$ are not completely spacelike. Isotony doesn't come to the rescue. - -However, what does come to the rescue is the quasilocal algebra $\mathfrak{U}$. By definition $\mathfrak{U}(\mathbf{B_1}) \subset \mathfrak{U}$ and $\mathfrak{U}(\mathbf{B_2}) \subset \mathfrak{U}$. So $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ are in a single, larger algebra $\mathfrak{U}$. We can thus interpret the statement +Isotony is the natural tool for this. If, say, $\mathbf{B_1} \subset \mathbf{B_2}$, isotony +places $\mathfrak{U}(\mathbf{B_1})$ inside $\mathfrak{U}(\mathbf{B_2})$, so that both live in the +single algebra $\mathfrak{U}(\mathbf{B_2})$. That case is of no use here, since nested regions are +never completely spacelike. But isotony does not require one of the two regions to contain the +other: it suffices to find a \emph{third} basis element $\mathbf{B}$ containing both. In Minkowski +spacetime such a $\mathbf{B}$ always exists --- any two double cones $I^+(p_1) \cap I^-(q_1)$ and +$I^+(p_2) \cap I^-(q_2)$ lie inside a larger double cone $I^+(p) \cap I^-(q)$, by taking $p$ +far enough in the past of $p_1$ and $p_2$ and $q$ far enough in the future of $q_1$ and $q_2$. +Isotony then places $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ inside the single, +larger algebra $\mathfrak{U}(\mathbf{B})$, in which products and sums of their elements are +defined. We can thus interpret the statement \begin{quote} If $\mathbf{B_1}$ and $\mathbf{B_2}$ are completely spacelike with respect to each other, then $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute. @@ -14,7 +22,16 @@ \section{Quasilocal Algebra}\label{sctn:quasilocal-algebra} as the statement \begin{quote} - If $\mathbf{B_1}$ and $\mathbf{B_2}$ are completely spacelike with respect to each other, then $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute in the quasilocal algebra $\mathfrak{U}$. + If $\mathbf{B_1}$ and $\mathbf{B_2}$ are completely spacelike with respect to each other, then $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute in $\mathfrak{U}(\mathbf{B})$ for every basis element $\mathbf{B}$ containing both $\mathbf{B_1}$ and $\mathbf{B_2}$. \end{quote} +Two remarks. First, the choice of $\mathbf{B}$ does not matter: because the isotony embeddings +are injective and compatible, commutation in one containing algebra implies commutation in every +other. Second, the quasilocal algebra $\mathfrak{U}$ of the next chapter contains every +$\mathfrak{U}(\mathbf{B})$, so the statement can equally be read as commutation in +$\mathfrak{U}$; the two readings are equivalent. The formulation through a containing +$\mathfrak{U}(\mathbf{B})$ is preferred because it uses only the net and isotony, and because it +is the formulation that survives in curved spacetime (Chapter 9), where a containing basis element +need not exist and the quasilocal algebra need not exist either. + Clarification. diff --git a/blueprint/src/sections/sec9/axiom-3-local-commutativity.tex b/blueprint/src/sections/sec9/axiom-3-local-commutativity.tex index 18a38d2..1c59992 100644 --- a/blueprint/src/sections/sec9/axiom-3-local-commutativity.tex +++ b/blueprint/src/sections/sec9/axiom-3-local-commutativity.tex @@ -16,7 +16,7 @@ \subsection{Quasilocal Algebra?} \end{figure} What is obvious is that there exists no basis element $\mathbf{B}$ of the Alexandrov topology that contains both $\mathbf{B}_1$ and $\mathbf{B}_2$ as it would have to extend beyond the singularity off of $M_{BH}$ which is of course not allowed. -As a result of this, one can not employ the isotony axiom, the only tool one has, to place $\mathfrak{U}(\mathbf{B}_1)$ and $\mathfrak{U}(\mathbf{B}_2)$ in a larger algebra, an algebra in which addition and multiplication of their elements would make sense. +As a result of this, one can not employ the isotony axiom, the only tool one has, to place $\mathfrak{U}(\mathbf{B}_1)$ and $\mathfrak{U}(\mathbf{B}_2)$ in a larger algebra, an algebra in which addition and multiplication of their elements would make sense. This is exactly where curved spacetime parts ways with Minkowski spacetime: there, any two basis elements lie in a common, larger basis element (Section~\ref{sctn:quasilocal-algebra}), so isotony always places their algebras in a common algebra, and Axiom 3 can be stated in it. This implies that the set-theoretic union of all $\mathfrak{U}(\mathbf{B})$ over all basis elements $\mathbf{B}$ of the Alexandrov topology on $M_{BH}$ is not a normed *-algebra. This in turn implies that the construction of the quasilocal algebra fails\footnote{This argument can be formalized. However, doing so doesn't provide any insight beyond what's presented by this informal argument. Hence, we omitted the formal argument.} for $M_{BH}$. diff --git a/notes/agents/plan-axiom3-common-diamond-2026-10-08.md b/notes/agents/plan-axiom3-common-diamond-2026-10-08.md new file mode 100644 index 0000000..dea8e96 --- /dev/null +++ b/notes/agents/plan-axiom3-common-diamond-2026-10-08.md @@ -0,0 +1,283 @@ +# Plan: restate Minkowski Axiom 3 through a common containing diamond + +Date: 2026-10-08. Toolchain: Lean / Mathlib v4.34.1. Proposed branch: `numina/axiom3-common-diamond` +off `numina/aqft-in-lean`. Status: **approved 2026-10-08** (D1–D5 as recommended, including the Chapter 5 wording +and the D5 Chapter 9 sentence). Nothing is changed until +the user approves: `LocalCommutativity` is an axiom definition, so it is a protected design +decision under CLAUDE.md. + +## Goal + +Restate Definition 485 (`def:local-commutativity`, Minkowski Axiom 3) so that commutation happens +in the local algebra 𝔘(B) of any Alexandrov diamond B containing both B₁ and B₂, using the Axiom 2 +isotony maps. This is the same shape as Definition 628 (`def:local-commutativity-in-curved-spacetime`). +The current form, commutation in the quasilocal algebra 𝔘, becomes a proved equivalent. + +## Mathematical content + +Write `LC_new` and `LC_old` for the two forms. + +- **`LC_new`.** For all diamonds B₁, B₂, B with B₁, B₂ completely spacelike and B₁, B₂ ⊆ B, and + all a ∈ 𝔘(B₁), b ∈ 𝔘(B₂): i_{B₁B}(a) and i_{B₂B}(b) commute in 𝔘(B). +- **`LC_old`** (today's Lean). There is a quasilocal algebra Q such that, for B₁, B₂ completely + spacelike, ι_{B₁}(a) and ι_{B₂}(b) commute in Q. +- **The two are equivalent.** Everything needed is already proved. + - **The bridge.** `QuasilocalAlgebra.commute_ι_iff_commute_map` (LocalCommutativity.lean): for any + Q and any diamond B ⊇ B₁ ∪ B₂, ι_{B₁}(a), ι_{B₂}(b) commute in Q iff i_{B₁B}(a), i_{B₂B}(b) commute + in 𝔘(B). + - **old ⇒ new.** Apply the bridge to the given Q, for every B. + - **new ⇒ old.** Take Q from `exists_quasilocalAlgebra` (QuasilocalExistence.lean, built from + the net alone). Pick a common B by upward directedness (`Spacetime.alexandrovBasis_directed`) and + apply the bridge. +- **The two forms are equivalent, not one strictly stronger than the other.** The gain is + foundational, not logical. The axiom then speaks only of the net and isotony, with no + quantifier over ambient C*-algebras. It no longer depends on `def:quasilocal-algebra`. It reads in + parallel with the curved Axiom 3, so the one real difference is visible: whether a common + containing diamond always exists. It does in Minkowski space; it may not in curved spacetime + (the Schwarzschild example of Chapter 9). +- **"For every common B" vs "for some common B".** In Minkowski space these are equivalent, by + directedness and injectivity of the isotony maps. The recommendation is "for every", so the Lean + definition is the curved one with only the basis-set predicate swapped. + +## Decisions needed + +**D1. Quantifier over the containing diamond.** +- (a) **Recommended:** "for every diamond B ⊇ B₁ ∪ B₂", identical in shape to Definition 628. +- (b) "for some diamond B ⊇ B₁ ∪ B₂". This is closer to "they live in a common algebra", but + it is no longer parallel to Definition 628. + +**D2. Placement of Axiom 3 in §10.6.** Axiom 3 would no longer use `def:quasilocal-algebra`. +- (a) **Recommended:** keep its current position, after `def:quasilocal-algebra`. Only the + dependency edge is dropped, the chapter is not reordered, and item numbers stay put except for + the one new lemma. +- (b) Move Axiom 3 directly after Axiom 2 (Isotony). This reads more naturally, but it reorders + your chapter, and every item number between the two positions shifts. + +**D3. The equivalence as a new node.** +- **Recommended:** add `lmm:local-commutativity-iff-quasilocal`, "Local Commutativity in the + Quasilocal Algebra". It says Axiom 3 holds iff, for some (equivalently every) quasilocal + algebra, the images of completely spacelike local algebras commute. That keeps the old form + citable. +- The existing `lmm:local-commutativity-any-quasilocal` (Lean `commute_ι`) stays as the "⇒, in + every quasilocal algebra" direction, with a simpler proof. +- The alternative is to fold the equivalence into that existing lemma (no new node, so no + renumbering), at the cost of a two-part statement. + +**D4. Chapter 5 wording.** Chapter 5 is your motivation chapter. It is not formalised, but it is +read before Axiom 3. Its §5 "Quasilocal Algebra" argues that isotony cannot place 𝔘(B₁) and +𝔘(B₂) in a common algebra. That considers only B₁ ⊂ B₂ and misses a third diamond B containing +both, which always exists in Minkowski space. Under the new Axiom 3 the argument becomes +incorrect, not merely unused. +- (a) **Recommended:** replace the "Naively …" / "However, what does come to the rescue …" + paragraphs and the two quotes with the proposed wording below. +- (b) Keep the text and add one corrective sentence. +- (c) Leave Chapter 5 unchanged. Not recommended, because it would contradict Axiom 3. + +**D5. Chapter 9 cross-reference (optional).** Chapter 9 §"Axiom 3 (Local Commutativity)" says +the absence of a common containing diamond is why curved spacetime differs. With the new +Minkowski axiom that contrast is exact. +- **Recommended:** add one sentence to Chapter 9 pointing back to Chapter 5's "a common containing + diamond always exists in Minkowski space". +- Alternatively, leave Chapter 9 as is. + +## Proposed Chapter 5 wording (for D4(a)) + +File: `blueprint/src/sections/sec5/quasilocal-algebra.tex`. Keep the section heading and the +opening two paragraphs ("The second point that requires clarification is a bit more subtle." and +"When one requires that … in which products and sums of their elements are defined."). Replace +everything from "Naively one might hope that isotony would come to the rescue." up to, but not +including, "Clarification." with: + +```latex +Isotony is the natural tool for this. If, say, $\mathbf{B_1} \subset \mathbf{B_2}$, isotony +places $\mathfrak{U}(\mathbf{B_1})$ inside $\mathfrak{U}(\mathbf{B_2})$, so that both live in the +single algebra $\mathfrak{U}(\mathbf{B_2})$. That case is of no use here, since nested regions are +never completely spacelike. But isotony does not require one of the two regions to contain the +other: it suffices to find a \emph{third} basis element $\mathbf{B}$ containing both. In Minkowski +spacetime such a $\mathbf{B}$ always exists --- any two double cones $I^+(p_1) \cap I^-(q_1)$ and +$I^+(p_2) \cap I^-(q_2)$ lie inside a larger double cone $I^+(p) \cap I^-(q)$, by taking $p$ +far enough in the past of $p_1$ and $p_2$ and $q$ far enough in the future of $q_1$ and $q_2$. +Isotony then places $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ inside the single, +larger algebra $\mathfrak{U}(\mathbf{B})$, in which products and sums of their elements are +defined. We can thus interpret the statement + +\begin{quote} + If $\mathbf{B_1}$ and $\mathbf{B_2}$ are completely spacelike with respect to each other, then $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute. +\end{quote} + +as the statement + +\begin{quote} + If $\mathbf{B_1}$ and $\mathbf{B_2}$ are completely spacelike with respect to each other, then $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute in $\mathfrak{U}(\mathbf{B})$ for every basis element $\mathbf{B}$ containing both $\mathbf{B_1}$ and $\mathbf{B_2}$. +\end{quote} + +Two remarks. First, the choice of $\mathbf{B}$ does not matter: because the isotony embeddings +are injective and compatible, commutation in one containing algebra implies commutation in every +other. Second, the quasilocal algebra $\mathfrak{U}$ of the next chapter contains every +$\mathfrak{U}(\mathbf{B})$, so the statement can equally be read as commutation in +$\mathfrak{U}$; the two readings are equivalent. The formulation through a containing +$\mathfrak{U}(\mathbf{B})$ is preferred because it uses only the net and isotony, and because it +is the formulation that survives in curved spacetime (Chapter 9), where a containing basis element +need not exist and the quasilocal algebra need not exist either. +``` + +Notes on this wording: +- The containment claim is upward directedness of the Minkowski diamonds, which the project proves + as `lmm:minkowski-diamonds-upward-directed`. Chapter 5 is not formalised and does not cite + Chapter 10 labels, so the wording states the fact informally. +- The two `quote` environments mirror the existing text's structure. The second quote is the + wording the new Definition 485 formalises. +- Check after the edit: the next section of Chapter 5 and the beginning of Chapter 6 may say + "commute in the quasilocal algebra" or motivate 𝔘 by Axiom 3. Chapter 6's motivation for 𝔘 + (Axiom 4: all observables of interest) is independent of Axiom 3 and stays valid. Stage 1 + includes a grep of Chapters 5–6 for such sentences, and any change is proposed to you before it + is made. + +## Stages + +Each stage: build, lint, `checkdecls`, commit on the branch, and a progress entry below. + +### Stage 1: blueprint (writer, then reviewer; targeted edits) + +1. **`def:local-commutativity`** (haag-kastler-axioms.tex, around line 384): new statement modelled + on Definition 628. + - Two diamonds B₁, B₂; Axiom 2's isotony family i. If B₁, B₂ are completely spacelike, then for + every diamond B ⊇ B₁ ∪ B₂ (per D1), i_{B₁B}(a₁) and i_{B₂B}(a₂) commute in 𝔘(B). + - Add one sentence: by upward directedness such a B always exists (cite + `lmm:minkowski-diamonds-upward-directed` by `\ref`). + - `\uses`: remove `def:quasilocal-algebra`, keep the rest. + - Remove the ι / cocone paragraphs from this node (see item 2). + - Replace "Note that the curved counterpart … has a different shape …" with the single real + difference: in Minkowski a containing diamond always exists, in curved spacetime it may not. + - Keep the final pointer, reworded: the equivalent form in 𝔘 is + `lmm:local-commutativity-iff-quasilocal`. +2. **`def:quasilocal-algebra`** (around line 372): absorb the canonical embeddings ι_B and the + cocone condition ι_{B₂} ∘ i_{B₁B₂} = ι_{B₁}. These are already fields of the Lean structure + `QuasilocalAlgebra` and are now stated nowhere else. Fix its sentence "as recorded in + \ref{def:local-commutativity}". +3. **New `lmm:local-commutativity-iff-quasilocal`** (after `lmm:local-commutativity-any-quasilocal`, + per D3): statement as in D3. The proof uses the bridge, `thrm:quasilocal-algebra-exists` and + `lmm:minkowski-diamonds-upward-directed`. +4. **`lmm:local-commutativity-any-quasilocal`:** the premise becomes "Suppose the net satisfies + Axiom 3". The proof is the bridge in one direction: no second quasilocal algebra and no + directedness needed. +5. **Einstein causality** (around line 767): "Local commutativity … is an algebraic statement + about the quasilocal algebra" becomes "about the local algebras; in the quasilocal algebra it + takes the form of `lmm:local-commutativity-any-quasilocal`". Add that lemma to the proof + `\uses` of `thrm:einstein-causality`. +6. **Curved chapter:** grep for sentences contrasting the two Axiom 3 forms. The known passages + (lines 45, 141 and 286: "there is no quasilocal algebra here") stay true. Change only + sentences that describe the Minkowski axiom as "in the quasilocal algebra". +7. **Chapter 5** per D4, and **Chapter 9** per D5. Exact wording is above, and the user approves + it before the writer applies it. +8. **`\leanok`.** Strip the stale `\leanok` from `def:local-commutativity` (its Lean changes in + Stage 2). It is re-recorded after the Lean is updated. + +The reviewer checks: faithfulness to D1–D5, the equivalence proof's grounding (the three Lean +names above), no other node depending on the removed `def:quasilocal-algebra` edge, and that no +prose still says Axiom 3 lives in 𝔘. + +### Stage 2: Lean statements (orchestrator; protected definition) + +**`Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean`:** +- New definition (D1(a)): + ```lean + def LocalCommutativity (U : LocalNet) (i : Isotony U) : Prop := + ∀ ⦃B₁ B₂ B : Set StandardMinkowskiSpacetime.Carrier⦄ + (hB₁ : IsAlexandrovBasisSet B₁) (hB₂ : IsAlexandrovBasisSet B₂) + (hB : IsAlexandrovBasisSet B), + Spacetime.IsCompletelySpacelike StandardMinkowskiSpacetime + standardMinkowskiTimeOrientation B₁ B₂ → + (h₁ : B₁ ⊆ B) → (h₂ : B₂ ⊆ B) → + ∀ (a : U.algebra B₁) (b : U.algebra B₂), + Commute (i.map hB₁ hB h₁ a) (i.map hB₂ hB h₂ b) + ``` +- Module docstring and definition docstring rewritten: commutation in a common containing diamond, + parallel to `HaagKastlerCurved.LocalCommutativity`. +- `QuasilocalAlgebra.commute_ι_iff_commute_map`: unchanged. +- `LocalCommutativity.commute_ι` (`lmm:local-commutativity-any-quasilocal`): signature unchanged; + the proof becomes directedness plus one use of the bridge. +- New `localCommutativity_iff_exists_commute_ι` (`lmm:local-commutativity-iff-quasilocal`): + `LocalCommutativity U i ↔ ∃ Q : QuasilocalAlgebra U i, ∀ ⦃B₁ B₂⦄ hB₁ hB₂, IsCompletelySpacelike … → + ∀ a b, Commute (Q.ι hB₁ a) (Q.ι hB₂ b)`, i.e. the old definition's body. + - The ← direction needs only the bridge; the → direction needs `exists_quasilocalAlgebra`. + - Import `Physicslib4.AQFT.HaagKastler.QuasilocalExistence`; no cycle, since QuasilocalExistence + imports only QuasilocalAlgebra and QuasilocalColimit. + - Optionally a companion `localCommutativity_iff_forall_commute_ι` (∀ Q). + +**`Physicslib4/AQFT/HaagKastler/Net.lean`:** +- `trivialLocalNet_localCommutativity` (line 345): the old proof builds `trivialQuasilocalAlgebra`. + The new one is "the algebras are ℂ, so everything commutes". +- `commute_ι_of_spacelike` (line 213) is unchanged, since it uses `commute_ι`. Its docstring ("Axiom + 3 only asserts this in *some* quasilocal algebra") is updated. + +**Unchanged:** +- the `HaagKastlerNet.localCommutativity` field; +- `einstein_causality` and the other consumers (they go through `commute_ι_of_spacelike`); +- `Spacetime/LorentzCausality.lean`, whose docstring wording stays accurate. + +**Verification:** a grep confirms that nothing else unfolds `LocalCommutativity`. Today only +`Net.lean:345` does. + +Statements are sorry'd, then the formalizer-reviewer runs. + +### Stage 3: proofs (provers, in parallel) + +- `LocalCommutativity.commute_ι`: about 4 lines. +- `localCommutativity_iff_exists_commute_ι`: about 10 lines, possibly split into its two + directions. +- `trivialLocalNet_localCommutativity`: about 3 lines (`mul_comm` in ℂ). + +### Stage 4: verify and audit + +- Build, lint, `checkdecls`, and `#print axioms` on the new and re-proved theorems. +- lean-auditor lists the protected change (one `def` body), confirms no consumer's statement + changed, and checks blueprint/Lean agreement for the touched nodes. + +### Stage 5: publish locally + +- Rebuild the PDF and the web blueprint, and check the PDF for new overfull lines (the print + preamble now wraps Lean names). +- Home page: + - `axioms.md`: the Axiom 3 row (Minkowski) and a note under it, "Axiom 3 is now stated in a + common containing diamond, parallel to the curved form; the quasilocal form is + Lemma N". + - `guide.md`: the Chapter 5 summary (line 26) currently repeats "isotony alone cannot place two + spacelike-separated regions in a common algebra". Rewrite it per D4. + - `sec10-6-haag-kastler.md`: the axioms overview, the new entry, and refreshed entries. + - If D3 adds a node: renumber items after it and update the counts (one more lemma; with D2(b), + more renumbering), as in earlier refreshes. +- Run a local Jekyll build and the link check, then commit and merge into `numina/aqft-in-lean`. + Don't push unless asked. + +## Risks + +- **Protected definition.** The body of `LocalCommutativity` changes. The field and every theorem + signature that uses it are unchanged, and the equivalence lemma makes the old form available + under its own name. +- **Universe/import plumbing.** `exists_quasilocalAlgebra` is stated for `LocalNet.{u}`. The + equivalence inherits that universe, and `LocalCommutativity` must stay universe-polymorphic in + the same way. Easy to check in Stage 2. +- **Chapter 5 is your text.** Stage 1 applies D4 only after you approve the wording above. + +## Estimate + +- One definition change, one new lemma (two declarations if the ∀ Q companion is added) and two + re-proofs, all short. +- About seven blueprint node edits plus Chapter 5 (and optionally Chapter 9) prose. +- One home-page refresh. + +## Progress log + +(empty) +- 2026-10-08, Stage 1 done (branch `numina/axiom3-common-diamond`). Chapter 5 rewritten with the + approved wording (applied verbatim from this plan). Writer edits in haag-kastler-axioms.tex: + def:local-commutativity restated through every common containing diamond (parallel to the + curved form; \leanok/\leanfile removed), ι/cocone material moved into def:quasilocal-algebra, + lmm:local-commutativity-any-quasilocal re-premised (\leanok removed), new + lmm:local-commutativity-iff-quasilocal, Einstein-causality prose and proof \uses, "rejected + alternative" sentence; Chapter 9 sentence added. Reviewer REVISE → fixed lines ≈50 and ≈369 + (no longer claim Axiom 3 needs the quasilocal algebra) and two optional \uses edges. Reviewer's + optional Chapter 5 refinements (common upper bound in the first remark; "nonempty nested + regions") NOT applied: they change user-approved wording, offered to the user. PDF builds + without errors or overfull lines. From 0c90000c0e9260b5220991a105e99c7fc881458c Mon Sep 17 00:00:00 2001 From: KellyJDavis Date: Thu, 8 Oct 2026 08:01:21 +0200 Subject: [PATCH 2/6] Chapter 5: tighten two remarks on the common containing diamond Co-Authored-By: Claude Opus 5.5 --- blueprint/src/sections/sec5/quasilocal-algebra.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/blueprint/src/sections/sec5/quasilocal-algebra.tex b/blueprint/src/sections/sec5/quasilocal-algebra.tex index 960ed9d..1192a5b 100644 --- a/blueprint/src/sections/sec5/quasilocal-algebra.tex +++ b/blueprint/src/sections/sec5/quasilocal-algebra.tex @@ -5,8 +5,8 @@ \section{Quasilocal Algebra}\label{sctn:quasilocal-algebra} Isotony is the natural tool for this. If, say, $\mathbf{B_1} \subset \mathbf{B_2}$, isotony places $\mathfrak{U}(\mathbf{B_1})$ inside $\mathfrak{U}(\mathbf{B_2})$, so that both live in the -single algebra $\mathfrak{U}(\mathbf{B_2})$. That case is of no use here, since nested regions are -never completely spacelike. But isotony does not require one of the two regions to contain the +single algebra $\mathfrak{U}(\mathbf{B_2})$. That case is of no use here, since nonempty nested regions +are never completely spacelike. But isotony does not require one of the two regions to contain the other: it suffices to find a \emph{third} basis element $\mathbf{B}$ containing both. In Minkowski spacetime such a $\mathbf{B}$ always exists --- any two double cones $I^+(p_1) \cap I^-(q_1)$ and $I^+(p_2) \cap I^-(q_2)$ lie inside a larger double cone $I^+(p) \cap I^-(q)$, by taking $p$ @@ -27,7 +27,7 @@ \section{Quasilocal Algebra}\label{sctn:quasilocal-algebra} Two remarks. First, the choice of $\mathbf{B}$ does not matter: because the isotony embeddings are injective and compatible, commutation in one containing algebra implies commutation in every -other. Second, the quasilocal algebra $\mathfrak{U}$ of the next chapter contains every +other (using a basis element containing both of them, which exists as above). Second, the quasilocal algebra $\mathfrak{U}$ of the next chapter contains every $\mathfrak{U}(\mathbf{B})$, so the statement can equally be read as commutation in $\mathfrak{U}$; the two readings are equivalent. The formulation through a containing $\mathfrak{U}(\mathbf{B})$ is preferred because it uses only the net and isotony, and because it From 311366846987ce7f1065cf36a18ed174108dc4b8 Mon Sep 17 00:00:00 2001 From: KellyJDavis Date: Thu, 8 Oct 2026 08:05:04 +0200 Subject: [PATCH 3/6] Restate Minkowski Axiom 3 in a common containing diamond (statements) LocalCommutativity now asks that completely spacelike local algebras commute in every Alexandrov diamond containing both, as in the curved axiom. State the equivalence with the former quasilocal form; proofs follow. Co-Authored-By: Claude Opus 5.5 --- .../AQFT/HaagKastler/EinsteinCausality.lean | 3 +- .../AQFT/HaagKastler/LocalCommutativity.lean | 96 ++++++++++--------- Physicslib4/AQFT/HaagKastler/Net.lean | 18 ++-- .../AQFT/HaagKastler/QuasilocalAlgebra.lean | 6 +- .../sections/sec10/haag-kastler-axioms.tex | 4 + .../plan-axiom3-common-diamond-2026-10-08.md | 9 ++ 6 files changed, 76 insertions(+), 60 deletions(-) diff --git a/Physicslib4/AQFT/HaagKastler/EinsteinCausality.lean b/Physicslib4/AQFT/HaagKastler/EinsteinCausality.lean index eb9dfc3..700471b 100644 --- a/Physicslib4/AQFT/HaagKastler/EinsteinCausality.lean +++ b/Physicslib4/AQFT/HaagKastler/EinsteinCausality.lean @@ -10,7 +10,8 @@ import Physicslib4.GNS.Construction # Einstein causality (microcausality) in a representation Axiom 3 (local commutativity) asserts that the local algebras of two completely -spacelike-separated regions commute inside the quasilocal algebra. This file +spacelike-separated regions commute in any local algebra containing both; by +`LocalCommutativity.commute_ι` they then commute inside the quasilocal algebra. This file pushes that algebraic statement into Hilbert space: under *any* `*`-representation `π` of the quasilocal algebra - in particular the GNS representation of any state - the local observables of spacelike-separated regions commute as bounded operators. diff --git a/Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean b/Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean index 75bb3d4..0404658 100644 --- a/Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean +++ b/Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean @@ -5,6 +5,7 @@ Authors: Lean Community -/ import Physicslib4.AQFT.HaagKastler.LocalAlgebras import Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra +import Physicslib4.AQFT.HaagKastler.QuasilocalExistence import Physicslib4.Spacetime.Causality import Physicslib4.Spacetime.MinkowskiDirected @@ -16,8 +17,13 @@ This file formalises the blueprint declaration axioms, section 10.6 of the AQFT-in-Lean blueprint): > If two Alexandrov-basis sets `𝐁₁`, `𝐁₂` are *completely spacelike* -> with respect to each other, then the local algebras `𝔘(𝐁₁)` and -> `𝔘(𝐁₂)` commute *inside the quasilocal algebra* `𝔘`. +> with respect to each other, then for every Alexandrov-basis set `𝐁` +> containing both, the isotony images of `𝔘(𝐁₁)` and `𝔘(𝐁₂)` commute in +> the local algebra `𝔘(𝐁)`. + +This is the curved-spacetime Axiom 3 (`HaagKastlerCurved.LocalCommutativity`) +with the Minkowski basis sets; the only difference is that in Minkowski +spacetime a containing basis set always exists (`alexandrovBasis_directed`). ## Main definitions @@ -30,27 +36,11 @@ axioms, section 10.6 of the AQFT-in-Lean blueprint): commutation of two local images in a quasilocal algebra is equivalent to commutation of their isotony images in a common local algebra `𝔘(B)`. * `Physicslib4.AQFT.HaagKastler.LocalCommutativity.commute_ι` - (`lmm:local-commutativity-any-quasilocal`): Axiom 3 holds in *every* - quasilocal algebra of the net, in particular in the canonical one. - -## Modelling notes - -* "Commuting in the quasilocal algebra" requires an ambient - C*-algebra `𝔘` containing every `𝔘(B)` as a subalgebra. Rather - than inlining that data, we quantify existentially over a - `QuasilocalAlgebra U` — the bundled structure that packages an - ambient C*-algebra together with the family of faithful unital - `*`-monomorphisms `ιB : 𝔘(B) →⋆ₐ[ℂ] 𝔘`. Axiom 3 then asserts - that, for *some* such ambient algebra, the images of any two - completely-spacelike local algebras commute pointwise. The choice of - ambient algebra does not matter: `LocalCommutativity.commute_ι` - shows the commutation then holds in *every* `QuasilocalAlgebra U i`. - -* The quasilocal algebra itself — including its density / completion - property — is *constructed* from the net by - `exists_quasilocalAlgebra` (`thrm:quasilocal-algebra-exists`); here - we only *use* the structure to phrase commutativity, and by - `LocalCommutativity.commute_ι` Axiom 3 holds in that canonical algebra. + (`lmm:local-commutativity-any-quasilocal`): Axiom 3 implies commutation in + every quasilocal algebra of the net, in particular in the canonical one. +* `Physicslib4.AQFT.HaagKastler.localCommutativity_iff_exists_commute_ι` + (`lmm:local-commutativity-iff-quasilocal`): Axiom 3 is equivalent to + commutation in some quasilocal algebra, the form used by earlier versions. -/ namespace Physicslib4 @@ -60,25 +50,26 @@ namespace HaagKastler open Physicslib4 /-- -**Axiom 3 (Local Commutativity).** A local net `U` satisfies *local -commutativity* if there exists a `QuasilocalAlgebra U` — i.e. an -ambient unital C*-algebra `Q.carrier` equipped with faithful unital -`*`-monomorphisms `Q.ι B : U.algebra B →⋆ₐ[ℂ] Q.carrier` for every -Alexandrov-basis set `B` — such that whenever two basis sets -`B₁`, `B₂` are completely spacelike with respect to each other, -the images `Q.ι B₁ (U.algebra B₁)` and `Q.ι B₂ (U.algebra B₂)` -commute pointwise inside `Q.carrier`. +**Axiom 3 (Local Commutativity).** A local net `U`, with Axiom 2 isotony family `i`, +satisfies *local commutativity* if whenever two Alexandrov-basis sets `B₁`, `B₂` are +completely spacelike and both contained in an Alexandrov-basis set `B`, their images in +`𝔘(B)` under the isotony embeddings `i.map` commute pointwise. + +In Minkowski spacetime a containing basis set always exists +(`Spacetime.alexandrovBasis_directed`), so the condition is never vacuous. The +equivalent form in a quasilocal algebra is `localCommutativity_iff_exists_commute_ι`. Blueprint reference: `def:local-commutativity`. -/ def LocalCommutativity (U : LocalNet) (i : Isotony U) : Prop := - ∃ Q : QuasilocalAlgebra U i, - ∀ ⦃B₁ B₂ : Set StandardMinkowskiSpacetime.Carrier⦄ - (hB₁ : IsAlexandrovBasisSet B₁) (hB₂ : IsAlexandrovBasisSet B₂), - Spacetime.IsCompletelySpacelike StandardMinkowskiSpacetime - standardMinkowskiTimeOrientation B₁ B₂ → - ∀ (a : U.algebra B₁) (b : U.algebra B₂), - Commute (Q.ι hB₁ a) (Q.ι hB₂ b) + ∀ ⦃B₁ B₂ B : Set StandardMinkowskiSpacetime.Carrier⦄ + (hB₁ : IsAlexandrovBasisSet B₁) (hB₂ : IsAlexandrovBasisSet B₂) + (hB : IsAlexandrovBasisSet B), + Spacetime.IsCompletelySpacelike StandardMinkowskiSpacetime + standardMinkowskiTimeOrientation B₁ B₂ → + (h₁ : B₁ ⊆ B) → (h₂ : B₂ ⊆ B) → + ∀ (a : U.algebra B₁) (b : U.algebra B₂), + Commute (i.map hB₁ hB h₁ a) (i.map hB₂ hB h₂ b) /-- **Commutation in a quasilocal algebra is decided locally.** For basis sets `B₁, B₂ ⊆ B`, the images `Q.ι hB₁ a` and `Q.ι hB₂ b` commute in `Q.carrier` iff @@ -95,10 +86,9 @@ theorem QuasilocalAlgebra.commute_ι_iff_commute_map {U : LocalNet} {i : Isotony refine ⟨fun h => Q.ι_injective hB ?_, fun h => h.map (Q.ι hB)⟩ simpa only [map_mul] using h.eq -/-- **Axiom 3 holds in every quasilocal algebra.** `LocalCommutativity` asserts -commutation of completely-spacelike local algebras in *some* quasilocal algebra; -since commutation is decided inside a common local algebra `𝔘(B)` -(`QuasilocalAlgebra.commute_ι_iff_commute_map`), it then holds in *every* +/-- **Axiom 3 holds in every quasilocal algebra.** Commutation of completely-spacelike +local algebras in a common containing `𝔘(B)` transfers, through the cocone condition and +`QuasilocalAlgebra.commute_ι_iff_commute_map`, to commutation of their images in *every* `QuasilocalAlgebra U i`, in particular in the canonical one built by `exists_quasilocalAlgebra`. @@ -111,10 +101,24 @@ theorem LocalCommutativity.commute_ι {U : LocalNet} {i : Isotony U} standardMinkowskiTimeOrientation B₁ B₂) (a : U.algebra B₁) (b : U.algebra B₂) : Commute (Q.ι hB₁ a) (Q.ι hB₂ b) := by - obtain ⟨Q₀, hQ₀⟩ := h - obtain ⟨B, hB, h₁, h₂⟩ := Spacetime.alexandrovBasis_directed hB₁ hB₂ - exact (Q.commute_ι_iff_commute_map hB₁ hB₂ hB h₁ h₂ a b).2 - ((Q₀.commute_ι_iff_commute_map hB₁ hB₂ hB h₁ h₂ a b).1 (hQ₀ hB₁ hB₂ hs a b)) + sorry + +/-- **Axiom 3 in the quasilocal algebra.** A net satisfies Axiom 3 if and only if there is a +quasilocal algebra in which the images of any two completely-spacelike local algebras commute. +This was the form of Axiom 3 in earlier versions of the blueprint; by +`LocalCommutativity.commute_ι` the commutation then holds in every quasilocal algebra. + +Blueprint reference: `lmm:local-commutativity-iff-quasilocal`. -/ +theorem localCommutativity_iff_exists_commute_ι (U : LocalNet) (i : Isotony U) : + LocalCommutativity U i ↔ + ∃ Q : QuasilocalAlgebra U i, + ∀ ⦃B₁ B₂ : Set StandardMinkowskiSpacetime.Carrier⦄ + (hB₁ : IsAlexandrovBasisSet B₁) (hB₂ : IsAlexandrovBasisSet B₂), + Spacetime.IsCompletelySpacelike StandardMinkowskiSpacetime + standardMinkowskiTimeOrientation B₁ B₂ → + ∀ (a : U.algebra B₁) (b : U.algebra B₂), + Commute (Q.ι hB₁ a) (Q.ι hB₂ b) := by + sorry end HaagKastler end AQFT diff --git a/Physicslib4/AQFT/HaagKastler/Net.lean b/Physicslib4/AQFT/HaagKastler/Net.lean index 7950e62..6383992 100644 --- a/Physicslib4/AQFT/HaagKastler/Net.lean +++ b/Physicslib4/AQFT/HaagKastler/Net.lean @@ -74,8 +74,9 @@ structure HaagKastlerNet where `*`-monomorphisms of the corresponding local algebras (Axiom 2). -/ isotony : Isotony U - /-- *Local commutativity*: local algebras of completely-spacelike - basis sets commute inside the quasilocal algebra (Axiom 3). -/ + /-- *Local commutativity*: for completely-spacelike basis sets `B₁`, `B₂` and every + basis set `B ⊇ B₁ ∪ B₂`, the isotony images of `𝔘(B₁)` and `𝔘(B₂)` commute in `𝔘(B)` + (Axiom 3). -/ localCommutativity : LocalCommutativity U isotony /-- *Lorentz covariance*: the inhomogeneous Lorentz group acts on the net and the action commutes with isotony (Axiom 5). -/ @@ -211,9 +212,9 @@ theorem covEquiv_isotony (L : InhomogeneousLorentzGroup) N.lorentzCovariance.choose_spec.2.2 L hB₁ hB₂ h a /-- **Local commutativity.** The images in the canonical quasilocal algebra -`quasilocal` of two completely-spacelike basis algebras commute. Axiom 3 only -asserts this in *some* quasilocal algebra; `LocalCommutativity.commute_ι` transfers -it to every one. -/ +`quasilocal` of two completely-spacelike basis algebras commute. Axiom 3 asserts +commutation in a common containing local algebra; `LocalCommutativity.commute_ι` transfers +it to every quasilocal algebra. -/ theorem commute_ι_of_spacelike ⦃B₁ B₂ : Set StandardMinkowskiSpacetime.Carrier⦄ (hB₁ : IsAlexandrovBasisSet B₁) (hB₂ : IsAlexandrovBasisSet B₂) (hs : Spacetime.IsCompletelySpacelike StandardMinkowskiSpacetime @@ -343,11 +344,8 @@ noncomputable def trivialQuasilocalAlgebra : ι_inclusion := fun _ _ _ _ _ _ => rfl theorem trivialLocalNet_localCommutativity : - LocalCommutativity trivialLocalNet trivialLocalNetIsotony := - ⟨trivialQuasilocalAlgebra, by - intro B₁ B₂ hB₁ hB₂ _ a b - exact @mul_comm ℂ _ (trivialQuasilocalAlgebra.ι hB₁ a) - (trivialQuasilocalAlgebra.ι hB₂ b)⟩ + LocalCommutativity trivialLocalNet trivialLocalNetIsotony := by + sorry theorem trivialLocalNet_lorentzCovariance : LorentzCovariance trivialLocalNet trivialLocalNetIsotony := by diff --git a/Physicslib4/AQFT/HaagKastler/QuasilocalAlgebra.lean b/Physicslib4/AQFT/HaagKastler/QuasilocalAlgebra.lean index 13f989b..4717b9d 100644 --- a/Physicslib4/AQFT/HaagKastler/QuasilocalAlgebra.lean +++ b/Physicslib4/AQFT/HaagKastler/QuasilocalAlgebra.lean @@ -113,9 +113,9 @@ structure QuasilocalAlgebra (U : LocalNet.{u}) (i : Isotony U) where the images of the local algebras, so any quasilocal algebra is already of their size, and a free universe would only add copies of the same algebra higher up. It does buy something important -- a free universe is constrained by no field, - so it could not be inferred, and `LocalCommutativity` would become a *family* of - `Prop`s indexed by a universe, making the content of Axiom 3 depend on that - index. -/ + so it could not be inferred, and the quasilocal form of Axiom 3 + (`localCommutativity_iff_exists_commute_ι`) would become a *family* of `Prop`s indexed + by a universe, so that its equivalence with Axiom 3 would depend on that index. -/ carrier : Type u /-- The `CStarAlgebra` instance on `carrier`. -/ instCStarAlgebra : CStarAlgebra carrier diff --git a/blueprint/src/sections/sec10/haag-kastler-axioms.tex b/blueprint/src/sections/sec10/haag-kastler-axioms.tex index 04e2f41..25a71e6 100644 --- a/blueprint/src/sections/sec10/haag-kastler-axioms.tex +++ b/blueprint/src/sections/sec10/haag-kastler-axioms.tex @@ -394,6 +394,7 @@ \subsection{The Quasilocal Colimit} \label{def:local-commutativity} \lean{Physicslib4.AQFT.HaagKastler.LocalCommutativity} \uses{def:alexandrov-topology, def:minkowski-spacetime, def:chronological-future-and-chronological-past, def:local-algebras, def:isotony, def:completely-spacelike, lmm:minkowski-diamonds-upward-directed} + \leanok Let $\mathbf{B}_1$ and $\mathbf{B}_2$ be any two basis elements of the Alexandrov topology on Minkowski spacetime, i.e. any two sets of the form $I^+(p_1) \cap I^-(q_1)$ and $I^+(p_2) \cap I^-(q_2)$. Assume Axiom 2 (\ref{def:isotony}), and let $i_{\mathbf{B}_1\mathbf{B}_2}$ denote its chosen isotony family. If $\mathbf{B_1}$ and $\mathbf{B_2}$ are completely spacelike, then for any Alexandrov topology basis element $\mathbf{B}$ such that $\mathbf{B_1}, \mathbf{B_2} \subseteq \mathbf{B}$ the algebras $\mathfrak{U}(\mathbf{B_1})$ and $\mathfrak{U}(\mathbf{B_2})$ commute in the C*-algebra $\mathfrak{U}(\mathbf{B})$: for any $a_1$ in $\mathfrak{U}(\mathbf{B_1})$ and $a_2$ in $\mathfrak{U}(\mathbf{B_2})$, @@ -413,6 +414,7 @@ \subsection{The Quasilocal Colimit} \label{lmm:local-commutativity-any-quasilocal} \lean{Physicslib4.AQFT.HaagKastler.LocalCommutativity.commute_ι, Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra.commute_ι_iff_commute_map} \uses{def:local-commutativity, def:quasilocal-algebra, def:completely-spacelike, lmm:minkowski-diamonds-upward-directed} + \leanok Suppose the net satisfies Axiom 3 (\ref{def:local-commutativity}). Let $\mathfrak{U}'$ be any quasilocal algebra of the net (\ref{def:quasilocal-algebra}), with canonical embeddings $\iota'_{\mathbf{B}}$. Then for any completely spacelike basis sets $\mathbf{B}_1$, $\mathbf{B}_2$ and any $a_1 \in \mathfrak{U}(\mathbf{B}_1)$, $a_2 \in \mathfrak{U}(\mathbf{B}_2)$, \begin{align} \iota'_{\mathbf{B}_1}(a_1)\, \iota'_{\mathbf{B}_2}(a_2) = \iota'_{\mathbf{B}_2}(a_2)\, \iota'_{\mathbf{B}_1}(a_1) @@ -426,7 +428,9 @@ \subsection{The Quasilocal Colimit} \begin{lemma}[Local Commutativity in the Quasilocal Algebra] \label{lmm:local-commutativity-iff-quasilocal} + \lean{Physicslib4.AQFT.HaagKastler.localCommutativity_iff_exists_commute_ι} \uses{def:local-commutativity, def:quasilocal-algebra, def:completely-spacelike} + \leanok A net satisfies Axiom 3 (\ref{def:local-commutativity}) if and only if there is a quasilocal algebra $\mathfrak{U}$ (\ref{def:quasilocal-algebra}), with canonical embeddings $\iota_{\mathbf{B}}$, in which, for all completely spacelike basis sets $\mathbf{B}_1$, $\mathbf{B}_2$ and all $a_1 \in \mathfrak{U}(\mathbf{B}_1)$, $a_2 \in \mathfrak{U}(\mathbf{B}_2)$, \begin{align} \iota_{\mathbf{B}_1}(a_1)\, \iota_{\mathbf{B}_2}(a_2) = \iota_{\mathbf{B}_2}(a_2)\, \iota_{\mathbf{B}_1}(a_1); diff --git a/notes/agents/plan-axiom3-common-diamond-2026-10-08.md b/notes/agents/plan-axiom3-common-diamond-2026-10-08.md index dea8e96..dc5da5c 100644 --- a/notes/agents/plan-axiom3-common-diamond-2026-10-08.md +++ b/notes/agents/plan-axiom3-common-diamond-2026-10-08.md @@ -281,3 +281,12 @@ Statements are sorry'd, then the formalizer-reviewer runs. optional Chapter 5 refinements (common upper bound in the first remark; "nonempty nested regions") NOT applied: they change user-approved wording, offered to the user. PDF builds without errors or overfull lines. +- 2026-10-08, Chapter 5 refinements applied (user-approved): "nonempty nested regions" and the + common-upper-bound parenthetical (0c90000). +- 2026-10-08, Stage 2 done: `LocalCommutativity` body rewritten (orchestrator; approved) to the + common-containing-diamond form, binder-for-binder parallel to the curved definition; module and + definition docstrings rewritten; new `localCommutativity_iff_exists_commute_ι` (right side = the + old body; imports QuasilocalExistence, no cycle); `LocalCommutativity.commute_ι` signature kept. + Sorry'd: commute_ι, the iff, `trivialLocalNet_localCommutativity` (Net.lean). Everything else + builds unchanged. Formalizer review: statements and universes PASS; three stale docstrings fixed + (Net.lean field, EinsteinCausality module doc, QuasilocalAlgebra.carrier universe rationale). From b37cbe16d2f994a7bbd47fe73a3c9fcd4ea67444 Mon Sep 17 00:00:00 2001 From: KellyJDavis Date: Thu, 8 Oct 2026 08:12:52 +0200 Subject: [PATCH 4/6] Prove the common-diamond form of Axiom 3 implies the quasilocal form and conversely Co-Authored-By: Claude Opus 5.5 --- Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean | 9 +++++++-- Physicslib4/AQFT/HaagKastler/Net.lean | 3 ++- blueprint/src/sections/sec10/haag-kastler-axioms.tex | 2 ++ notes/agents/plan-axiom3-common-diamond-2026-10-08.md | 4 ++++ 4 files changed, 15 insertions(+), 3 deletions(-) diff --git a/Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean b/Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean index 0404658..78aad59 100644 --- a/Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean +++ b/Physicslib4/AQFT/HaagKastler/LocalCommutativity.lean @@ -101,7 +101,8 @@ theorem LocalCommutativity.commute_ι {U : LocalNet} {i : Isotony U} standardMinkowskiTimeOrientation B₁ B₂) (a : U.algebra B₁) (b : U.algebra B₂) : Commute (Q.ι hB₁ a) (Q.ι hB₂ b) := by - sorry + obtain ⟨B, hB, h₁, h₂⟩ := Spacetime.alexandrovBasis_directed hB₁ hB₂ + exact (Q.commute_ι_iff_commute_map hB₁ hB₂ hB h₁ h₂ a b).2 (h hB₁ hB₂ hB hs h₁ h₂ a b) /-- **Axiom 3 in the quasilocal algebra.** A net satisfies Axiom 3 if and only if there is a quasilocal algebra in which the images of any two completely-spacelike local algebras commute. @@ -118,7 +119,11 @@ theorem localCommutativity_iff_exists_commute_ι (U : LocalNet) (i : Isotony U) standardMinkowskiTimeOrientation B₁ B₂ → ∀ (a : U.algebra B₁) (b : U.algebra B₂), Commute (Q.ι hB₁ a) (Q.ι hB₂ b) := by - sorry + refine ⟨fun h => ?_, ?_⟩ + · obtain ⟨Q⟩ := exists_quasilocalAlgebra U i + exact ⟨Q, fun B₁ B₂ hB₁ hB₂ hs a b => h.commute_ι Q hB₁ hB₂ hs a b⟩ + · rintro ⟨Q, hQ⟩ B₁ B₂ B hB₁ hB₂ hB hs h₁ h₂ a b + exact (Q.commute_ι_iff_commute_map hB₁ hB₂ hB h₁ h₂ a b).1 (hQ hB₁ hB₂ hs a b) end HaagKastler end AQFT diff --git a/Physicslib4/AQFT/HaagKastler/Net.lean b/Physicslib4/AQFT/HaagKastler/Net.lean index 6383992..b1e4a6a 100644 --- a/Physicslib4/AQFT/HaagKastler/Net.lean +++ b/Physicslib4/AQFT/HaagKastler/Net.lean @@ -345,7 +345,8 @@ noncomputable def trivialQuasilocalAlgebra : theorem trivialLocalNet_localCommutativity : LocalCommutativity trivialLocalNet trivialLocalNetIsotony := by - sorry + intro B₁ B₂ B hB₁ hB₂ hB _ h₁ h₂ a b + exact @mul_comm ℂ _ _ _ theorem trivialLocalNet_lorentzCovariance : LorentzCovariance trivialLocalNet trivialLocalNetIsotony := by diff --git a/blueprint/src/sections/sec10/haag-kastler-axioms.tex b/blueprint/src/sections/sec10/haag-kastler-axioms.tex index 25a71e6..f4f6faa 100644 --- a/blueprint/src/sections/sec10/haag-kastler-axioms.tex +++ b/blueprint/src/sections/sec10/haag-kastler-axioms.tex @@ -422,6 +422,7 @@ \subsection{The Quasilocal Colimit} in $\mathfrak{U}'$. In particular these images commute in the canonical quasilocal algebra of \ref{thrm:quasilocal-algebra-exists}. \end{lemma} \begin{proof} + \leanok \uses{def:local-commutativity, def:quasilocal-algebra, def:isotony, lmm:minkowski-diamonds-upward-directed} Let $\mathbf{B}_1$, $\mathbf{B}_2$ be completely spacelike. By upward directedness of the Alexandrov diamonds (\ref{lmm:minkowski-diamonds-upward-directed}) pick a basis set $\mathbf{B}$ with $\mathbf{B}_1 \cup \mathbf{B}_2 \subseteq \mathbf{B}$. By the cocone condition of \ref{def:quasilocal-algebra}, $\iota'_{\mathbf{B}_k}(a_k) = \iota'_{\mathbf{B}}(i_{\mathbf{B}_k\mathbf{B}}(a_k))$ for $k = 1, 2$, where $i_{\mathbf{B}_k\mathbf{B}}$ are the isotony embeddings of Axiom 2 (\ref{def:isotony}). Since $\iota'_{\mathbf{B}}$ is an injective $*$-homomorphism, $\iota'_{\mathbf{B}_1}(a_1)$ and $\iota'_{\mathbf{B}_2}(a_2)$ commute if and only if $i_{\mathbf{B}_1\mathbf{B}}(a_1)$ and $i_{\mathbf{B}_2\mathbf{B}}(a_2)$ commute in $\mathfrak{U}(\mathbf{B})$, and that is what Axiom 3 (\ref{def:local-commutativity}) gives for this $\mathbf{B}$. \end{proof} @@ -438,6 +439,7 @@ \subsection{The Quasilocal Colimit} and in that case this holds in every quasilocal algebra of the net. \end{lemma} \begin{proof} + \leanok \uses{lmm:local-commutativity-any-quasilocal, thrm:quasilocal-algebra-exists, lmm:minkowski-diamonds-upward-directed, def:isotony, def:quasilocal-algebra} ($\Rightarrow$) A quasilocal algebra exists by \ref{thrm:quasilocal-algebra-exists}, and \ref{lmm:local-commutativity-any-quasilocal} gives the commutation in it, and indeed in every quasilocal algebra. diff --git a/notes/agents/plan-axiom3-common-diamond-2026-10-08.md b/notes/agents/plan-axiom3-common-diamond-2026-10-08.md index dc5da5c..7d4d95e 100644 --- a/notes/agents/plan-axiom3-common-diamond-2026-10-08.md +++ b/notes/agents/plan-axiom3-common-diamond-2026-10-08.md @@ -290,3 +290,7 @@ Statements are sorry'd, then the formalizer-reviewer runs. Sorry'd: commute_ι, the iff, `trivialLocalNet_localCommutativity` (Net.lean). Everything else builds unchanged. Formalizer review: statements and universes PASS; three stale docstrings fixed (Net.lean field, EinsteinCausality module doc, QuasilocalAlgebra.carrier universe rationale). +- 2026-10-08, Stage 3 done: commute_ι (2 lines: directedness + bridge), the equivalence (4 lines: + exists_quasilocalAlgebra + commute_ι; bridge backwards), trivialLocalNet_localCommutativity + (mul_comm in ℂ). No sorry; build and lint clean. trivialQuasilocalAlgebra still used by + QuasilocalIntertwiner.lean. From f72b8fa214e9ce94869138883ef6d2f98275a26d Mon Sep 17 00:00:00 2001 From: KellyJDavis Date: Thu, 8 Oct 2026 08:25:24 +0200 Subject: [PATCH 5/6] Refresh the home page for Axiom 3 in a common containing diamond; add the audit memo Co-Authored-By: Claude Opus 5.5 --- home_page/axioms.md | 27 +- home_page/guide.md | 14 +- home_page/index.md | 30 +- home_page/sec10-3-unbounded.md | 6 +- home_page/sec10-4-stone.md | 4 +- home_page/sec10-5-spacetime.md | 20 +- home_page/sec10-6-haag-kastler.md | 363 +++++++++--------- home_page/sec10-7-curved.md | 164 ++++---- home_page/sec10-8-general-covariance.md | 12 +- .../audit-axiom3-common-diamond-2026-10-08.md | 204 ++++++++++ .../plan-axiom3-common-diamond-2026-10-08.md | 11 + 11 files changed, 536 insertions(+), 319 deletions(-) create mode 100644 notes/agents/audits/audit-axiom3-common-diamond-2026-10-08.md diff --git a/home_page/axioms.md b/home_page/axioms.md index 1a06504..5df4157 100644 --- a/home_page/axioms.md +++ b/home_page/axioms.md @@ -10,32 +10,33 @@ usemathjax: true Axiom 6 (Primitivity) from the original 1964 Haag–Kastler paper is not carried into the sharpened axiom set — see Chapter 8 for the discussion of why it is dropped. The five sharpened axioms in each setting, bundled together as a `HaagKastlerNet`, are what is actually formalised. -**Minkowski spacetime** — bundled as `Physicslib4.AQFT.HaagKastler.HaagKastlerNet` (Definition 507): +**Minkowski spacetime** — bundled as `Physicslib4.AQFT.HaagKastler.HaagKastlerNet` (Definition 508): | Axiom | Node | Lean | |---|---|---| | 1. Local Algebras | Definition 467 | `Physicslib4.AQFT.HaagKastler.LocalNet` | | 2. Isotony | Definition 468 | `Physicslib4.AQFT.HaagKastler.Isotony` | | 3. Local Commutativity | Definition 485 | `Physicslib4.AQFT.HaagKastler.LocalCommutativity` | -| 4. Quasilocal Completeness | Definition 488 | `Physicslib4.AQFT.HaagKastler.ObservableCorrespondence` | -| 5. Lorentz Covariance | Definition 506 | `Physicslib4.AQFT.HaagKastler.LorentzCovariance` | +| 4. Quasilocal Completeness | Definition 489 | `Physicslib4.AQFT.HaagKastler.ObservableCorrespondence` | +| 5. Lorentz Covariance | Definition 507 | `Physicslib4.AQFT.HaagKastler.LorentzCovariance` | -**Curved spacetime** — bundled as `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` (Definition 632): +**Curved spacetime** — bundled as `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` (Definition 633): | Axiom | Node | Lean | |---|---|---| -| 1. Local Algebras | Definition 626 | `Physicslib4.AQFT.HaagKastlerCurved.LocalNet` | -| 2. Isotony | Definition 627 | `Physicslib4.AQFT.HaagKastlerCurved.Isotony` | -| 3. Local Commutativity | Definition 628 | `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` | -| 4. Local Completeness | Definition 630 | `Physicslib4.AQFT.HaagKastlerCurved.LocalAlgebra` | -| 5. Isometric Covariance | Definition 631 | `Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance` | +| 1. Local Algebras | Definition 627 | `Physicslib4.AQFT.HaagKastlerCurved.LocalNet` | +| 2. Isotony | Definition 628 | `Physicslib4.AQFT.HaagKastlerCurved.Isotony` | +| 3. Local Commutativity | Definition 629 | `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` | +| 4. Local Completeness | Definition 631 | `Physicslib4.AQFT.HaagKastlerCurved.LocalAlgebra` | +| 5. Isometric Covariance | Definition 632 | `Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance` | -**General Covariance (Definition 676, `Physicslib4.AQFT.HaagKastlerCurved.IsGenerallyCovariant`)** is deliberately *not* a sixth axiom. Axioms 1–5 each constrain a single net over a single fixed spacetime, whereas general covariance relates two nets over two spacetimes; it is therefore a property of the section $$L \mapsto \mathfrak{U}_L$$ assigning a net to every Lorentzian spacetime, not an extra field of the net structure. +**General Covariance (Definition 677, `Physicslib4.AQFT.HaagKastlerCurved.IsGenerallyCovariant`)** is deliberately *not* a sixth axiom. Axioms 1–5 each constrain a single net over a single fixed spacetime, whereas general covariance relates two nets over two spacetimes; it is therefore a property of the section $$L \mapsto \mathfrak{U}_L$$ assigning a net to every Lorentzian spacetime, not an extra field of the net structure. -Two changes to the axioms are worth calling out for readers coming from an earlier version of this blueprint: +Three changes to the axioms are worth calling out for readers coming from an earlier version of this blueprint: -- **Isotony now supplies its embeddings as data.** Axiom 2 (Definitions 468 and 627) fixes the family $$i_{\mathbf{B}_1 \mathbf{B}_2}$$ together with identity and composition laws, making $$\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$$ a functor on the inclusion order. Axiom 3 consumes that family rather than choosing witnesses of its own, and Axiom 5 (Definitions 506 and 631) now states its coherence condition for the same family, so that covariance-compatibility of a quasilocal algebra is a lemma (Lemma 587) rather than a hypothesis. This is what makes the quasilocal colimit well posed in Minkowski spacetime, and it removes the coherence side-hypotheses that curved-spacetime statements about nested regions previously had to carry. -- **Axiom 4 has been split.** The mathematical claim that a quasilocal algebra exists is now Theorem 489 (`Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra`), proved from the colimit-and-completion chain; what remains as Axiom 4 (Definition 488) is the bridge principle relating physical observables to quasilocal ones, which has — by design — no mathematical consumers. +- **Isotony now supplies its embeddings as data.** Axiom 2 (Definitions 468 and 628) fixes the family $$i_{\mathbf{B}_1 \mathbf{B}_2}$$ together with identity and composition laws, making $$\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$$ a functor on the inclusion order. Axiom 3 consumes that family rather than choosing witnesses of its own, and Axiom 5 (Definitions 507 and 632) now states its coherence condition for the same family, so that covariance-compatibility of a quasilocal algebra is a lemma (Lemma 588) rather than a hypothesis. This is what makes the quasilocal colimit well posed in Minkowski spacetime, and it removes the coherence side-hypotheses that curved-spacetime statements about nested regions previously had to carry. +- **Minkowski Axiom 3 is stated in a common containing diamond.** Definition 485 now requires completely spacelike local algebras to commute in every local algebra $$\mathfrak{U}(\mathbf{B})$$ containing both, through the Axiom 2 isotony maps, exactly as Definition 629 does in curved spacetime; the two differ only in that a containing diamond always exists in Minkowski spacetime. The former statement, commutation in the quasilocal algebra, is now Lemma 487 (`Physicslib4.AQFT.HaagKastler.localCommutativity_iff_exists_commute_ι`), proved equivalent. +- **Axiom 4 has been split.** The mathematical claim that a quasilocal algebra exists is now Theorem 490 (`Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra`), proved from the colimit-and-completion chain; what remains as Axiom 4 (Definition 489) is the bridge principle relating physical observables to quasilocal ones, which has — by design — no mathematical consumers. [Home](./) · [§10.1 GNS](sec10-1-gns.html) · [§10.2 Spectral](sec10-2-spectral.html) · [§10.3 Unbounded](sec10-3-unbounded.html) · [§10.4 Stone](sec10-4-stone.html) · [§10.5 Spacetime](sec10-5-spacetime.html) · [§10.6 Minkowski](sec10-6-haag-kastler.html) · [§10.7 Curved](sec10-7-curved.html) · [§10.8 Covariance](sec10-8-general-covariance.html) · [Axioms](axioms.html) · [Guide](guide.html) diff --git a/home_page/guide.md b/home_page/guide.md index 4ffe655..e1e900b 100644 --- a/home_page/guide.md +++ b/home_page/guide.md @@ -13,7 +13,7 @@ The blueprint is 341 pages long and splits cleanly in two. **Chapters 1–9 are mathematical background and are not formalised in Lean.** They motivate and analyse each of the original Haag–Kastler axioms in turn, and then generalise them to curved spacetime. Along the way they cite twelve supporting results, numbered 1 through 12 — Gelfand–Naimark, the Bounded Linear Transformation Theorem, the existence of a Lorentz metric, and so on. These are quoted from the literature where needed; none of them carries a Lean declaration. -**Chapter 10 collects the formalisation-ready content, and it is the content of Chapter 10 that is formalised in Lean.** Its items are numbered consecutively, running from Definition 13 through Definition 676, and comprise **661 declarations in total: 150 definitions, 156 theorems, 252 lemmas, 99 propositions, and 4 corollaries**, mapped onto **1,145 distinct Lean declarations** (where Mathlib already supplies a result, the blueprint names the Mathlib declaration directly). Three further numbered items — Conventions 22, 213 and 260, standing hypotheses of the two spectral-theory sections — share the numbering but are conventions rather than declarations, and are not counted. Chapter 10 is divided into eight top-level sections, §10.1 through §10.8. +**Chapter 10 collects the formalisation-ready content, and it is the content of Chapter 10 that is formalised in Lean.** Its items are numbered consecutively, running from Definition 13 through Definition 677, and comprise **662 declarations in total: 150 definitions, 156 theorems, 253 lemmas, 99 propositions, and 4 corollaries**, mapped onto **1,146 distinct Lean declarations** (where Mathlib already supplies a result, the blueprint names the Mathlib declaration directly). Three further numbered items — Conventions 22, 213 and 260, standing hypotheses of the two spectral-theory sections — share the numbering but are conventions rather than declarations, and are not counted. Chapter 10 is divided into eight top-level sections, §10.1 through §10.8. ## Chapters 1–9: unpacking the original axioms @@ -23,7 +23,7 @@ Chapters 1–9 unpack and analyse the original Haag–Kastler axioms one by one: - **Chapter 2 (Axiom 0 – Minkowski Space, p. 5)** examines the role of Minkowski spacetime and compares the standard, indiscrete, Euclidean, and Alexandrov topologies on it, settling on the Alexandrov topology—generated by the "double cone" sets $$I^+(p) \cap I^-(q)$$—as the physically natural choice, which happens to coincide with the Euclidean topology on $$\mathbb{R}^4$$. - **Chapter 3 (Axiom 1 – Local Algebras, p. 7)** discusses the notion of regions of measurement and the assignment of a C\*-algebra to each such region, arguing that the "physically natural" regions are double cones $$I^+(p) \cap I^-(q)$$ rather than arbitrary open sets with compact closure, and rejecting the causal (light-cone) alternative $$J^+(p) \cap J^-(q)$$ because it would force single points to carry algebras, which conflicts with fields being operator-valued distributions. - **Chapter 4 (Axiom 2 – Isotony, p. 10)** introduces the GNS Construction in motivating context, studies the isotony condition—the requirement that inclusions of spacetime regions induce \*-monomorphisms of the corresponding algebras—and settles the "common unit" convention flagged in the original axiom by assigning the empty region the algebra $$\mathbb{C}1$$, so that every local algebra automatically contains a unit. -- **Chapter 5 (Axiom 3 – Local Commutativity, p. 13)** introduces the notion of completely spacelike separated regions and the quasilocal algebra, and studies the requirement that observables localised in spacelike separated regions commute, clarifying that this commutation is to be understood inside the quasilocal algebra $$\mathfrak{U}$$, since isotony alone cannot place two spacelike-separated (and hence non-nested) regions in a common algebra. +- **Chapter 5 (Axiom 3 – Local Commutativity, p. 13)** introduces the notion of completely spacelike separated regions and the quasilocal algebra, and studies the requirement that observables localised in spacelike separated regions commute, clarifying where this commutation takes place: since any two diamonds in Minkowski spacetime lie in a common, larger diamond, isotony places the two local algebras in a common local algebra $$\mathfrak{U}(\mathbf{B})$$, and commutation is required there, equivalently in the quasilocal algebra $$\mathfrak{U}$$. This is the formulation that carries over to curved spacetime (Chapter 9), where a common containing diamond need not exist. - **Chapter 6 (Axiom 4 – Quasilocal Algebra, p. 15)** analyses the construction of the quasilocal algebra as the completion of the set-theoretic union of all local algebras, and the axiom that $$\mathfrak{U}$$ contains all observables of interest—which is shown to mean that $$\pi_\omega(\mathfrak{U})$$ is strongly dense in the von Neumann algebra $$\pi_\omega(\mathfrak{U})''$$ it generates, so that any "missing" observable in $$\pi_\omega(\mathfrak{U})'' \setminus \pi_\omega(\mathfrak{U})$$ is experimentally indistinguishable from one in $$\mathfrak{U}$$. - **Chapter 7 (Axiom 5 – Lorentz Covariance, p. 18)** studies the action of the inhomogeneous Lorentz group (connected to the identity) on the net of local algebras and the covariance requirement, and shows, via the Bounded Linear Transformation Theorem, how this norm-one action extends uniquely from the dense union of local algebras to the whole quasilocal algebra. - **Chapter 8 (Axiom 6 – Primitivity, p. 20)** examines faithful and irreducible representations, noting that every unital C\*-algebra already has a faithful representation (Gelfand–Naimark) so primitivity is a genuinely extra condition; this axiom is ultimately abandoned in the sharpened formulation, following later presentations by Haag himself, on the grounds that its physical motivation is thin. @@ -35,16 +35,16 @@ The lower numbers, 1 through 12, label supporting theorems and definitions intro ## The section pages -Only Chapter 10 is formalised in Lean. Its items are numbered consecutively from Definition 13 through Definition 676 — 661 declarations, plus the three standing conventions of §10.2 and §10.3. Each section page below summarises its section and lists **every** item, in numerical order, under the blueprint subsection in which it appears. Each entry links to the node in the web blueprint and names the principal Lean declaration that realises it; where a node maps onto several declarations, the count of the remainder is shown. The lists are derived mechanically from the blueprint's own `\lean` and `\leanok` annotations. +Only Chapter 10 is formalised in Lean. Its items are numbered consecutively from Definition 13 through Definition 678 — 662 declarations, plus the three standing conventions of §10.2 and §10.3. Each section page below summarises its section and lists **every** item, in numerical order, under the blueprint subsection in which it appears. Each entry links to the node in the web blueprint and names the principal Lean declaration that realises it; where a node maps onto several declarations, the count of the remainder is shown. The lists are derived mechanically from the blueprint's own `\lean` and `\leanok` annotations. - [§10.1 GNS Construction (pp. 29–38)](sec10-1-gns.html) - [§10.2 Spectral Theorems (pp. 38–155)](sec10-2-spectral.html) - [§10.3 Unbounded Spectral Theorems (pp. 155–235)](sec10-3-unbounded.html) - [§10.4 Stone's Theorem (pp. 235–249)](sec10-4-stone.html) -- [§10.5 Spacetime and causal structure (pp. 249–286)](sec10-5-spacetime.html) -- [§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 286–327)](sec10-6-haag-kastler.html) -- [§10.7 Haag–Kastler Axioms in curved spacetime (pp. 327–337)](sec10-7-curved.html) -- [§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 337–338)](sec10-8-general-covariance.html) +- [§10.5 Spacetime and causal structure (pp. 249–287)](sec10-5-spacetime.html) +- [§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 287–328)](sec10-6-haag-kastler.html) +- [§10.7 Haag–Kastler Axioms in curved spacetime (pp. 328–338)](sec10-7-curved.html) +- [§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 338–339)](sec10-8-general-covariance.html) ## Where the Lean lives diff --git a/home_page/index.md b/home_page/index.md index 9556adc..7128960 100644 --- a/home_page/index.md +++ b/home_page/index.md @@ -29,11 +29,11 @@ The blueprint is 341 pages long and splits cleanly in two. **Chapters 1–9 are mathematical background and are not formalised in Lean.** They motivate and analyse each of the original Haag–Kastler axioms in turn, and then generalise them to curved spacetime. Along the way they cite twelve supporting results, numbered 1 through 12 — Gelfand–Naimark, the Bounded Linear Transformation Theorem, the existence of a Lorentz metric, and so on. These are quoted from the literature where needed; none of them carries a Lean declaration. -**Chapter 10 collects the formalisation-ready content, and it is the content of Chapter 10 that is formalised in Lean.** Its items are numbered consecutively, running from Definition 13 through Definition 676, and comprise **661 declarations in total: 150 definitions, 156 theorems, 252 lemmas, 99 propositions, and 4 corollaries**, mapped onto **1,145 distinct Lean declarations** (where Mathlib already supplies a result, the blueprint names the Mathlib declaration directly). Three further numbered items — Conventions 22, 213 and 260, standing hypotheses of the two spectral-theory sections — share the numbering but are conventions rather than declarations, and are not counted. Chapter 10 is divided into eight top-level sections, §10.1 through §10.8. +**Chapter 10 collects the formalisation-ready content, and it is the content of Chapter 10 that is formalised in Lean.** Its items are numbered consecutively, running from Definition 13 through Definition 677, and comprise **662 declarations in total: 150 definitions, 156 theorems, 253 lemmas, 99 propositions, and 4 corollaries**, mapped onto **1,146 distinct Lean declarations** (where Mathlib already supplies a result, the blueprint names the Mathlib declaration directly). Three further numbered items — Conventions 22, 213 and 260, standing hypotheses of the two spectral-theory sections — share the numbering but are conventions rather than declarations, and are not counted. Chapter 10 is divided into eight top-level sections, §10.1 through §10.8. -**All 661 are formalised, statements and proofs alike, and the Lean contains no `sorry`.** Of the 511 theorems, lemmas, propositions and corollaries in Chapter 10, 471 carry a written proof in the blueprint; the other 40, in §10.2, §10.3 and §10.4, are standard results quoted from the literature (Heine–Borel, Stone–Weierstrass, dominated convergence, the Bochner integral and the like) that the blueprint states in full but deliberately does not prove. All 511 proofs are formalised in Lean, including the 40 quoted results, which are linked to the Mathlib declarations that supply them. +**All 662 are formalised, statements and proofs alike, and the Lean contains no `sorry`.** Of the 512 theorems, lemmas, propositions and corollaries in Chapter 10, 472 carry a written proof in the blueprint; the other 40, in §10.2, §10.3 and §10.4, are standard results quoted from the literature (Heine–Borel, Stone–Weierstrass, dominated convergence, the Bochner integral and the like) that the blueprint states in full but deliberately does not prove. All 512 proofs are formalised in Lean, including the 40 quoted results, which are linked to the Mathlib declarations that supply them. -At a glance, the 661 declarations break down by top-level section as follows (click a section to see its items): +At a glance, the 662 declarations break down by top-level section as follows (click a section to see its items): | Section | Topic | Pages | Definitions | Theorems | Lemmas | Propositions | Corollaries | Total | Formalised | |---|---|---|---|---|---|---|---|---|---| @@ -41,11 +41,11 @@ At a glance, the 661 declarations break down by top-level section as follows (cl | [§10.2](sec10-2-spectral.html) | Spectral Theorems (bounded) | 38–155 | 25 | 24 | 27 | 67 | 1 | 144 | 144 | | [§10.3](sec10-3-unbounded.html) | Unbounded Spectral Theorems | 155–235 | 27 | 13 | 50 | 30 | 3 | 123 | 123 | | [§10.4](sec10-4-stone.html) | Stone's Theorem | 235–249 | 3 | 2 | 34 | 2 | 0 | 41 | 41 | -| [§10.5](sec10-5-spacetime.html) | Spacetime and causal structure | 249–286 | 34 | 17 | 86 | 0 | 0 | 137 | 137 | -| [§10.6](sec10-6-haag-kastler.html) | Haag–Kastler Axioms (Minkowski) | 286–327 | 40 | 70 | 49 | 0 | 0 | 159 | 159 | -| [§10.7](sec10-7-curved.html) | Haag–Kastler Axioms (curved spacetime) | 327–337 | 17 | 29 | 3 | 0 | 0 | 49 | 49 | -| [§10.8](sec10-8-general-covariance.html) | General Covariance | 337–338 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -| **Total** | | | **150** | **156** | **252** | **99** | **4** | **661** | **661** | +| [§10.5](sec10-5-spacetime.html) | Spacetime and causal structure | 249–287 | 34 | 17 | 86 | 0 | 0 | 137 | 137 | +| [§10.6](sec10-6-haag-kastler.html) | Haag–Kastler Axioms (Minkowski) | 287–328 | 40 | 70 | 50 | 0 | 0 | 160 | 160 | +| [§10.7](sec10-7-curved.html) | Haag–Kastler Axioms (curved spacetime) | 328–338 | 17 | 29 | 3 | 0 | 0 | 49 | 49 | +| [§10.8](sec10-8-general-covariance.html) | General Covariance | 338–339 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | +| **Total** | | | **150** | **156** | **253** | **99** | **4** | **662** | **662** | ## Explore @@ -56,24 +56,24 @@ At a glance, the 661 declarations break down by top-level section as follows (cl - [§10.2 Spectral Theorems (pp. 38–155)](sec10-2-spectral.html) - [§10.3 Unbounded Spectral Theorems (pp. 155–235)](sec10-3-unbounded.html) - [§10.4 Stone's Theorem (pp. 235–249)](sec10-4-stone.html) - - [§10.5 Spacetime and causal structure (pp. 249–286)](sec10-5-spacetime.html) - - [§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 286–327)](sec10-6-haag-kastler.html) - - [§10.7 Haag–Kastler Axioms in curved spacetime (pp. 327–337)](sec10-7-curved.html) - - [§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 337–338)](sec10-8-general-covariance.html) + - [§10.5 Spacetime and causal structure (pp. 249–287)](sec10-5-spacetime.html) + - [§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 287–328)](sec10-6-haag-kastler.html) + - [§10.7 Haag–Kastler Axioms in curved spacetime (pp. 328–338)](sec10-7-curved.html) + - [§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 338–339)](sec10-8-general-covariance.html) ## Formalisation status The blueprint annotates every node with the Lean declarations that realise it, so the status of each node is a matter of record rather than of estimate. Every node in Chapter 10 is formalised. There is one place where the Lean is deliberately shaped differently from the prose; it is flagged in the blueprint text itself and collected here so that it is not discovered by surprise. The project contains no `sorry`. -**Recently completed: Stone's Theorem.** The new [§10.4](sec10-4-stone.html) (Definition 289 through Theorem 329, pp. 235–249) proves Stone's Theorem, the correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators, and four new §10.3 nodes supply what it needs: a self-adjoint operator has no proper symmetric extension (Lemma 189), the spectral measure of a self-adjoint operator (Definition 286), its functional calculus (Definition 287), and the agreement of that calculus with the bounded one for bounded functions (Lemma 288). All 39 nodes are formalised and proved; the theorem itself is `Physicslib4.Spectral.Stone.stone` in `Physicslib4/Spectral/Stone/Theorem.lean`. With Stone's Theorem in Lean, positive energy (the spectrum condition behind vacuum and ground states) is now stated with a genuine unbounded generator: a one-parameter unitary group has positive energy when it is strongly continuous and its generator is a positive operator, equivalently $$V(t) = e^{itA}$$ with $$A$$ positive and self-adjoint (Theorem 602, `Physicslib4.AQFT.isPositiveEnergy_iff_exists_exp`). The former bounded-generator restriction is gone, and the project records no remaining restrictions. +**Recently completed: Stone's Theorem.** The new [§10.4](sec10-4-stone.html) (Definition 289 through Theorem 329, pp. 236–249) proves Stone's Theorem, the correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators, and four new §10.3 nodes supply what it needs: a self-adjoint operator has no proper symmetric extension (Lemma 189), the spectral measure of a self-adjoint operator (Definition 286), its functional calculus (Definition 287), and the agreement of that calculus with the bounded one for bounded functions (Lemma 288). All 39 nodes are formalised and proved; the theorem itself is `Physicslib4.Spectral.Stone.stone` in `Physicslib4/Spectral/Stone/Theorem.lean`. With Stone's Theorem in Lean, positive energy (the spectrum condition behind vacuum and ground states) is now stated with a genuine unbounded generator: a one-parameter unitary group has positive energy when it is strongly continuous and its generator is a positive operator, equivalently $$V(t) = e^{itA}$$ with $$A$$ positive and self-adjoint (Theorem 603, `Physicslib4.AQFT.isPositiveEnergy_iff_exists_exp`). The former bounded-generator restriction is gone, and the project records no remaining restrictions. **Recently completed: smoothness of the inverse musical isomorphism.** Lemma 358 (§10.5, p. 255), the last unformalised node, states that $$x \mapsto \flat_x^{-1}$$ is a smooth section of $$\mathrm{Hom}(T^*M, TM)$$. It is formalised as `Physicslib4.Geometry.PseudoRiemannianMetric.contMDiff_flatEquiv_symm` in `Physicslib4/Geometry/PseudoRiemannian/Basic.lean`, following the blueprint: in local trivialisations the coordinate form of $$\flat_x^{-1}$$ is the inverse of the coordinate form of the metric, and inversion of continuous linear maps is smooth. -**Recently completed: the von Neumann density theorem.** Theorem 505, "Quasilocal Observables are Strongly Dense in the Bicommutant" (§10.6.2, p. 305), formalised as `Physicslib4.AQFT.HaagKastler.dense_range_in_bicommutant` in `Physicslib4/AQFT/HaagKastler/StrongDensity.lean`, is now proved. Mathlib has the strong operator topology (`PointwiseConvergenceCLM`) and the double commutant property of bundled von Neumann algebras, but not the density theorem itself, so the project proves it locally, following Murphy, Lemma 4.1.4, in the new §10.6.2 (Lemmas 495–504, Definition 498): a closed subspace invariant under an operator and its adjoint has a commuting projection (Lemma 495, in `Physicslib4/Operators/ReducingSubspace.lean`), so the closed cyclic subspace of a vector reduces the representation (Lemma 496) and the one-vector case follows (Lemma 497); a finite amplification $$\pi^\iota$$ on $$\ell^2(\iota; H)$$ (Definition 498, Lemmas 499–502, in `Physicslib4/Operators/DensityTheorem.lean`) carries the bicommutant diagonally into its own bicommutant, which upgrades the one-vector case to finitely many vectors (Lemma 503); and a neighbourhood basis for the strong operator topology (Lemma 504) turns that into density. The Kaplansky density theorem, which would additionally keep approximants self-adjoint and norm-bounded, remains out of scope; nothing in the project needs it. +**Recently completed: the von Neumann density theorem.** Theorem 506, "Quasilocal Observables are Strongly Dense in the Bicommutant" (§10.6.2, p. 307), formalised as `Physicslib4.AQFT.HaagKastler.dense_range_in_bicommutant` in `Physicslib4/AQFT/HaagKastler/StrongDensity.lean`, is now proved. Mathlib has the strong operator topology (`PointwiseConvergenceCLM`) and the double commutant property of bundled von Neumann algebras, but not the density theorem itself, so the project proves it locally, following Murphy, Lemma 4.1.4, in the new §10.6.2 (Lemmas 496–505, Definition 499): a closed subspace invariant under an operator and its adjoint has a commuting projection (Lemma 496, in `Physicslib4/Operators/ReducingSubspace.lean`), so the closed cyclic subspace of a vector reduces the representation (Lemma 497) and the one-vector case follows (Lemma 498); a finite amplification $$\pi^\iota$$ on $$\ell^2(\iota; H)$$ (Definition 499, Lemmas 500–503, in `Physicslib4/Operators/DensityTheorem.lean`) carries the bicommutant diagonally into its own bicommutant, which upgrades the one-vector case to finitely many vectors (Lemma 504); and a neighbourhood basis for the strong operator topology (Lemma 505) turns that into density. The Kaplansky density theorem, which would additionally keep approximants self-adjoint and norm-bounded, remains out of scope; nothing in the project needs it. **Formalised, but the Lean is shaped differently from the prose (one place).** The node below carries Lean declarations and a formalised proof; the caveat is one of presentation, not of coverage or content. -- **Axiom 5 in curved spacetime (Definition 631).** "Isometries connected to the identity" and "identity-component isometries preserving the future orientation" describe the same group, but the inclusion of the former in the latter rests on a Myers–Steenrod-type rigidity result not yet in Mathlib. The Lean therefore intersects the identity component with the explicitly orientation-preserving subgroup. This is an implementation choice and does not alter the mathematical content of the axiom. +- **Axiom 5 in curved spacetime (Definition 632).** "Isometries connected to the identity" and "identity-component isometries preserving the future orientation" describe the same group, but the inclusion of the former in the latter rests on a Myers–Steenrod-type rigidity result not yet in Mathlib. The Lean therefore intersects the identity component with the explicitly orientation-preserving subgroup. This is an implementation choice and does not alter the mathematical content of the axiom. ## Useful links diff --git a/home_page/sec10-3-unbounded.md b/home_page/sec10-3-unbounded.md index b2e499e..1759a08 100644 --- a/home_page/sec10-3-unbounded.md +++ b/home_page/sec10-3-unbounded.md @@ -1,5 +1,5 @@ --- -title: "§10.3 Unbounded Spectral Theorems (pp. 155–249)" +title: "§10.3 Unbounded Spectral Theorems (pp. 155–235)" usemathjax: true --- @@ -7,7 +7,7 @@ usemathjax: true ← [§10.2 Spectral](sec10-2-spectral.html) · [§10.4 Stone](sec10-4-stone.html) → -# §10.3 Unbounded Spectral Theorems (pp. 155–249) +# §10.3 Unbounded Spectral Theorems (pp. 155–235) 123 declarations (items 164–288, together with Conventions 213 and 260), extending §10.2 to the Spectral Theorem for Unbounded, Self-Adjoint Operators (Theorem 285) and the functional calculus it supplies. The blueprint motivates this with the few genuinely unbounded self-adjoint operators that arise in AQFT, the momentum operator being the standard example. The section reuses §10.2 without restating it, follows the same Hall presentation, and builds up unbounded operators from scratch. Its single subsection, §10.3.1 Spectral Theorem: Unbounded Self-Adjoint Operators (p. 156), proceeds in six stages. @@ -166,7 +166,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · The Spectral Theorem for Bounded Normal Operators › The Two-Variable Spectral Mapping Theorem (p. 210) — 2 items +§10.3.1 · The Spectral Theorem for Bounded Normal Operators › The Two-Variable Spectral Mapping Theorem (p. 211) — 2 items - [**Theorem 257**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:hall-10.23) — Spectral Mapping for Polynomials in $$A$$ and $$A^*$$ · `Physicslib4.Spectral.Unbounded.spectrum_mvApply` - [**Corollary 258**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#crllr:norm-of-polynomial-in-a-astar) — Norm of a Polynomial in $$A$$ and $$A^*$$ · `Physicslib4.Spectral.Unbounded.norm_mvApply` diff --git a/home_page/sec10-4-stone.md b/home_page/sec10-4-stone.md index 4e94434..6dd69cf 100644 --- a/home_page/sec10-4-stone.md +++ b/home_page/sec10-4-stone.md @@ -11,7 +11,7 @@ usemathjax: true 41 declarations (items 289–329), all formalised and proved. The section proves **Stone's Theorem** (Theorem 329): every strongly continuous one-parameter unitary group $$U$$ on a Hilbert space $$\mathbf{H}$$ is $$U(t) = e^{itA}$$ for a unique self-adjoint operator $$A$$, its infinitesimal generator. Together with the converse direction (Proposition 313) this is a one-to-one correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators. In AQFT it is what turns the translation symmetry of a net into an energy–momentum operator, and it is what lets positive energy (`IsPositiveEnergy`, §10.6) be stated with a genuine unbounded generator. The section builds on the Spectral Theorem for Unbounded, Self-Adjoint Operators (Theorem 285) and the functional calculus it supplies (Definitions 286–287, Lemma 288), and proceeds in five subsections. -- *One-parameter unitary groups (p. 235, items 289–300).* A one-parameter unitary group (Definition 289), strong continuity and its reduction to continuity at $$0$$ (Lemma 290), the adjoint $$U(t)^* = U(-t)$$ (Lemma 291), and the infinitesimal generator (Definition 292), with its characterisation (Lemma 293) and the symmetry identity it satisfies (Lemma 294). Three generators are computed directly: the trivial group has generator $$0$$ (Lemma 295), conjugating a group by a unitary $$W$$ conjugates its generator (Lemmas 296–297), and for bounded self-adjoint $$P$$ the group $$\exp(itP)$$ has generator $$P$$ (Lemmas 298–300); these feed the positive-energy lemmas of §10.6. +- *One-parameter unitary groups (p. 236, items 289–300).* A one-parameter unitary group (Definition 289), strong continuity and its reduction to continuity at $$0$$ (Lemma 290), the adjoint $$U(t)^* = U(-t)$$ (Lemma 291), and the infinitesimal generator (Definition 292), with its characterisation (Lemma 293) and the symmetry identity it satisfies (Lemma 294). Three generators are computed directly: the trivial group has generator $$0$$ (Lemma 295), conjugating a group by a unitary $$W$$ conjugates its generator (Lemmas 296–297), and for bounded self-adjoint $$P$$ the group $$\exp(itP)$$ has generator $$P$$ (Lemmas 298–300); these feed the positive-energy lemmas of §10.6. - *Imported results (p. 239, items 301–305).* Standard analysis stated without proof and linked to Mathlib: the Bochner integral (Theorem 301), smooth approximate identities (Lemma 302), facts about differentiation (Proposition 303) and about convolution (Lemma 305). The scalar equation $$y' = cy$$ (Lemma 304) is proved from them. - *From a self-adjoint operator to a unitary group (p. 241, items 306–313).* For self-adjoint $$A$$ the bounded functions $$e^{it\lambda}$$ give unitaries (Lemma 306) forming the group $$e^{itA}$$ (Definition 307), which satisfies the group law (Lemma 308) and is strongly continuous by dominated convergence (Lemma 309). A difference-quotient norm identity (Lemmas 310–311) gives the derivative at $$0$$ on the domain of $$A$$ (Lemma 312), and maximal symmetry of a self-adjoint operator identifies the generator with $$A$$ (Proposition 313). - *Two intermediate results (p. 244, items 314–320).* Orbits $$t \mapsto U(t)\psi$$ are differentiable on the generator's domain (Lemma 314), the group commutes with its generator and preserves its domain (Lemmas 315–316), and averaging an orbit against a smooth bump, written as a convolution (Lemmas 317–319), shows that the generator is densely defined (Lemma 320). @@ -26,7 +26,7 @@ The closing Summary of the blueprint section, the bijection between groups and g Each blueprint subsection below is collapsed; click a heading to see its items. Every entry links to its node in the web blueprint and names the principal Lean declaration; the four imported results name the Mathlib declarations that supply them.
-§10.4.1 One-Parameter Unitary Groups (p. 235) — 12 items +§10.4.1 One-Parameter Unitary Groups (p. 236) — 12 items - [**Definition 289**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-10.11) — One-Parameter Unitary Group · `Physicslib4.Spectral.Stone.IsOneParameterUnitaryGroup` (+1 more) - [**Lemma 290**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:strongly-continuous-iff-at-zero) — Strong Continuity is Strong Continuity at $$0$$ · `Physicslib4.Spectral.Stone.isStronglyContinuous_iff_tendsto_zero` diff --git a/home_page/sec10-5-spacetime.md b/home_page/sec10-5-spacetime.md index 54f2beb..d689313 100644 --- a/home_page/sec10-5-spacetime.md +++ b/home_page/sec10-5-spacetime.md @@ -1,5 +1,5 @@ --- -title: "§10.5 Spacetime and causal structure (pp. 249–286)" +title: "§10.5 Spacetime and causal structure (pp. 249–287)" usemathjax: true --- @@ -7,19 +7,19 @@ usemathjax: true ← [§10.4 Stone](sec10-4-stone.html) · [§10.6 Minkowski](sec10-6-haag-kastler.html) → -# §10.5 Spacetime and causal structure (pp. 249–286) +# §10.5 Spacetime and causal structure (pp. 249–287) 137 items (Definitions 330–466), building the entire causal and topological apparatus the axioms are indexed on. The trips underlying the chronological and causal relations $$\ll$$ and $$\prec$$ are genuine geodesics of the Levi-Civita connection of the metric, which the section constructs for an arbitrary pseudo-Riemannian metric on top of Mathlib's covariant derivatives (Mathlib itself has only a Riemannian Levi-Civita connection and no geodesics). It proceeds in nine layers. - *Spacetime, tangent-vector causality, and curves (pp. 249–254, items 330–353).* Gives precise definitions of spacetime (Definition 330) and standard Minkowski spacetime (Definition 331); classifies tangent vectors as timelike, spacelike, or null (Definition 332) and proves the trichotomy (Lemma 333), the reverse Cauchy–Schwarz and reverse triangle inequalities for timelike vectors (Lemmas 334–335), and the cone geometry—orientation of pointing vectors, the sign lemma, definiteness of the spacelike complement of a timelike vector, and convexity of the cones (Lemmas 338–341)—alongside time orientations (Definition 336) and future- and past-pointing vectors (Definition 337). Then paths, curves, and oriented curves are defined as equivalence classes of paths up to reparametrisation (Definitions 342–347), with causal type and future/past orientation each shown well-defined on the quotient (Theorems 346 and 348) and a forgetful projection from oriented to unoriented curves (Theorem 349). Endpoints (Definition 350), shown to be well defined on curves and smooth curves (Lemma 351), come with two point-set lemmas—an extremal parameter lies in the frontier, and two endpoints force a compact parameter interval (Lemmas 352–353). -- *The Levi-Civita connection and geodesics (pp. 254–258, items 354–369).* A new layer. Pseudo-Riemannian metrics (Definition 354) are smooth, symmetric, non-degenerate families of bilinear forms, with no signature condition, so everything here applies to every spacetime. Covariant derivatives are Mathlib's `IsCovariantDerivativeOn`; metric compatibility and the Levi-Civita condition (compatible and torsion-free) are defined on top of them (Definitions 355–356). The musical isomorphism $$v \mapsto g_x(v, \cdot)$$ is a linear isomorphism at each point (Lemma 357), and its inverse is a smooth section of $$\mathrm{Hom}(T^*M, TM)$$ (Lemma 358). The Koszul formula (Lemma 359), the tensoriality of the Koszul expression (Lemma 360), and the verification that the Koszul connection is a covariant derivative (Lemma 361) that is Levi-Civita (Lemma 362) give existence and uniqueness of the Levi-Civita connection (Theorem 363). Five lemmas about smooth paths—a function vanishing along a path has zero derivative along it, the covariant derivative along a curve is local, local vector fields globalise, a smooth path has a local left inverse, and its velocity extends to a smooth vector field (Lemmas 364–368)—make the geodesic condition chart-free, well defined and non-vacuous. A geodesic (Definition 369) is then an affinely parametrised smooth path $$\mu$$ with $$(\nabla_X X)(\mu(s)) = 0$$ at every interior parameter, for every smooth vector field $$X$$ extending the velocity near $$s$$. -- *Trips, futures and pasts (pp. 258–260, items 370–378).* Trips and causal trips (Definitions 370–371) are finite chains of future-oriented timelike (respectively causal) geodesic segments, geodesic in the sense of Definition 369, with matching past and future endpoints. Chronological and causal precedence are transitive (Theorem 372); the causality condition (Definition 373) makes them a strict partial order (Theorem 374); and chronological and causal futures and pasts (Definitions 375–376) come with their basic inclusion and monotonicity properties (Lemmas 377–378). +- *The Levi-Civita connection and geodesics (pp. 255–258, items 354–369).* A new layer. Pseudo-Riemannian metrics (Definition 354) are smooth, symmetric, non-degenerate families of bilinear forms, with no signature condition, so everything here applies to every spacetime. Covariant derivatives are Mathlib's `IsCovariantDerivativeOn`; metric compatibility and the Levi-Civita condition (compatible and torsion-free) are defined on top of them (Definitions 355–356). The musical isomorphism $$v \mapsto g_x(v, \cdot)$$ is a linear isomorphism at each point (Lemma 357), and its inverse is a smooth section of $$\mathrm{Hom}(T^*M, TM)$$ (Lemma 358). The Koszul formula (Lemma 359), the tensoriality of the Koszul expression (Lemma 360), and the verification that the Koszul connection is a covariant derivative (Lemma 361) that is Levi-Civita (Lemma 362) give existence and uniqueness of the Levi-Civita connection (Theorem 363). Five lemmas about smooth paths—a function vanishing along a path has zero derivative along it, the covariant derivative along a curve is local, local vector fields globalise, a smooth path has a local left inverse, and its velocity extends to a smooth vector field (Lemmas 364–368)—make the geodesic condition chart-free, well defined and non-vacuous. A geodesic (Definition 369) is then an affinely parametrised smooth path $$\mu$$ with $$(\nabla_X X)(\mu(s)) = 0$$ at every interior parameter, for every smooth vector field $$X$$ extending the velocity near $$s$$. +- *Trips, futures and pasts (pp. 259–260, items 370–378).* Trips and causal trips (Definitions 370–371) are finite chains of future-oriented timelike (respectively causal) geodesic segments, geodesic in the sense of Definition 369, with matching past and future endpoints. Chronological and causal precedence are transitive (Theorem 372); the causality condition (Definition 373) makes them a strict partial order (Theorem 374); and chronological and causal futures and pasts (Definitions 375–376) come with their basic inclusion and monotonicity properties (Lemmas 377–378). - *§10.5.1 Causal diamonds (p. 260, items 379–393).* Introduces the causal diamond $$J^+(p) \cap J^-(q)$$ and the chronological (Alexandrov) diamond $$I^+(p) \cap I^-(q)$$ (Definition 379), with the structural properties of the causal diamond—monotonicity under endpoint spread, causal convexity, and that nonemptiness forces $$p \prec q$$ (Lemma 380)—and the containment of chronological diamonds in causal ones, together with the identification of the Alexandrov basis as exactly the chronological diamonds (Theorem 381). It then develops spacelike separation of points and of regions (Definitions 382–383, Lemmas 384–385), the spacelike complement $$\mathbf{B}^\perp$$ (Definition 386) and its order structure—antitone, extensive on the double complement, with the triple complement collapsing, making complementation the Galois connection attached to the spacelike-separation relation (Lemma 387). The double complement is packaged as the causal closure operator (Definition 388, Lemma 389), whose fixed points are the causally complete regions (Definition 390). These form a complete lattice—meets are intersections, joins are causal closures of unions—on which the causal complement is an order-reversing involution (Theorem 391); the set-level De Morgan laws (Lemma 392) then lift to full binary and infinitary De Morgan laws on the lattice (Theorem 393). The blueprint is careful to record what does *not* hold: the full orthocomplement law $$\mathbf{B} \wedge \mathbf{B}^\perp = \bot$$ fails at this generality, because the trip-based causal relation is irreflexive and so a point is spacelike-separated from itself. -- *§10.5.2 Causal convexity (p. 264, items 394–396).* Defines a causally convex region as one containing every point causally between two of its own points (Definition 394), shows causal diamonds are causally convex (Lemma 395), and shows every spacelike complement—hence every causally complete region—is causally convex (Theorem 396). +- *§10.5.2 Causal convexity (p. 265, items 394–396).* Defines a causally convex region as one containing every point causally between two of its own points (Definition 394), shows causal diamonds are causally convex (Lemma 395), and shows every spacelike complement—hence every causally complete region—is causally convex (Theorem 396). - *§10.5.3 Causal convexity: closure structure, and the Alexandrov basis theorems (p. 265, items 397–420).* The causally convex regions are shown to form a closure system (Lemma 397), giving the causal-convex hull (Definition 398) as a genuine closure operator whose closed sets are exactly the causally convex regions (Lemmas 399, Theorem 400). The subsection then turns to the Alexandrov topology (Definition 401): every diamond is open (Lemma 402); chronological futures and pasts are open under an explicit "no endpoints" hypothesis (Lemma 403) and unconditionally on standard Minkowski spacetime, where the coordinate-cone description discharges the hypothesis (Lemma 404). On standard Minkowski spacetime the componentwise directional derivative is a Levi-Civita connection (Lemma 405), hence the Levi-Civita connection is flat (Lemma 406) and affinely parametrised straight lines are geodesics (Lemma 407), so trips there are the familiar straight-line segments. Minkowski spacetime and Lorentzian spacetime are then defined (Definitions 408–409, the latter as a spacetime whose Alexandrov topology is Hausdorff), with a bundled form of spacelike separation and basis openness (Lemma 410). A point lying in no diamond is pathological: its only Alexandrov neighbourhood is the whole space (Lemma 411), and the Hausdorff assumption rules this out, forcing the diamonds to cover the space (Lemma 412). Given downward-directedness the diamonds form a genuine topological basis (Theorem 413); past and future interpolation on standard Minkowski (Lemmas 414–415) supply downward-directedness there (Lemma 416), and common chronological predecessors and successors supply upward-directedness (Lemmas 417–419), so on standard Minkowski the diamonds are an unconditional basis (Theorem 420). Upward-directedness is what the quasilocal colimit of §10.6.1 later consumes. -- *§10.5.4 Dilations are causal automorphisms but not isometries (p. 271, items 421–423).* The dilations $$x \mapsto \lambda x$$ preserve the Minkowski cones (Lemma 421) and are therefore causal automorphisms (Theorem 422), yet scale the metric by $$\lambda^2$$ and so are not isometries for $$\lambda \neq 1$$ (Theorem 423). This is the counterexample that forces the general-covariance morphism of §10.8 to be specified geometrically rather than causally. +- *§10.5.4 Dilations are causal automorphisms but not isometries (p. 272, items 421–423).* The dilations $$x \mapsto \lambda x$$ preserve the Minkowski cones (Lemma 421) and are therefore causal automorphisms (Theorem 422), yet scale the metric by $$\lambda^2$$ and so are not isometries for $$\lambda \neq 1$$ (Theorem 423). This is the counterexample that forces the general-covariance morphism of §10.8 to be specified geometrically rather than causally. - *§10.5.5 Isometries and basis-set preservation (p. 272, items 424–434).* Single-metric transport: isometries preserve the causal classification of tangent vectors (Lemma 424), paths have unique differentials and well-defined pushforwards (Lemmas 425–426), and isometries respect the Levi-Civita structure: pullbacks of vector fields and of the metric pairing, their derivatives and Lie brackets transform naturally (Lemmas 427–429), so an isometry preserves the Levi-Civita connection (Lemma 430) and maps geodesics to geodesics and back (Lemma 431). Future-orientation-preserving isometries preserve chronology (Lemma 432) and map Alexandrov-basis diamonds to diamonds (Lemma 433), which is exactly the well-definedness condition for the Axiom 5 action (Lemma 434). -- *§10.5.6 Pullback metrics and cross-metric isometries (p. 274, items 435–466).* The single-metric lemmas above compare a spacetime with itself; general covariance instead compares two different metrics on one carrier, so the transport statements are redone cross-metric. This subsection defines the pullback $$\psi^* g$$ of a spacetime metric as a bundled family of continuous bilinear forms (Definition 435) and verifies every obligation in turn: the differential of a diffeomorphism is a linear equivalence (Lemma 436), with the two round-trip cancellation identities that Mathlib does not supply for a global `Diffeomorph` proved by hand (Lemmas 437–439); the pullback metric is symmetric, non-degenerate, Lorentzian, and a smooth section of the bilinear-form bundle (Lemmas 440–443), so the pullback of a spacetime is a spacetime (Theorem 444). Two-sided preservation of future orientation is then defined (Definition 445) and the pullback time orientation is shown to be bundle-smooth, nowhere vanishing, and everywhere timelike (Lemmas 446–449), with transport of future-pointing timelike and null vectors (Lemmas 450–451) giving two-sided orientation preservation (Lemma 452). Cross-metric isometries are defined (Definition 453) and shown closed under inverses (Lemma 454), to preserve causal classification (Lemma 455), to push paths forward preserving the timelike/causal conditions and endpoints (Lemmas 456–459), and to transport chronological precedence and the chronological future and past (Lemmas 460–462), hence to preserve Alexandrov-basis sets (Lemma 463). A general topological lemma—a bijection matching generating families is a homeomorphism (Lemma 464)—then identifies the pullback Alexandrov topology (Lemma 465) and yields that the pullback of a Lorentzian spacetime is again a Lorentzian spacetime (Theorem 466). Without Theorem 466 the phrase "the net over $$\psi^*(M,g)$$" in §10.8 would have no referent. +- *§10.5.6 Pullback metrics and cross-metric isometries (p. 275, items 435–466).* The single-metric lemmas above compare a spacetime with itself; general covariance instead compares two different metrics on one carrier, so the transport statements are redone cross-metric. This subsection defines the pullback $$\psi^* g$$ of a spacetime metric as a bundled family of continuous bilinear forms (Definition 435) and verifies every obligation in turn: the differential of a diffeomorphism is a linear equivalence (Lemma 436), with the two round-trip cancellation identities that Mathlib does not supply for a global `Diffeomorph` proved by hand (Lemmas 437–439); the pullback metric is symmetric, non-degenerate, Lorentzian, and a smooth section of the bilinear-form bundle (Lemmas 440–443), so the pullback of a spacetime is a spacetime (Theorem 444). Two-sided preservation of future orientation is then defined (Definition 445) and the pullback time orientation is shown to be bundle-smooth, nowhere vanishing, and everywhere timelike (Lemmas 446–449), with transport of future-pointing timelike and null vectors (Lemmas 450–451) giving two-sided orientation preservation (Lemma 452). Cross-metric isometries are defined (Definition 453) and shown closed under inverses (Lemma 454), to preserve causal classification (Lemma 455), to push paths forward preserving the timelike/causal conditions and endpoints (Lemmas 456–459), and to transport chronological precedence and the chronological future and past (Lemmas 460–462), hence to preserve Alexandrov-basis sets (Lemma 463). A general topological lemma—a bijection matching generating families is a homeomorphism (Lemma 464)—then identifies the pullback Alexandrov topology (Lemma 465) and yields that the pullback of a Lorentzian spacetime is again a Lorentzian spacetime (Theorem 466). Without Theorem 466 the phrase "the net over $$\psi^*(M,g)$$" in §10.8 would have no referent. **Where the Lean lives:** `Physicslib4/Geometry/PseudoRiemannian/` (`Basic`, `LeviCivita`, `Flat`), `Physicslib4/Spacetime/` (`Causality`, `Curves`, `AlongPath`, `CausalComplement`, `CausalStructure`, `Minkowski`, `MinkowskiDirected`, `LorentzianSpacetime`, `IsometryCausality`, …) @@ -104,7 +104,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.5.2 Causal convexity (p. 264) — 3 items +§10.5.2 Causal convexity (p. 265) — 3 items - [**Definition 394**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causally-convex-region) — Causally convex region · `Physicslib4.Spacetime.IsCausallyConvex` - [**Lemma 395**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-diamond-causally-convex) — Causal diamonds are causally convex · `Physicslib4.Spacetime.causalDiamond_isCausallyConvex` @@ -143,7 +143,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.5.4 Dilations are causal automorphisms but not isometries (p. 271) — 3 items +§10.5.4 Dilations are causal automorphisms but not isometries (p. 272) — 3 items - [**Lemma 421**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-dilation-cone) — Dilations Preserve the Minkowski Cones · `Physicslib4.minkowskiForwardCone_smul` (+1 more) - [**Theorem 422**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:minkowski-dilation-causal-automorphism) — Dilations are Causal Automorphisms · `Physicslib4.alexandrovBasis_image_smul` @@ -169,7 +169,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.5.6 Pullback metrics and cross-metric isometries (p. 274) — 32 items +§10.5.6 Pullback metrics and cross-metric isometries (p. 275) — 32 items - [**Definition 435**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pullback-metric) — Pullback of a Spacetime Metric · `Physicslib4.Spacetime.bilinearPrecomp` (+3 more) - [**Lemma 436**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-diffeo-linear-equiv) — The Differential of a Diffeomorphism is a Linear Equivalence · `Physicslib4.Spacetime.Diffeo` (+4 more) diff --git a/home_page/sec10-6-haag-kastler.md b/home_page/sec10-6-haag-kastler.md index 3d3dfd3..0f38306 100644 --- a/home_page/sec10-6-haag-kastler.md +++ b/home_page/sec10-6-haag-kastler.md @@ -1,5 +1,5 @@ --- -title: "§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 286–327)" +title: "§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 287–328)" usemathjax: true --- @@ -7,25 +7,25 @@ usemathjax: true ← [§10.5 Spacetime](sec10-5-spacetime.html) · [§10.7 Curved](sec10-7-curved.html) → -# §10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 286–327) - -The largest section by definition and theorem count (159 items, Definitions 467–625). Each axiom is stated as a definition so that it appears as a node in the declaration graph. - -- *The axioms and the quasilocal colimit (§10.6 and §10.6.1, pp. 286–303, items 467–494).* Axiom 1 (Local Algebras, Definition 467) assigns an abstract C\*-algebra to every Alexandrov-basis set, with $$\emptyset \mapsto \mathbb{C}1$$. Axiom 2 (Isotony, Definition 468) now supplies the family of unital \*-monomorphisms $$i_{\mathbf{B}_1 \mathbf{B}_2}$$ **as chosen data**, subject to injectivity, an identity law, and a composition law—so that $$\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$$ is a functor on the inclusion order of basis sets. The blueprint is explicit that the family cannot be an existence statement (the identity and composition conditions are equations between the maps themselves) and that the inclusion hypothesis is non-strict, matching the Lean. §10.6.1 then cashes this in: the diamonds are directed under inclusion (Lemma 469), the isotony family is a directed system in Mathlib's sense (Lemma 470), and the direct limit carries a well-defined norm (Lemmas 471–473) making it a normed \*-algebra (Lemma 474) satisfying the C\*-inequality (Lemma 475) but not, in general, complete. A block of results stated for an arbitrary normed \*-algebra under standing hypotheses (Definition 476) then carries the structure through completion—the involution and its extension (Definition 477, Lemma 478), the completion coercion as a bundled \*-algebra homomorphism (Lemma 479), and the passage of the C\*-inequality, the normed $$\mathbb{C}$$-algebra structure, and finally the C\*-algebra property to the completion (Lemmas 480–482)—yielding that the completion of the quasilocal colimit is a C\*-algebra (Lemma 483). The quasilocal algebra $$\mathfrak{U}$$ is defined as that colimit-then-completion (Definition 484); the blueprint records why the shortcut of realising $$\mathfrak{U}$$ as a closed \*-subalgebra of an ambient C\*-algebra is rejected—it would assume an ambient algebra containing copies of every $$\mathfrak{U}(\mathbf{B})$$, for which the physics supplies no justification. Axiom 3 (Local Commutativity, Definition 485) follows, together with the new Lemma 486: local commutativity transfers from any one quasilocal algebra to every other, so the choice of quasilocal algebra in Axiom 3 is immaterial and the canonical $$\mathfrak{U}$$ of a net can be used throughout. The quasilocal observable (Definition 487) follows. **Axiom 4 is now split in two.** Axiom 4 (Quasilocal Completeness, Definition 488) is presented as a bridge principle—the one axiom joining physical reality to the formalism, asserting the one-way inclusion that every physical observable corresponds to a quasilocal observable, and therefore having no mathematical consumers. The mathematical content previously conflated with it is separated out as a theorem: every local net satisfying Axiom 2 admits a quasilocal algebra, with an injective cocone of canonical embeddings whose images are dense (Theorem 489, with the supporting Lemmas 490–494). The indexing over Alexandrov-basis sets only is essential and not stylistic, since the algebras attached to non-basis subsets are junk fibres about which the axioms say nothing. -- *§10.6.2 The von Neumann Density Theorem (pp. 303–306, items 495–505).* A new subsection of general operator theory for a unital \*-homomorphism $$\pi : \mathfrak{A} \to \mathcal{B}(H)$$, whose only consumer is Theorem 505, the density result that keeps the bridge principle tenable in the presence of larger bicommutants: $$\pi_\omega(\mathfrak{U})$$ is strongly dense in $$\pi_\omega(\mathfrak{U})$$. Mathlib lacks the density theorem, so it is proved here following Murphy, Lemma 4.1.4. A closed subspace invariant under an operator and its adjoint has a commuting orthogonal projection (Lemma 495), so the closed cyclic subspace $$\overline{\pi(\mathfrak{A})\xi}$$ reduces the representation (Lemma 496) and every $$T \in \pi(\mathfrak{A})$$ maps $$\xi$$ into it (Lemma 497). The finite amplification $$\pi^\iota$$ on $$\ell^2(\iota; H)$$ (Definition 498) is a unital \*-homomorphism (Lemma 499); the block entries of an operator in its commutant lie in $$\pi(\mathfrak{A})'$$ (Lemma 500), an operator on $$\ell^2(\iota; H)$$ is recovered from its block entries (Lemma 501), and so $$\mathrm{diag}(T)$$ lies in the amplified bicommutant (Lemma 502). Applying the one-vector case to the amplification approximates $$T$$ on any finite set of vectors (Lemma 503), and a neighbourhood basis for the strong operator topology (Lemma 504) turns this into the density theorem (Theorem 505), now fully proved. The Kaplansky density theorem, which would also keep approximants self-adjoint, remains out of scope; nothing needs it. -- *§10.6.3 Lorentz Covariance and the Haag–Kastler Net (p. 306, items 506–507).* Axiom 5 (Lorentz Covariance, Definition 506), whose coherence condition is now stated for the Axiom 2 isotony family itself, and the bundled `HaagKastlerNet` (Definition 507) close the axiomatic block. -- *§10.6.4 Einstein Causality (p. 307, item 508).* Derives the operator form of local commutativity in any \*-representation of the quasilocal algebra (Theorem 508). -- *§10.6.5 Local von Neumann Algebras (p. 307, items 509–521).* Defines the local von Neumann algebra $$R(\mathbf{B}) = \pi(\mathfrak{U}(\mathbf{B}))''$$ of a region in a representation (Definition 509), registers it as a first-class `VonNeumannAlgebra` via the bicommutant-of-a-self-adjoint-set lemma (Lemma 510, Definition 511), and proves Microcausality (Theorem 512) and Isotony of the von Neumann net (Theorem 513), together with their bundled forms (Theorem 514) and the region-indexed assignment $$\mathbf{B} \mapsto R(\mathbf{B})$$ as an order-preserving map—the net of von Neumann algebras itself (Definition 515). It then proves the Statistical Independence (Schlieder) property in set-level and bundled forms (Theorems 516–517): if the cyclic vector of one region is cyclic for its local observables, it is separating for the local von Neumann algebra of any spacelike-separated region. Additive-free locality (Theorem 518) expresses locality through the spacelike complement without attaching an algebra to the unbounded complement itself. Geometric Covariance (Theorem 519) shows conjugation by the implementing unitary carries $$R(\mathbf{B})$$ onto $$R(L \cdot \mathbf{B})$$; being a factor is therefore constant along the Lorentz orbit of a region (Theorem 520), and the set equality upgrades to a first-class \*-algebra isomorphism of the bundled algebras (Theorem 521). -- *§10.6.6 Relative Commutants of Nested Local Algebras (p. 310, items 522–529).* A new block organising the theory of inclusions $$R(\mathbf{B}_1) \subseteq R(\mathbf{B}_2)$$ around the relative commutant $$R(\mathbf{B}_1)' \cap R(\mathbf{B}_2)$$ (Definition 522). It proves antitonicity of the commutant (Theorem 523), that the relative commutant lies in the larger algebra and commutes with the smaller (Theorems 524–525), and that it always contains the centre of the ambient algebra (Theorem 526). An inclusion is irreducible when its relative commutant is trivial (Definition 527); an irreducible inclusion forces the ambient algebra to be a factor (Theorem 528), and the trivial self-inclusion is irreducible exactly when the algebra is a factor (Theorem 529). -- *§10.6.7 Irreducibility and Schur's Lemma (p. 311, items 530–552).* Introduces irreducible representations via their commutant (Definition 530) and establishes the Topological Schur Lemma for cyclic representations (Theorem 531) together with the operator-theoretic bridge identifying commutant scalars with coefficients proportional to the state (Theorem 532). Defines pure states (Definition 533) and proves "pure implies irreducible" directly (Theorem 534), then the full equivalence Pure $$\iff$$ Irreducible (Theorem 537) via a GNS Radon–Nikodym theorem realising every dominated positive functional as an operator in the commutant (Theorems 535–536). An irreducible representation generates a factor (Theorem 538) and, more sharply, all of $$\mathcal{B}(H)$$ (Theorem 539), with a bundled density form (Theorem 540); consequently the GNS representation of a pure state generates a factor (Theorem 541) and all of $$\mathcal{B}(H)$$ (Theorem 542). The norm of a positive linear functional on a unital C\*-algebra equals its value on the unit (Theorem 543), which supports Pure $$\iff$$ Extreme Point of the state space (Definition 544, Theorem 545), the underlying convexity and the bridge to Mathlib's extreme-points API (Theorem 546), and weak-\* compactness of the state space (Theorem 550), which supplies the existence of pure states via Krein–Milman. The pullback of a state along a unital \*-homomorphism is introduced as its own object (Definition 547) with functoriality (Theorem 548) and the invariance of purity under a \*-isomorphism (Theorem 549). The section closes by specialising to the quasilocal algebra (Theorems 551–552). -- *§10.6.8 Unitary Equivalence and Superselection (p. 315, items 553–554).* Defines unitary equivalence of representations (Definition 553) and shows irreducibility and factoriality are unitary invariants, transported by the cross-space conjugation induced by the implementing unitary (Theorem 554). -- *§10.6.9 GNS Covariance (p. 315, items 555–560).* A new block. Cyclicity pulls back along a surjective \*-homomorphism (Lemma 555), so GNS data transports covariantly along a \*-isomorphism of the algebras (Theorem 556), restated as a unitary equivalence (Theorem 557). Pullback along a surjection preserves the image algebra (Lemma 558), whence superselection type transports along a \*-isomorphism (Theorem 559): the whole sector structure is an invariant of the algebra, not of its presentation. Applied to the covariance equivalence $$\alpha_L : \mathfrak{U}(\mathbf{B}) \simeq \mathfrak{U}(L \cdot \mathbf{B})$$ supplied by Axiom 5, this says the superselection type of a local state is constant along the Lorentz orbit of its region (Theorem 560). -- *§10.6.10 Disjointness and Quasi-Equivalence (p. 317, items 561–578).* Defines disjointness via the vanishing of all intertwiners (Definition 561) and the coarser quasi-equivalence via a \*-isomorphism of generated von Neumann algebras (Definition 562). Proves Schur's Lemma in the form of the Irreducible Dichotomy—two irreducible representations are either disjoint or unitarily equivalent (Theorem 563)—together with Schur multiplicity (Lemma 564) and the triviality of the endomorphism algebra of an irreducible representation (Lemma 565). The commutant is packaged as the self-intertwiner (gauge) von Neumann algebra, trivial exactly when the representation is irreducible (Theorem 566), with double-commutant duality (Theorem 567) and the factor/triviality duality between an algebra and its commutant (Theorem 568). A supporting run develops the centre from scratch: abelian $$\iff$$ self-commuting (Lemma 569), the centre $$Z(R) = R \cap R'$$ (Definition 570) and its being a von Neumann algebra (Lemma 571), the general two-algebra intersection lemma that the relative commutants of §10.6.6 need (Lemma 572), the centre is abelian (Theorem 573), $$R$$ is abelian iff it equals its centre (Lemma 574), a factor is abelian iff it is the scalars (Theorem 575), an algebra and its commutant share a centre (Theorem 576), and factoriality is triviality of the centre (Theorem 577). The Pure-State Dichotomy underlying superselection sectors (Theorem 578) closes the block. -- *§10.6.11 Direct Sums, Amplification, and Reducibility (p. 319, items 579–582).* Defines the direct-sum representation on the $$\ell^2$$-direct sum (Definition 579), shows each summand embeds as a subrepresentation whose projection lies in the commutant of the sum (Theorem 580), defines the $$\iota$$-fold amplification (Definition 581), and proves a direct sum with at least two nonzero summands is reducible, so a multiply-amplified representation is never irreducible (Theorem 582). -- *§10.6.12 Covariant States and the Covariance Action (p. 320, items 583–609).* Defines covariant families of local states (Definition 583) with their composition law (Lemma 584), and the lift of the fibrewise covariance action to a \*-automorphism of the quasilocal algebra (Definition 585), with uniqueness (Lemma 586) and existence for every quasilocal algebra (Theorem 588), no covariance-compatibility hypothesis being needed because every quasilocal algebra is automatically covariance-compatible (Lemma 587, new), and the trivial net as a special case (Theorem 589). The covariant quasilocal algebra of a net is then simply its canonical quasilocal algebra together with the covariance action (Definition 590, realised in Lean as `HaagKastlerNet.action`, so the covariance dynamics are stated directly for the net), which is a genuine group action (Lemma 591). Invariant states are defined (Definition 592) and shown to be implemented by GNS unitaries (Theorem 593); a state that is both invariant and pure yields a GNS representation that is simultaneously covariant and irreducible (Theorem 594). The spectrum condition follows. Positive energy (Definition 595) asks that a one-parameter unitary group be strongly continuous with positive generator; the trivial group has positive energy (Lemma 596), the generator is unique (Lemma 597), positivity and positive energy are unitarily invariant (Lemmas 598–599), a positive-energy group is strongly continuous (Lemma 600), the bounded case $$\exp(itP)$$ is included (Lemma 601), and, by Stone's Theorem, positive energy says exactly that $$V(t) = e^{itA}$$ with $$A$$ positive and self-adjoint (Theorem 602), all collected in Theorem 603. A vacuum state (Definition 604) is an invariant state whose translation groups have positive energy, with the consequences of invariance alone (Theorem 605), the future-timelike translation subgroup (Definition 606), and the vacuum state with that concrete predicate substituted in, leaving no free parameter (Definition 607). Purity is preserved by any \*-automorphism and is therefore covariance-invariant (Theorem 608), and GNS data transports along the quasilocal covariance action (Theorem 609). -- *§10.6.13 The Separating Vector of a Faithful State (p. 325, item 610).* The cyclic vector of a faithful state is also separating for the image of the representation (Theorem 610)—the basic datum of Tomita–Takesaki modular theory. This holds in any representation reproducing a faithful state, not only the canonical GNS one. -- *§10.6.14 The KMS Condition and Thermal Equilibrium (p. 325, items 611–620).* Introduces one-parameter automorphism groups (Definition 611) and KMS states (Definition 612) as the algebraic characterisation of thermal equilibrium. The condition is phrased purely as an analyticity statement about correlation functions, so—unlike the spectrum condition—it needs no unbounded-operator theory. The KMS state set is convex (Theorem 613); a boundary-coincidence argument at $$a = 1$$ (Lemma 614) together with the Strip-Liouville Principle (Definition 615), proved at positive inverse temperature via an $$i\beta$$-periodic entire extension (Theorem 616) and Liouville's theorem (Theorem 617), yields that KMS states are automatically invariant under the time evolution (Theorem 618); uniqueness on the strip from boundary values (Theorem 619) gives uniqueness of the analytic completion of a KMS correlation function (Theorem 620). -- *§10.6.15 KMS States for the Covariance Flow (p. 326, items 621–625).* A one-parameter subgroup of the inhomogeneous Lorentz group induces a one-parameter automorphism group on the quasilocal algebra via the covariance lift (Definition 621, Lemma 622); KMS states for that flow are defined accordingly (Definition 623) and shown convex (Theorem 624). The zero-temperature ($$\beta \to \infty$$) counterpart—a ground state for a covariance flow, whose GNS-implementing unitary group has positive energy—is recorded alongside it (Definition 625). +# §10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 287–328) + +The largest section by definition and theorem count (160 items, Definitions 467–626). Each axiom is stated as a definition so that it appears as a node in the declaration graph. + +- *The axioms and the quasilocal colimit (§10.6 and §10.6.1, pp. 287–305, items 467–495).* Axiom 1 (Local Algebras, Definition 467) assigns an abstract C\*-algebra to every Alexandrov-basis set, with $$\emptyset \mapsto \mathbb{C}1$$. Axiom 2 (Isotony, Definition 468) now supplies the family of unital \*-monomorphisms $$i_{\mathbf{B}_1 \mathbf{B}_2}$$ **as chosen data**, subject to injectivity, an identity law, and a composition law—so that $$\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$$ is a functor on the inclusion order of basis sets. The blueprint is explicit that the family cannot be an existence statement (the identity and composition conditions are equations between the maps themselves) and that the inclusion hypothesis is non-strict, matching the Lean. §10.6.1 then cashes this in: the diamonds are directed under inclusion (Lemma 469), the isotony family is a directed system in Mathlib's sense (Lemma 470), and the direct limit carries a well-defined norm (Lemmas 471–473) making it a normed \*-algebra (Lemma 474) satisfying the C\*-inequality (Lemma 475) but not, in general, complete. A block of results stated for an arbitrary normed \*-algebra under standing hypotheses (Definition 476) then carries the structure through completion—the involution and its extension (Definition 477, Lemma 478), the completion coercion as a bundled \*-algebra homomorphism (Lemma 479), and the passage of the C\*-inequality, the normed $$\mathbb{C}$$-algebra structure, and finally the C\*-algebra property to the completion (Lemmas 480–482)—yielding that the completion of the quasilocal colimit is a C\*-algebra (Lemma 483). The quasilocal algebra $$\mathfrak{U}$$ is defined as that colimit-then-completion (Definition 484); the blueprint records why the shortcut of realising $$\mathfrak{U}$$ as a closed \*-subalgebra of an ambient C\*-algebra is rejected—it would assume an ambient algebra containing copies of every $$\mathfrak{U}(\mathbf{B})$$, for which the physics supplies no justification. Axiom 3 (Local Commutativity, Definition 485) follows. It is stated, exactly as in curved spacetime, through the isotony maps: completely spacelike local algebras commute in every local algebra $$\mathfrak{U}(\mathbf{B})$$ containing both, and such a $$\mathbf{B}$$ always exists because the diamonds are directed. It then holds in every quasilocal algebra (Lemma 486), and it is equivalent to commutation in some quasilocal algebra, the form used by earlier versions of this blueprint (Lemma 487). The quasilocal observable (Definition 488) follows. **Axiom 4 is now split in two.** Axiom 4 (Quasilocal Completeness, Definition 489) is presented as a bridge principle—the one axiom joining physical reality to the formalism, asserting the one-way inclusion that every physical observable corresponds to a quasilocal observable, and therefore having no mathematical consumers. The mathematical content previously conflated with it is separated out as a theorem: every local net satisfying Axiom 2 admits a quasilocal algebra, with an injective cocone of canonical embeddings whose images are dense (Theorem 490, with the supporting Lemmas 491–495). The indexing over Alexandrov-basis sets only is essential and not stylistic, since the algebras attached to non-basis subsets are junk fibres about which the axioms say nothing. +- *§10.6.2 The von Neumann Density Theorem (pp. 305–307, items 496–506).* A new subsection of general operator theory for a unital \*-homomorphism $$\pi : \mathfrak{A} \to \mathcal{B}(H)$$, whose only consumer is Theorem 506, the density result that keeps the bridge principle tenable in the presence of larger bicommutants: $$\pi_\omega(\mathfrak{U})$$ is strongly dense in $$\pi_\omega(\mathfrak{U})$$. Mathlib lacks the density theorem, so it is proved here following Murphy, Lemma 4.1.4. A closed subspace invariant under an operator and its adjoint has a commuting orthogonal projection (Lemma 496), so the closed cyclic subspace $$\overline{\pi(\mathfrak{A})\xi}$$ reduces the representation (Lemma 497) and every $$T \in \pi(\mathfrak{A})$$ maps $$\xi$$ into it (Lemma 498). The finite amplification $$\pi^\iota$$ on $$\ell^2(\iota; H)$$ (Definition 499) is a unital \*-homomorphism (Lemma 500); the block entries of an operator in its commutant lie in $$\pi(\mathfrak{A})'$$ (Lemma 501), an operator on $$\ell^2(\iota; H)$$ is recovered from its block entries (Lemma 502), and so $$\mathrm{diag}(T)$$ lies in the amplified bicommutant (Lemma 503). Applying the one-vector case to the amplification approximates $$T$$ on any finite set of vectors (Lemma 504), and a neighbourhood basis for the strong operator topology (Lemma 505) turns this into the density theorem (Theorem 506), now fully proved. The Kaplansky density theorem, which would also keep approximants self-adjoint, remains out of scope; nothing needs it. +- *§10.6.3 Lorentz Covariance and the Haag–Kastler Net (p. 308, items 507–508).* Axiom 5 (Lorentz Covariance, Definition 507), whose coherence condition is now stated for the Axiom 2 isotony family itself, and the bundled `HaagKastlerNet` (Definition 508) close the axiomatic block. +- *§10.6.4 Einstein Causality (p. 308, item 509).* Derives the operator form of local commutativity in any \*-representation of the quasilocal algebra (Theorem 509). +- *§10.6.5 Local von Neumann Algebras (p. 308, items 510–522).* Defines the local von Neumann algebra $$R(\mathbf{B}) = \pi(\mathfrak{U}(\mathbf{B}))''$$ of a region in a representation (Definition 510), registers it as a first-class `VonNeumannAlgebra` via the bicommutant-of-a-self-adjoint-set lemma (Lemma 511, Definition 512), and proves Microcausality (Theorem 513) and Isotony of the von Neumann net (Theorem 514), together with their bundled forms (Theorem 515) and the region-indexed assignment $$\mathbf{B} \mapsto R(\mathbf{B})$$ as an order-preserving map—the net of von Neumann algebras itself (Definition 516). It then proves the Statistical Independence (Schlieder) property in set-level and bundled forms (Theorems 517–518): if the cyclic vector of one region is cyclic for its local observables, it is separating for the local von Neumann algebra of any spacelike-separated region. Additive-free locality (Theorem 519) expresses locality through the spacelike complement without attaching an algebra to the unbounded complement itself. Geometric Covariance (Theorem 520) shows conjugation by the implementing unitary carries $$R(\mathbf{B})$$ onto $$R(L \cdot \mathbf{B})$$; being a factor is therefore constant along the Lorentz orbit of a region (Theorem 521), and the set equality upgrades to a first-class \*-algebra isomorphism of the bundled algebras (Theorem 522). +- *§10.6.6 Relative Commutants of Nested Local Algebras (p. 311, items 523–530).* A new block organising the theory of inclusions $$R(\mathbf{B}_1) \subseteq R(\mathbf{B}_2)$$ around the relative commutant $$R(\mathbf{B}_1)' \cap R(\mathbf{B}_2)$$ (Definition 523). It proves antitonicity of the commutant (Theorem 524), that the relative commutant lies in the larger algebra and commutes with the smaller (Theorems 525–526), and that it always contains the centre of the ambient algebra (Theorem 527). An inclusion is irreducible when its relative commutant is trivial (Definition 528); an irreducible inclusion forces the ambient algebra to be a factor (Theorem 529), and the trivial self-inclusion is irreducible exactly when the algebra is a factor (Theorem 530). +- *§10.6.7 Irreducibility and Schur's Lemma (p. 312, items 531–553).* Introduces irreducible representations via their commutant (Definition 531) and establishes the Topological Schur Lemma for cyclic representations (Theorem 532) together with the operator-theoretic bridge identifying commutant scalars with coefficients proportional to the state (Theorem 533). Defines pure states (Definition 534) and proves "pure implies irreducible" directly (Theorem 535), then the full equivalence Pure $$\iff$$ Irreducible (Theorem 538) via a GNS Radon–Nikodym theorem realising every dominated positive functional as an operator in the commutant (Theorems 536–537). An irreducible representation generates a factor (Theorem 539) and, more sharply, all of $$\mathcal{B}(H)$$ (Theorem 540), with a bundled density form (Theorem 541); consequently the GNS representation of a pure state generates a factor (Theorem 542) and all of $$\mathcal{B}(H)$$ (Theorem 543). The norm of a positive linear functional on a unital C\*-algebra equals its value on the unit (Theorem 544), which supports Pure $$\iff$$ Extreme Point of the state space (Definition 545, Theorem 546), the underlying convexity and the bridge to Mathlib's extreme-points API (Theorem 547), and weak-\* compactness of the state space (Theorem 551), which supplies the existence of pure states via Krein–Milman. The pullback of a state along a unital \*-homomorphism is introduced as its own object (Definition 548) with functoriality (Theorem 549) and the invariance of purity under a \*-isomorphism (Theorem 550). The section closes by specialising to the quasilocal algebra (Theorems 552–553). +- *§10.6.8 Unitary Equivalence and Superselection (p. 316, items 554–555).* Defines unitary equivalence of representations (Definition 554) and shows irreducibility and factoriality are unitary invariants, transported by the cross-space conjugation induced by the implementing unitary (Theorem 555). +- *§10.6.9 GNS Covariance (p. 316, items 556–561).* A new block. Cyclicity pulls back along a surjective \*-homomorphism (Lemma 556), so GNS data transports covariantly along a \*-isomorphism of the algebras (Theorem 557), restated as a unitary equivalence (Theorem 558). Pullback along a surjection preserves the image algebra (Lemma 559), whence superselection type transports along a \*-isomorphism (Theorem 560): the whole sector structure is an invariant of the algebra, not of its presentation. Applied to the covariance equivalence $$\alpha_L : \mathfrak{U}(\mathbf{B}) \simeq \mathfrak{U}(L \cdot \mathbf{B})$$ supplied by Axiom 5, this says the superselection type of a local state is constant along the Lorentz orbit of its region (Theorem 561). +- *§10.6.10 Disjointness and Quasi-Equivalence (p. 318, items 562–579).* Defines disjointness via the vanishing of all intertwiners (Definition 562) and the coarser quasi-equivalence via a \*-isomorphism of generated von Neumann algebras (Definition 563). Proves Schur's Lemma in the form of the Irreducible Dichotomy—two irreducible representations are either disjoint or unitarily equivalent (Theorem 564)—together with Schur multiplicity (Lemma 565) and the triviality of the endomorphism algebra of an irreducible representation (Lemma 566). The commutant is packaged as the self-intertwiner (gauge) von Neumann algebra, trivial exactly when the representation is irreducible (Theorem 567), with double-commutant duality (Theorem 568) and the factor/triviality duality between an algebra and its commutant (Theorem 569). A supporting run develops the centre from scratch: abelian $$\iff$$ self-commuting (Lemma 570), the centre $$Z(R) = R \cap R'$$ (Definition 571) and its being a von Neumann algebra (Lemma 572), the general two-algebra intersection lemma that the relative commutants of §10.6.6 need (Lemma 573), the centre is abelian (Theorem 574), $$R$$ is abelian iff it equals its centre (Lemma 575), a factor is abelian iff it is the scalars (Theorem 576), an algebra and its commutant share a centre (Theorem 577), and factoriality is triviality of the centre (Theorem 578). The Pure-State Dichotomy underlying superselection sectors (Theorem 579) closes the block. +- *§10.6.11 Direct Sums, Amplification, and Reducibility (p. 321, items 580–583).* Defines the direct-sum representation on the $$\ell^2$$-direct sum (Definition 580), shows each summand embeds as a subrepresentation whose projection lies in the commutant of the sum (Theorem 581), defines the $$\iota$$-fold amplification (Definition 582), and proves a direct sum with at least two nonzero summands is reducible, so a multiply-amplified representation is never irreducible (Theorem 583). +- *§10.6.12 Covariant States and the Covariance Action (p. 321, items 584–610).* Defines covariant families of local states (Definition 584) with their composition law (Lemma 585), and the lift of the fibrewise covariance action to a \*-automorphism of the quasilocal algebra (Definition 586), with uniqueness (Lemma 587) and existence for every quasilocal algebra (Theorem 589), no covariance-compatibility hypothesis being needed because every quasilocal algebra is automatically covariance-compatible (Lemma 588, new), and the trivial net as a special case (Theorem 590). The covariant quasilocal algebra of a net is then simply its canonical quasilocal algebra together with the covariance action (Definition 591, realised in Lean as `HaagKastlerNet.action`, so the covariance dynamics are stated directly for the net), which is a genuine group action (Lemma 592). Invariant states are defined (Definition 593) and shown to be implemented by GNS unitaries (Theorem 594); a state that is both invariant and pure yields a GNS representation that is simultaneously covariant and irreducible (Theorem 595). The spectrum condition follows. Positive energy (Definition 596) asks that a one-parameter unitary group be strongly continuous with positive generator; the trivial group has positive energy (Lemma 597), the generator is unique (Lemma 598), positivity and positive energy are unitarily invariant (Lemmas 599–600), a positive-energy group is strongly continuous (Lemma 601), the bounded case $$\exp(itP)$$ is included (Lemma 602), and, by Stone's Theorem, positive energy says exactly that $$V(t) = e^{itA}$$ with $$A$$ positive and self-adjoint (Theorem 603), all collected in Theorem 604. A vacuum state (Definition 605) is an invariant state whose translation groups have positive energy, with the consequences of invariance alone (Theorem 606), the future-timelike translation subgroup (Definition 607), and the vacuum state with that concrete predicate substituted in, leaving no free parameter (Definition 608). Purity is preserved by any \*-automorphism and is therefore covariance-invariant (Theorem 609), and GNS data transports along the quasilocal covariance action (Theorem 610). +- *§10.6.13 The Separating Vector of a Faithful State (p. 326, item 611).* The cyclic vector of a faithful state is also separating for the image of the representation (Theorem 611)—the basic datum of Tomita–Takesaki modular theory. This holds in any representation reproducing a faithful state, not only the canonical GNS one. +- *§10.6.14 The KMS Condition and Thermal Equilibrium (p. 326, items 612–621).* Introduces one-parameter automorphism groups (Definition 612) and KMS states (Definition 613) as the algebraic characterisation of thermal equilibrium. The condition is phrased purely as an analyticity statement about correlation functions, so—unlike the spectrum condition—it needs no unbounded-operator theory. The KMS state set is convex (Theorem 614); a boundary-coincidence argument at $$a = 1$$ (Lemma 615) together with the Strip-Liouville Principle (Definition 616), proved at positive inverse temperature via an $$i\beta$$-periodic entire extension (Theorem 617) and Liouville's theorem (Theorem 618), yields that KMS states are automatically invariant under the time evolution (Theorem 619); uniqueness on the strip from boundary values (Theorem 620) gives uniqueness of the analytic completion of a KMS correlation function (Theorem 621). +- *§10.6.15 KMS States for the Covariance Flow (p. 328, items 622–626).* A one-parameter subgroup of the inhomogeneous Lorentz group induces a one-parameter automorphism group on the quasilocal algebra via the covariance lift (Definition 622, Lemma 623); KMS states for that flow are defined accordingly (Definition 624) and shown convex (Theorem 625). The zero-temperature ($$\beta \to \infty$$) counterpart—a ground state for a covariance flow, whose GNS-implementing unitary group has positive energy—is recorded alongside it (Definition 626). **Where the Lean lives:** `Physicslib4/AQFT/HaagKastler/`, `Physicslib4/GNS/` (`Irreducibility`, `Superselection`, `RadonNikodym`, `ExtremeState`, …), `Physicslib4/Operators/` (`ReducingSubspace`, `DensityTheorem`), `Physicslib4/AQFT/KMS.lean`, `Physicslib4/Analysis/StripPeriodicExtension.lean` @@ -34,7 +34,7 @@ The largest section by definition and theorem count (159 items, Definitions 467 Each blueprint subsection below is collapsed; click a heading to see its items. Every entry links to its node in the web blueprint and names the principal Lean declaration.
-§10.6 The axioms (p. 286) — 2 items +§10.6 The axioms (p. 287) — 2 items - [**Definition 467**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-algebras) — Axiom 1: Local Algebras · `Physicslib4.AQFT.HaagKastler.LocalNet` - [**Definition 468**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isotony) — Axiom 2: Isotony · `Physicslib4.AQFT.HaagKastler.Isotony` @@ -42,7 +42,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.6.1 The Quasilocal Colimit (p. 288) — 26 items +§10.6.1 The Quasilocal Colimit (p. 288) — 27 items - [**Lemma 469**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-diamonds-isDirected) — Alexandrov Diamonds are Directed under Inclusion · `Physicslib4.AQFT.HaagKastler.Diamond` (+2 more) - [**Lemma 470**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isotony-directed-system) — The Isotony Family is a Directed System · `Physicslib4.AQFT.HaagKastler.transitionHom` (+1 more) @@ -62,229 +62,230 @@ Each blueprint subsection below is collapsed; click a heading to see its items. - [**Definition 484**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-algebra) — Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra` - [**Definition 485**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-commutativity) — Axiom 3: Local Commutativity · `Physicslib4.AQFT.HaagKastler.LocalCommutativity` - [**Lemma 486**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-commutativity-any-quasilocal) — Local Commutativity Holds in Every Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.LocalCommutativity.commute_ι` (+1 more) -- [**Definition 487**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-observable) — Quasilocal Observable · `Physicslib4.AQFT.HaagKastler.IsQuasilocalObservable` -- [**Definition 488**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-completeness) — Axiom 4: Quasilocal Completeness · `Physicslib4.AQFT.HaagKastler.ObservableCorrespondence` -- [**Theorem 489**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-algebra-exists) — Existence of a Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra` -- [**Lemma 490**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding) — The Canonical Embeddings into the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.colimitStarOf` (+1 more) -- [**Lemma 491**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding-injective) — The Canonical Embeddings are Injective · `Physicslib4.AQFT.HaagKastler.quasilocalEmbedding_injective` -- [**Lemma 492**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding-cocone) — The Canonical Embeddings Form a Cocone · `Physicslib4.AQFT.HaagKastler.quasilocalEmbedding_transitionHom` -- [**Lemma 493**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-union-of-insertions) — The Colimit is the Union of the Images of its Insertions · `Physicslib4.AQFT.HaagKastler.exists_eq_colimitStarOf` -- [**Lemma 494**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embeddings-dense) — The Images of the Canonical Embeddings are Dense · `Physicslib4.AQFT.HaagKastler.dense_iUnion_range_quasilocalEmbedding` +- [**Lemma 487**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-commutativity-iff-quasilocal) — Local Commutativity in the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.localCommutativity_iff_exists_commute_ι` +- [**Definition 488**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-observable) — Quasilocal Observable · `Physicslib4.AQFT.HaagKastler.IsQuasilocalObservable` +- [**Definition 489**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-completeness) — Axiom 4: Quasilocal Completeness · `Physicslib4.AQFT.HaagKastler.ObservableCorrespondence` +- [**Theorem 490**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-algebra-exists) — Existence of a Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra` +- [**Lemma 491**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding) — The Canonical Embeddings into the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.colimitStarOf` (+1 more) +- [**Lemma 492**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding-injective) — The Canonical Embeddings are Injective · `Physicslib4.AQFT.HaagKastler.quasilocalEmbedding_injective` +- [**Lemma 493**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding-cocone) — The Canonical Embeddings Form a Cocone · `Physicslib4.AQFT.HaagKastler.quasilocalEmbedding_transitionHom` +- [**Lemma 494**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-union-of-insertions) — The Colimit is the Union of the Images of its Insertions · `Physicslib4.AQFT.HaagKastler.exists_eq_colimitStarOf` +- [**Lemma 495**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embeddings-dense) — The Images of the Canonical Embeddings are Dense · `Physicslib4.AQFT.HaagKastler.dense_iUnion_range_quasilocalEmbedding`
-§10.6.2 The von Neumann Density Theorem (p. 303) — 11 items - -- [**Lemma 495**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:reducing-subspace-projection-commutes) — Reducing Subspaces Give Commuting Projections · `Physicslib4.commute_starProjection_of_invariant` -- [**Lemma 496**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cyclic-subspace-reduces) — Cyclic Subspaces Reduce the Representation · `Physicslib4.starProjection_cyclic_mem_centralizer` -- [**Lemma 497**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-single-vector) — The Bicommutant on a Single Vector · `Physicslib4.apply_mem_closure_of_mem_bicommutant` -- [**Definition 498**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:finite-amplification) — Finite Amplification · `Physicslib4.diagAmplification` (+2 more) -- [**Lemma 499**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:finite-amplification-star-hom) — The Finite Amplification is a Unital \*-Homomorphism · `Physicslib4.diagAmplification` -- [**Lemma 500**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:commutant-of-amplification-entries) — Block Entries of the Amplified Commutant · `Physicslib4.blockEntry_mem_centralizer` -- [**Lemma 501**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:amplification-block-expansion) — Block Expansion on the Amplified Space · `Physicslib4.apply_eq_sum_blockEntry` -- [**Lemma 502**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:diagonal-in-amplified-bicommutant) — Diagonal Operators in the Amplified Bicommutant · `Physicslib4.ampDiag_mem_bicommutant` -- [**Lemma 503**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-finite-vectors) — The Bicommutant on Finitely Many Vectors · `Physicslib4.exists_forall_norm_sub_lt_of_mem_bicommutant` -- [**Lemma 504**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:strong-neighbourhood-basis) — A Neighbourhood Basis for the Strong Operator Topology · `Physicslib4.AQFT.HaagKastler.hasBasis_nhds_ofFun` -- [**Theorem 505**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-strongly-dense) — Quasilocal Observables are Strongly Dense in the Bicommutant · `Physicslib4.AQFT.HaagKastler.dense_range_in_bicommutant` — the von Neumann density theorem, proved locally from Lemmas 448–457 (see [Formalisation status](./#formalisation-status)) +§10.6.2 The von Neumann Density Theorem (p. 305) — 11 items + +- [**Lemma 496**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:reducing-subspace-projection-commutes) — Reducing Subspaces Give Commuting Projections · `Physicslib4.commute_starProjection_of_invariant` +- [**Lemma 497**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cyclic-subspace-reduces) — Cyclic Subspaces Reduce the Representation · `Physicslib4.starProjection_cyclic_mem_centralizer` +- [**Lemma 498**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-single-vector) — The Bicommutant on a Single Vector · `Physicslib4.apply_mem_closure_of_mem_bicommutant` +- [**Definition 499**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:finite-amplification) — Finite Amplification · `Physicslib4.diagAmplification` (+2 more) +- [**Lemma 500**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:finite-amplification-star-hom) — The Finite Amplification is a Unital \*-Homomorphism · `Physicslib4.diagAmplification` +- [**Lemma 501**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:commutant-of-amplification-entries) — Block Entries of the Amplified Commutant · `Physicslib4.blockEntry_mem_centralizer` +- [**Lemma 502**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:amplification-block-expansion) — Block Expansion on the Amplified Space · `Physicslib4.apply_eq_sum_blockEntry` +- [**Lemma 503**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:diagonal-in-amplified-bicommutant) — Diagonal Operators in the Amplified Bicommutant · `Physicslib4.ampDiag_mem_bicommutant` +- [**Lemma 504**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-finite-vectors) — The Bicommutant on Finitely Many Vectors · `Physicslib4.exists_forall_norm_sub_lt_of_mem_bicommutant` +- [**Lemma 505**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:strong-neighbourhood-basis) — A Neighbourhood Basis for the Strong Operator Topology · `Physicslib4.AQFT.HaagKastler.hasBasis_nhds_ofFun` +- [**Theorem 506**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-strongly-dense) — Quasilocal Observables are Strongly Dense in the Bicommutant · `Physicslib4.AQFT.HaagKastler.dense_range_in_bicommutant` — the von Neumann density theorem, proved locally from Lemmas 448–457 (see [Formalisation status](./#formalisation-status))
-§10.6.3 Lorentz Covariance and the Haag–Kastler Net (p. 306) — 2 items +§10.6.3 Lorentz Covariance and the Haag–Kastler Net (p. 307) — 2 items -- [**Definition 506**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:lorentz-covariance) — Axiom 5: Lorentz Covariance · `Physicslib4.AQFT.HaagKastler.LorentzCovariance` -- [**Definition 507**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:haag-kastler-net) — Haag–Kastler Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet` +- [**Definition 507**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:lorentz-covariance) — Axiom 5: Lorentz Covariance · `Physicslib4.AQFT.HaagKastler.LorentzCovariance` +- [**Definition 508**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:haag-kastler-net) — Haag–Kastler Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet`
-§10.6.4 Einstein Causality (p. 307) — 1 item +§10.6.4 Einstein Causality (p. 308) — 1 item -- [**Theorem 508**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:einstein-causality) — Einstein Causality in a Representation · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.einstein_causality` (+1 more) +- [**Theorem 509**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:einstein-causality) — Einstein Causality in a Representation · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.einstein_causality` (+1 more)
-§10.6.5 Local von Neumann Algebras (p. 307) — 13 items - -- [**Definition 509**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann) — Local von Neumann Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localOperators` (+1 more) -- [**Lemma 510**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-of-selfadjoint-is-von-neumann) — The Bicommutant of a Self-Adjoint Set is a von Neumann Algebra · `Physicslib4.GNS.vonNeumannOfSelfAdjoint` -- [**Definition 511**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-algebra) — $$R(\mathbf{B})$$ as a von Neumann Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra` -- [**Theorem 512**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-microcausality) — Microcausality at the von Neumann Level · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_subset_centralizer` -- [**Theorem 513**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-isotony) — Isotony of the von Neumann Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_mono` -- [**Theorem 514**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-bundled-order) — Bundled von Neumann Microcausality and Isotony · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_le_commutant` (+1 more) -- [**Definition 515**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-net) — The Net of von Neumann Algebras · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.vonNeumannNet` -- [**Theorem 516**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence) — Statistical Independence (Schlieder Property) · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_separating` (+1 more) -- [**Theorem 517**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-bundled) — Statistical Independence, bundled · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_separating` -- [**Theorem 518**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:additive-free-locality) — Additive-Free Locality via the Spacelike Complement · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_le_commutant_of_subset_spacelikeComplement` -- [**Theorem 519**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-geometric-covariance) — Geometric Covariance of the von Neumann Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.lieConj_image_localVonNeumann` (+1 more) -- [**Theorem 520**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-factor-orbit) — Orbit-Invariance of Factoriality · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_isFactor_smul` -- [**Theorem 521**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-covariance-iso) — Geometric Covariance as a von Neumann Algebra Isomorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannEquiv` +§10.6.5 Local von Neumann Algebras (p. 309) — 13 items + +- [**Definition 510**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann) — Local von Neumann Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localOperators` (+1 more) +- [**Lemma 511**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-of-selfadjoint-is-von-neumann) — The Bicommutant of a Self-Adjoint Set is a von Neumann Algebra · `Physicslib4.GNS.vonNeumannOfSelfAdjoint` +- [**Definition 512**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-algebra) — $$R(\mathbf{B})$$ as a von Neumann Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra` +- [**Theorem 513**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-microcausality) — Microcausality at the von Neumann Level · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_subset_centralizer` +- [**Theorem 514**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-isotony) — Isotony of the von Neumann Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_mono` +- [**Theorem 515**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-bundled-order) — Bundled von Neumann Microcausality and Isotony · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_le_commutant` (+1 more) +- [**Definition 516**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-net) — The Net of von Neumann Algebras · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.vonNeumannNet` +- [**Theorem 517**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence) — Statistical Independence (Schlieder Property) · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_separating` (+1 more) +- [**Theorem 518**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-bundled) — Statistical Independence, bundled · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_separating` +- [**Theorem 519**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:additive-free-locality) — Additive-Free Locality via the Spacelike Complement · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_le_commutant_of_subset_spacelikeComplement` +- [**Theorem 520**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-geometric-covariance) — Geometric Covariance of the von Neumann Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.lieConj_image_localVonNeumann` (+1 more) +- [**Theorem 521**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-factor-orbit) — Orbit-Invariance of Factoriality · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_isFactor_smul` +- [**Theorem 522**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-covariance-iso) — Geometric Covariance as a von Neumann Algebra Isomorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannEquiv`
-§10.6.6 Relative Commutants of Nested Local Algebras (p. 310) — 8 items +§10.6.6 Relative Commutants of Nested Local Algebras (p. 311) — 8 items -- [**Definition 522**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:relative-commutant) — Relative Commutant of a Nested Pair · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant` -- [**Theorem 523**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-antitone) — Antitonicity of the Commutant · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.commutant_le_commutant_of_le` -- [**Theorem 524**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-le-right) — Relative Commutant Lies in the Larger Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant_le_right` -- [**Theorem 525**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-coe-commutant) — Relative Commutant Commutes with the Smaller Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant_coe_subset_commutant` -- [**Theorem 526**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-center) — Relative Commutant Contains the Centre · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.center_le_relativeCommutant` -- [**Definition 527**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-inclusion) — Irreducible Inclusion · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsIrreducibleInclusion` -- [**Theorem 528**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-inclusion-factor) — An Irreducible Inclusion has Factor Ambient · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isFactor_of_isIrreducibleInclusion` -- [**Theorem 529**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:self-inclusion-factor) — Self-Inclusion is Irreducible iff Factor · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor` +- [**Definition 523**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:relative-commutant) — Relative Commutant of a Nested Pair · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant` +- [**Theorem 524**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-antitone) — Antitonicity of the Commutant · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.commutant_le_commutant_of_le` +- [**Theorem 525**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-le-right) — Relative Commutant Lies in the Larger Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant_le_right` +- [**Theorem 526**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-coe-commutant) — Relative Commutant Commutes with the Smaller Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant_coe_subset_commutant` +- [**Theorem 527**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-center) — Relative Commutant Contains the Centre · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.center_le_relativeCommutant` +- [**Definition 528**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-inclusion) — Irreducible Inclusion · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsIrreducibleInclusion` +- [**Theorem 529**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-inclusion-factor) — An Irreducible Inclusion has Factor Ambient · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isFactor_of_isIrreducibleInclusion` +- [**Theorem 530**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:self-inclusion-factor) — Self-Inclusion is Irreducible iff Factor · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor`
-§10.6.7 Irreducibility and Schur's Lemma (p. 311) — 23 items - -- [**Definition 530**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-representation) — Irreducible Representation · `Physicslib4.GNS.IsIrreducible` -- [**Theorem 531**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:schur-lemma) — Topological Schur Lemma · `Physicslib4.GNS.eq_smul_one_of_commute_of_cyclic` -- [**Theorem 532**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-scalar-iff) — Commutant Scalar iff Proportional Coefficient · `Physicslib4.GNS.isScalar_iff_coeff_proportional` -- [**Definition 533**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pure-state) — Pure State · `Physicslib4.GNS.IsPure` -- [**Theorem 534**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-implies-irreducible) — Pure Implies Irreducible · `Physicslib4.GNS.isIrreducible_of_isPure` -- [**Theorem 535**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-form-bound) — The GNS Radon–Nikodym Form is Bounded · `Physicslib4.GNS.gns_form_norm_le` (+1 more) -- [**Theorem 536**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-radon-nikodym-operator) — The GNS Radon–Nikodym Operator · `Physicslib4.GNS.rnOp` (+3 more) -- [**Theorem 537**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible) — Pure $$\iff$$ Irreducible · `Physicslib4.GNS.isPure_iff_isIrreducible` -- [**Theorem 538**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-factor) — An Irreducible Representation Generates a Factor · `Physicslib4.GNS.center_gnsVonNeumann_eq_of_isIrreducible` -- [**Theorem 539**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-generates-all) — Irreducibility $$\iff$$ Generating $$\mathcal{B}(H)$$ · `Physicslib4.GNS.isIrreducible_iff_gnsVonNeumann_eq_univ` (+1 more) -- [**Theorem 540**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-generates-all-bundled) — Bundled Density Form of Irreducibility · `Physicslib4.GNS.coe_gnsVonNeumannAlgebra_eq_univ_of_isIrreducible` (+1 more) -- [**Theorem 541**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-factor) — The GNS Representation of a Pure State is a Factor · `Physicslib4.GNS.exists_gns_factor_of_isPure` -- [**Theorem 542**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-generates-all) — The GNS Representation of a Pure State Generates $$\mathcal{B}(H)$$ · `Physicslib4.GNS.exists_gns_generates_all_of_isPure` -- [**Theorem 543**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:norm-positive-functional) — Norm of a Positive Functional · `Physicslib4.GNS.norm_eq_re_apply_one_of_positive` -- [**Definition 544**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:extreme-state) — Extreme Point of the State Space · `Physicslib4.GNS.State.IsExtremePoint` -- [**Theorem 545**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme) — Pure $$\iff$$ Extreme Point · `Physicslib4.GNS.isPure_iff_isExtremePoint` -- [**Theorem 546**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-space-convex-bridge) — State-Space Convexity and the Extreme-Point Bridge · `Physicslib4.GNS.convex_stateSpace` (+1 more) -- [**Definition 547**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:state-pullback) — Pullback of a State · `Physicslib4.GNS.State.comp` -- [**Theorem 548**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-pullback-functorial) — Functoriality of the State Pullback · `Physicslib4.GNS.State.comp_id` (+1 more) -- [**Theorem 549**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-pullback-invariant) — Purity is Invariant under a \*-Isomorphism · `Physicslib4.GNS.isPure_comp_iff` -- [**Theorem 550**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-space-weak-compact) — Weak-\* Compactness of the State Space · `Physicslib4.GNS.isCompact_weakStateSet` -- [**Theorem 551**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme-quasilocal) — Pure $$\iff$$ Extreme Point on the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.pure_iff_extreme` -- [**Theorem 552**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible-quasilocal) — Pure $$\iff$$ Irreducible GNS on the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.exists_gns_pure_iff_irreducible` +§10.6.7 Irreducibility and Schur's Lemma (p. 312) — 23 items + +- [**Definition 531**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-representation) — Irreducible Representation · `Physicslib4.GNS.IsIrreducible` +- [**Theorem 532**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:schur-lemma) — Topological Schur Lemma · `Physicslib4.GNS.eq_smul_one_of_commute_of_cyclic` +- [**Theorem 533**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-scalar-iff) — Commutant Scalar iff Proportional Coefficient · `Physicslib4.GNS.isScalar_iff_coeff_proportional` +- [**Definition 534**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pure-state) — Pure State · `Physicslib4.GNS.IsPure` +- [**Theorem 535**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-implies-irreducible) — Pure Implies Irreducible · `Physicslib4.GNS.isIrreducible_of_isPure` +- [**Theorem 536**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-form-bound) — The GNS Radon–Nikodym Form is Bounded · `Physicslib4.GNS.gns_form_norm_le` (+1 more) +- [**Theorem 537**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-radon-nikodym-operator) — The GNS Radon–Nikodym Operator · `Physicslib4.GNS.rnOp` (+3 more) +- [**Theorem 538**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible) — Pure $$\iff$$ Irreducible · `Physicslib4.GNS.isPure_iff_isIrreducible` +- [**Theorem 539**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-factor) — An Irreducible Representation Generates a Factor · `Physicslib4.GNS.center_gnsVonNeumann_eq_of_isIrreducible` +- [**Theorem 540**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-generates-all) — Irreducibility $$\iff$$ Generating $$\mathcal{B}(H)$$ · `Physicslib4.GNS.isIrreducible_iff_gnsVonNeumann_eq_univ` (+1 more) +- [**Theorem 541**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-generates-all-bundled) — Bundled Density Form of Irreducibility · `Physicslib4.GNS.coe_gnsVonNeumannAlgebra_eq_univ_of_isIrreducible` (+1 more) +- [**Theorem 542**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-factor) — The GNS Representation of a Pure State is a Factor · `Physicslib4.GNS.exists_gns_factor_of_isPure` +- [**Theorem 543**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-generates-all) — The GNS Representation of a Pure State Generates $$\mathcal{B}(H)$$ · `Physicslib4.GNS.exists_gns_generates_all_of_isPure` +- [**Theorem 544**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:norm-positive-functional) — Norm of a Positive Functional · `Physicslib4.GNS.norm_eq_re_apply_one_of_positive` +- [**Definition 545**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:extreme-state) — Extreme Point of the State Space · `Physicslib4.GNS.State.IsExtremePoint` +- [**Theorem 546**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme) — Pure $$\iff$$ Extreme Point · `Physicslib4.GNS.isPure_iff_isExtremePoint` +- [**Theorem 547**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-space-convex-bridge) — State-Space Convexity and the Extreme-Point Bridge · `Physicslib4.GNS.convex_stateSpace` (+1 more) +- [**Definition 548**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:state-pullback) — Pullback of a State · `Physicslib4.GNS.State.comp` +- [**Theorem 549**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-pullback-functorial) — Functoriality of the State Pullback · `Physicslib4.GNS.State.comp_id` (+1 more) +- [**Theorem 550**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-pullback-invariant) — Purity is Invariant under a \*-Isomorphism · `Physicslib4.GNS.isPure_comp_iff` +- [**Theorem 551**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-space-weak-compact) — Weak-\* Compactness of the State Space · `Physicslib4.GNS.isCompact_weakStateSet` +- [**Theorem 552**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme-quasilocal) — Pure $$\iff$$ Extreme Point on the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.pure_iff_extreme` +- [**Theorem 553**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible-quasilocal) — Pure $$\iff$$ Irreducible GNS on the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.exists_gns_pure_iff_irreducible`
-§10.6.8 Unitary Equivalence and Superselection (p. 315) — 2 items +§10.6.8 Unitary Equivalence and Superselection (p. 316) — 2 items -- [**Definition 553**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:unitary-equivalence) — Unitary Equivalence of Representations · `Physicslib4.GNS.UnitaryEquiv` -- [**Theorem 554**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:unitary-equiv-invariants) — Irreducibility and Factoriality are Unitary Invariants · `Physicslib4.GNS.UnitaryEquiv.isIrreducible_iff` (+1 more) +- [**Definition 554**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:unitary-equivalence) — Unitary Equivalence of Representations · `Physicslib4.GNS.UnitaryEquiv` +- [**Theorem 555**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:unitary-equiv-invariants) — Irreducibility and Factoriality are Unitary Invariants · `Physicslib4.GNS.UnitaryEquiv.isIrreducible_iff` (+1 more)
-§10.6.9 GNS Covariance (p. 315) — 6 items +§10.6.9 GNS Covariance (p. 316) — 6 items -- [**Lemma 555**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cyclic-pullback-surjective) — Cyclicity pulls back along a surjective \*-homomorphism · `Physicslib4.GNS.isCyclicVector_comp_of_surjective` -- [**Theorem 556**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance) — GNS covariance under a \*-isomorphism · `Physicslib4.GNS.exists_unitary_of_gns_comp` -- [**Theorem 557**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-unitary-equiv) — The GNS representation of a pullback state · `Physicslib4.GNS.unitaryEquiv_comp_of_gns` -- [**Lemma 558**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-image-invariants) — Pullback along a surjection preserves the image algebra · `Physicslib4.GNS.range_comp_of_surjective` (+2 more) -- [**Theorem 559**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-sector-transport) — Superselection type transports along a \*-isomorphism · `Physicslib4.GNS.isIrreducible_iff_of_gns_comp` (+1 more) -- [**Theorem 560**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-local) — GNS covariance for local algebras · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_covEquiv` (+2 more) +- [**Lemma 556**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cyclic-pullback-surjective) — Cyclicity pulls back along a surjective \*-homomorphism · `Physicslib4.GNS.isCyclicVector_comp_of_surjective` +- [**Theorem 557**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance) — GNS covariance under a \*-isomorphism · `Physicslib4.GNS.exists_unitary_of_gns_comp` +- [**Theorem 558**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-unitary-equiv) — The GNS representation of a pullback state · `Physicslib4.GNS.unitaryEquiv_comp_of_gns` +- [**Lemma 559**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-image-invariants) — Pullback along a surjection preserves the image algebra · `Physicslib4.GNS.range_comp_of_surjective` (+2 more) +- [**Theorem 560**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-sector-transport) — Superselection type transports along a \*-isomorphism · `Physicslib4.GNS.isIrreducible_iff_of_gns_comp` (+1 more) +- [**Theorem 561**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-local) — GNS covariance for local algebras · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_covEquiv` (+2 more)
-§10.6.10 Disjointness and Quasi-Equivalence (p. 317) — 18 items - -- [**Definition 561**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:disjoint-representations) — Disjoint Representations · `Physicslib4.GNS.AreDisjoint` -- [**Definition 562**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasi-equivalence) — Quasi-Equivalence of Representations · `Physicslib4.GNS.QuasiEquiv` -- [**Theorem 563**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-dichotomy) — Schur's Lemma and the Irreducible Dichotomy · `Physicslib4.GNS.UnitaryEquiv.of_intertwines_of_isIrreducible` (+1 more) -- [**Lemma 564**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:schur-multiplicity) — Schur Multiplicity · `Physicslib4.GNS.eq_smul_of_intertwines_of_isIrreducible` -- [**Lemma 565**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:endomorphism-scalar) — Endomorphism Algebra of an Irreducible Representation · `Physicslib4.GNS.intertwines_self_iff_isScalar` -- [**Theorem 566**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-von-neumann) — The commutant (self-intertwiner) von Neumann algebra · `Physicslib4.GNS.commutantVonNeumann` (+2 more) -- [**Theorem 567**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:double-commutant-duality) — Double-Commutant Duality · `Physicslib4.GNS.commutant_gnsVonNeumannAlgebra` (+1 more) -- [**Theorem 568**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-factor-duality) — A Factor and its Commutant; Triviality Duality · `Physicslib4.GNS.isFactor_gnsVonNeumann_iff_isFactor_commutant` (+1 more) -- [**Lemma 569**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-abelian-self-commuting) — Abelian $$\iff$$ Self-Commuting · `Physicslib4.GNS.isAbelian_iff_le_commutant` -- [**Definition 570**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-center) — Centre of a von Neumann algebra · `Physicslib4.GNS.vonNeumannCenter` -- [**Lemma 571**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-inter-is-von-neumann) — The centre is a von Neumann algebra · `Physicslib4.GNS.bicommutant_inter_commutant_eq` -- [**Lemma 572**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-inter-general) — The intersection of two von Neumann algebras is a von Neumann algebra · `Physicslib4.GNS.bicommutant_inter_eq` -- [**Theorem 573**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-center-abelian) — The centre of a von Neumann algebra is abelian · `Physicslib4.GNS.vonNeumannCenter_isAbelian` -- [**Lemma 574**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-center-eq-self-iff-abelian) — $$R$$ is abelian iff it equals its centre · `Physicslib4.GNS.vonNeumannCenter_eq_self_iff_isAbelian` -- [**Theorem 575**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:factor-abelian-iff-scalars) — A factor is abelian iff it is the scalars · `Physicslib4.GNS.isAbelian_iff_eq_scalars_of_isFactor` -- [**Theorem 576**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:center-eq-commutant-center) — A von Neumann algebra and its commutant share a centre · `Physicslib4.GNS.vonNeumannCenter_eq_commutant` -- [**Theorem 577**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:factor-iff-center-scalars) — A von Neumann algebra is a factor iff its centre is the scalars · `Physicslib4.GNS.isFactor_iff_center_eq_scalars` -- [**Theorem 578**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-state-dichotomy) — The Pure-State Dichotomy · `Physicslib4.GNS.exists_gns_areDisjoint_or_unitaryEquiv_of_isPure` +§10.6.10 Disjointness and Quasi-Equivalence (p. 318) — 18 items + +- [**Definition 562**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:disjoint-representations) — Disjoint Representations · `Physicslib4.GNS.AreDisjoint` +- [**Definition 563**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasi-equivalence) — Quasi-Equivalence of Representations · `Physicslib4.GNS.QuasiEquiv` +- [**Theorem 564**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-dichotomy) — Schur's Lemma and the Irreducible Dichotomy · `Physicslib4.GNS.UnitaryEquiv.of_intertwines_of_isIrreducible` (+1 more) +- [**Lemma 565**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:schur-multiplicity) — Schur Multiplicity · `Physicslib4.GNS.eq_smul_of_intertwines_of_isIrreducible` +- [**Lemma 566**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:endomorphism-scalar) — Endomorphism Algebra of an Irreducible Representation · `Physicslib4.GNS.intertwines_self_iff_isScalar` +- [**Theorem 567**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-von-neumann) — The commutant (self-intertwiner) von Neumann algebra · `Physicslib4.GNS.commutantVonNeumann` (+2 more) +- [**Theorem 568**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:double-commutant-duality) — Double-Commutant Duality · `Physicslib4.GNS.commutant_gnsVonNeumannAlgebra` (+1 more) +- [**Theorem 569**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-factor-duality) — A Factor and its Commutant; Triviality Duality · `Physicslib4.GNS.isFactor_gnsVonNeumann_iff_isFactor_commutant` (+1 more) +- [**Lemma 570**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-abelian-self-commuting) — Abelian $$\iff$$ Self-Commuting · `Physicslib4.GNS.isAbelian_iff_le_commutant` +- [**Definition 571**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-center) — Centre of a von Neumann algebra · `Physicslib4.GNS.vonNeumannCenter` +- [**Lemma 572**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-inter-is-von-neumann) — The centre is a von Neumann algebra · `Physicslib4.GNS.bicommutant_inter_commutant_eq` +- [**Lemma 573**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-inter-general) — The intersection of two von Neumann algebras is a von Neumann algebra · `Physicslib4.GNS.bicommutant_inter_eq` +- [**Theorem 574**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-center-abelian) — The centre of a von Neumann algebra is abelian · `Physicslib4.GNS.vonNeumannCenter_isAbelian` +- [**Lemma 575**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-center-eq-self-iff-abelian) — $$R$$ is abelian iff it equals its centre · `Physicslib4.GNS.vonNeumannCenter_eq_self_iff_isAbelian` +- [**Theorem 576**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:factor-abelian-iff-scalars) — A factor is abelian iff it is the scalars · `Physicslib4.GNS.isAbelian_iff_eq_scalars_of_isFactor` +- [**Theorem 577**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:center-eq-commutant-center) — A von Neumann algebra and its commutant share a centre · `Physicslib4.GNS.vonNeumannCenter_eq_commutant` +- [**Theorem 578**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:factor-iff-center-scalars) — A von Neumann algebra is a factor iff its centre is the scalars · `Physicslib4.GNS.isFactor_iff_center_eq_scalars` +- [**Theorem 579**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-state-dichotomy) — The Pure-State Dichotomy · `Physicslib4.GNS.exists_gns_areDisjoint_or_unitaryEquiv_of_isPure`
-§10.6.11 Direct Sums, Amplification, and Reducibility (p. 319) — 4 items +§10.6.11 Direct Sums, Amplification, and Reducibility (p. 321) — 4 items -- [**Definition 579**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:direct-sum-representation) — Direct-Sum Representation · `Physicslib4.GNS.directSum` -- [**Theorem 580**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:direct-sum-subrepresentation) — Subrepresentations and Commutant of a Direct Sum · `Physicslib4.GNS.intertwines_single` (+1 more) -- [**Definition 581**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:amplification) — Amplification · `Physicslib4.GNS.amplification` -- [**Theorem 582**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:direct-sum-reducible) — Reducibility of a Direct Sum · `Physicslib4.GNS.not_isIrreducible_directSum` +- [**Definition 580**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:direct-sum-representation) — Direct-Sum Representation · `Physicslib4.GNS.directSum` +- [**Theorem 581**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:direct-sum-subrepresentation) — Subrepresentations and Commutant of a Direct Sum · `Physicslib4.GNS.intertwines_single` (+1 more) +- [**Definition 582**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:amplification) — Amplification · `Physicslib4.GNS.amplification` +- [**Theorem 583**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:direct-sum-reducible) — Reducibility of a Direct Sum · `Physicslib4.GNS.not_isIrreducible_directSum`
-§10.6.12 Covariant States and the Covariance Action (p. 320) — 27 items - -- [**Definition 583**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-state-family) — Covariant Family of Local States · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsCovariantFamily` -- [**Lemma 584**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-state-family-compose) — Composition of Covariance · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsCovariantFamily.comp` -- [**Definition 585**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-lift) — Quasilocal Covariance Automorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift` -- [**Lemma 586**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-lift-unique) — Uniqueness of the Quasilocal Lift · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift.unique` -- [**Lemma 587**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-covariance-compatible) — Every Quasilocal Algebra is Covariance-Compatible · `Physicslib4.AQFT.HaagKastler.isCovariantQuasilocal` -- [**Theorem 588**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-lift-exists) — Existence of the Quasilocal Lift · `Physicslib4.AQFT.HaagKastler.nonempty_quasilocalLift` -- [**Theorem 589**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-lift-trivial) — Existence for the Trivial Net · `Physicslib4.AQFT.HaagKastler.nonempty_trivialQuasilocalLift` -- [**Definition 590**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-quasilocal-algebra) — Covariant Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.action` -- [**Lemma 591**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-action-coherence) — Group-Action Coherence of the Covariance Automorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.action_one` (+1 more) -- [**Definition 592**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:invariant-state) — Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState` -- [**Theorem 593**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:invariant-state-gns-unitary) — GNS Unitary Implementation of an Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitary` -- [**Theorem 594**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-covariant-representation) — Irreducible Covariant Representation of a Pure Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_irreducible_covariant` -- [**Definition 595**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:positive-energy) — Positive Energy · `Physicslib4.AQFT.IsPositiveEnergy` -- [**Lemma 596**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-const) — The Trivial Group has Positive Energy · `Physicslib4.AQFT.isPositiveEnergy_const_refl` -- [**Lemma 597**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:exp-generator-unique) — Uniqueness of the Generator · `Physicslib4.AQFT.generator_eq_of_eq_expUnitary` (+1 more) -- [**Lemma 598**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-conj) — Unitary Conjugation Preserves Positivity · `Physicslib4.Spectral.Unbounded.IsPositive.of_conj` -- [**Lemma 599**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-conj) — Positive Energy is Unitarily Invariant · `Physicslib4.AQFT.IsPositiveEnergy.conj` -- [**Lemma 600**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-strong-continuous) — A Positive-Energy Group is Strongly Continuous · `Physicslib4.AQFT.IsPositiveEnergy.strongContinuous` -- [**Lemma 601**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-exp-bounded) — The Bounded Case · `Physicslib4.AQFT.isPositiveEnergy_exp` -- [**Theorem 602**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:positive-energy-iff-exp) — Positive Energy in Exponential Form · `Physicslib4.AQFT.isPositiveEnergy_iff_exists_exp` -- [**Theorem 603**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:positive-energy-api) — Properties of Positive Energy · `Physicslib4.AQFT.isPositiveEnergy_const_refl` (+5 more) -- [**Definition 604**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:vacuum-state) — Vacuum State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState` -- [**Theorem 605**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:vacuum-invariance-consequences) — Consequences of Invariance for a Vacuum State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState.invariant` (+1 more) -- [**Definition 606**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:future-timelike-translation) — Future-Timelike Translation Subgroup · `Physicslib4.AQFT.HaagKastler.translationSub` (+3 more) -- [**Definition 607**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:vacuum-state-concrete) — Vacuum State with the Concrete Spectrum Condition · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumStateConcrete` -- [**Theorem 608**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:purity-covariance-invariant) — Purity is Covariance-Invariant · `Physicslib4.GNS.isPure_precomp_iff` (+1 more) -- [**Theorem 609**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-quasilocal-action) — GNS covariance along the quasilocal action · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_action` (+2 more) +§10.6.12 Covariant States and the Covariance Action (p. 321) — 27 items + +- [**Definition 584**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-state-family) — Covariant Family of Local States · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsCovariantFamily` +- [**Lemma 585**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-state-family-compose) — Composition of Covariance · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsCovariantFamily.comp` +- [**Definition 586**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-lift) — Quasilocal Covariance Automorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift` +- [**Lemma 587**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-lift-unique) — Uniqueness of the Quasilocal Lift · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift.unique` +- [**Lemma 588**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-covariance-compatible) — Every Quasilocal Algebra is Covariance-Compatible · `Physicslib4.AQFT.HaagKastler.isCovariantQuasilocal` +- [**Theorem 589**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-lift-exists) — Existence of the Quasilocal Lift · `Physicslib4.AQFT.HaagKastler.nonempty_quasilocalLift` +- [**Theorem 590**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-lift-trivial) — Existence for the Trivial Net · `Physicslib4.AQFT.HaagKastler.nonempty_trivialQuasilocalLift` +- [**Definition 591**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-quasilocal-algebra) — Covariant Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.action` +- [**Lemma 592**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-action-coherence) — Group-Action Coherence of the Covariance Automorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.action_one` (+1 more) +- [**Definition 593**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:invariant-state) — Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState` +- [**Theorem 594**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:invariant-state-gns-unitary) — GNS Unitary Implementation of an Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitary` +- [**Theorem 595**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-covariant-representation) — Irreducible Covariant Representation of a Pure Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_irreducible_covariant` +- [**Definition 596**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:positive-energy) — Positive Energy · `Physicslib4.AQFT.IsPositiveEnergy` +- [**Lemma 597**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-const) — The Trivial Group has Positive Energy · `Physicslib4.AQFT.isPositiveEnergy_const_refl` +- [**Lemma 598**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:exp-generator-unique) — Uniqueness of the Generator · `Physicslib4.AQFT.generator_eq_of_eq_expUnitary` (+1 more) +- [**Lemma 599**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-conj) — Unitary Conjugation Preserves Positivity · `Physicslib4.Spectral.Unbounded.IsPositive.of_conj` +- [**Lemma 600**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-conj) — Positive Energy is Unitarily Invariant · `Physicslib4.AQFT.IsPositiveEnergy.conj` +- [**Lemma 601**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-strong-continuous) — A Positive-Energy Group is Strongly Continuous · `Physicslib4.AQFT.IsPositiveEnergy.strongContinuous` +- [**Lemma 602**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-exp-bounded) — The Bounded Case · `Physicslib4.AQFT.isPositiveEnergy_exp` +- [**Theorem 603**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:positive-energy-iff-exp) — Positive Energy in Exponential Form · `Physicslib4.AQFT.isPositiveEnergy_iff_exists_exp` +- [**Theorem 604**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:positive-energy-api) — Properties of Positive Energy · `Physicslib4.AQFT.isPositiveEnergy_const_refl` (+5 more) +- [**Definition 605**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:vacuum-state) — Vacuum State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState` +- [**Theorem 606**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:vacuum-invariance-consequences) — Consequences of Invariance for a Vacuum State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState.invariant` (+1 more) +- [**Definition 607**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:future-timelike-translation) — Future-Timelike Translation Subgroup · `Physicslib4.AQFT.HaagKastler.translationSub` (+3 more) +- [**Definition 608**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:vacuum-state-concrete) — Vacuum State with the Concrete Spectrum Condition · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumStateConcrete` +- [**Theorem 609**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:purity-covariance-invariant) — Purity is Covariance-Invariant · `Physicslib4.GNS.isPure_precomp_iff` (+1 more) +- [**Theorem 610**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-quasilocal-action) — GNS covariance along the quasilocal action · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_action` (+2 more)
-§10.6.13 The Separating Vector of a Faithful State (p. 325) — 1 item +§10.6.13 The Separating Vector of a Faithful State (p. 326) — 1 item -- [**Theorem 610**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:separating-faithful) — Separating Vector of a Faithful State · `Physicslib4.GNS.separating_of_faithful` (+1 more) +- [**Theorem 611**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:separating-faithful) — Separating Vector of a Faithful State · `Physicslib4.GNS.separating_of_faithful` (+1 more)
-§10.6.14 The KMS Condition and Thermal Equilibrium (p. 325) — 10 items - -- [**Definition 611**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:one-parameter-aut) — One-Parameter Automorphism Group · `Physicslib4.AQFT.IsOneParameterAut` -- [**Definition 612**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state) — KMS State · `Physicslib4.AQFT.IsKMSState` -- [**Theorem 613**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex) — The KMS State Set is Convex · `Physicslib4.AQFT.IsKMSState.convexCombo` -- [**Lemma 614**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:kms-correlation-one) — Boundary Coincidence for $$a = 1$$ · `Physicslib4.AQFT.IsKMSState.correlationOne` -- [**Definition 615**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:strip-liouville) — Strip-Liouville Principle · `Physicslib4.AQFT.StripLiouville` -- [**Theorem 616**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-periodic-extension) — $$i\beta$$-Periodic Entire Extension (Strip Schwarz Reflection) · `Physicslib4.exists_bounded_entire_extension_of_strip_periodic` -- [**Theorem 617**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-liouville-pos) — Strip-Liouville Holds for $$\beta > 0$$ · `Physicslib4.AQFT.stripLiouville_of_pos` -- [**Theorem 618**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-invariance) — KMS States are Invariant · `Physicslib4.AQFT.IsKMSState.invariant_of_pos` -- [**Theorem 619**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-uniqueness) — Uniqueness on the Strip from Boundary Values · `Physicslib4.eqOn_strip_of_eq_boundary` -- [**Theorem 620**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-correlation-unique) — Uniqueness of the KMS Correlation Function · `Physicslib4.AQFT.IsKMSState.correlation_eqOn` +§10.6.14 The KMS Condition and Thermal Equilibrium (p. 326) — 10 items + +- [**Definition 612**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:one-parameter-aut) — One-Parameter Automorphism Group · `Physicslib4.AQFT.IsOneParameterAut` +- [**Definition 613**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state) — KMS State · `Physicslib4.AQFT.IsKMSState` +- [**Theorem 614**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex) — The KMS State Set is Convex · `Physicslib4.AQFT.IsKMSState.convexCombo` +- [**Lemma 615**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:kms-correlation-one) — Boundary Coincidence for $$a = 1$$ · `Physicslib4.AQFT.IsKMSState.correlationOne` +- [**Definition 616**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:strip-liouville) — Strip-Liouville Principle · `Physicslib4.AQFT.StripLiouville` +- [**Theorem 617**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-periodic-extension) — $$i\beta$$-Periodic Entire Extension (Strip Schwarz Reflection) · `Physicslib4.exists_bounded_entire_extension_of_strip_periodic` +- [**Theorem 618**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-liouville-pos) — Strip-Liouville Holds for $$\beta > 0$$ · `Physicslib4.AQFT.stripLiouville_of_pos` +- [**Theorem 619**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-invariance) — KMS States are Invariant · `Physicslib4.AQFT.IsKMSState.invariant_of_pos` +- [**Theorem 620**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-uniqueness) — Uniqueness on the Strip from Boundary Values · `Physicslib4.eqOn_strip_of_eq_boundary` +- [**Theorem 621**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-correlation-unique) — Uniqueness of the KMS Correlation Function · `Physicslib4.AQFT.IsKMSState.correlation_eqOn`
-§10.6.15 KMS States for the Covariance Flow (p. 326) — 5 items +§10.6.15 KMS States for the Covariance Flow (p. 328) — 5 items -- [**Definition 621**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:flow-aut-covariance) — Covariance-Flow Automorphism Family · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.flowAut` -- [**Lemma 622**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:one-parameter-aut-flow-covariance) — A Lorentz One-Parameter Subgroup Induces a One-Parameter Group · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isOneParameterAut_flowAut` -- [**Definition 623**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state-for-flow-covariance) — KMS State for the Covariance Flow · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsKMSStateForFlow` -- [**Theorem 624**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex-for-flow-covariance) — Convexity of the Covariance-Flow KMS States · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsKMSStateForFlow.convexCombo` -- [**Definition 625**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:ground-state-for-flow-covariance) — Ground State for a Covariance Flow · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsGroundStateForFlow` (+2 more) +- [**Definition 622**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:flow-aut-covariance) — Covariance-Flow Automorphism Family · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.flowAut` +- [**Lemma 623**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:one-parameter-aut-flow-covariance) — A Lorentz One-Parameter Subgroup Induces a One-Parameter Group · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isOneParameterAut_flowAut` +- [**Definition 624**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state-for-flow-covariance) — KMS State for the Covariance Flow · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsKMSStateForFlow` +- [**Theorem 625**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex-for-flow-covariance) — Convexity of the Covariance-Flow KMS States · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsKMSStateForFlow.convexCombo` +- [**Definition 626**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:ground-state-for-flow-covariance) — Ground State for a Covariance Flow · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsGroundStateForFlow` (+2 more)
diff --git a/home_page/sec10-7-curved.md b/home_page/sec10-7-curved.md index bdbe8ac..54684ac 100644 --- a/home_page/sec10-7-curved.md +++ b/home_page/sec10-7-curved.md @@ -1,5 +1,5 @@ --- -title: "§10.7 Haag–Kastler Axioms in curved spacetime (pp. 327–337)" +title: "§10.7 Haag–Kastler Axioms in curved spacetime (pp. 328–338)" usemathjax: true --- @@ -7,17 +7,17 @@ usemathjax: true ← [§10.6 Minkowski](sec10-6-haag-kastler.html) · [§10.8 Covariance](sec10-8-general-covariance.html) → -# §10.7 Haag–Kastler Axioms in curved spacetime (pp. 327–337) +# §10.7 Haag–Kastler Axioms in curved spacetime (pp. 328–338) -49 items, Definitions 626–674. The axioms are restated for a Lorentzian spacetime: Local Algebras (Definition 626), Isotony (Definition 627, again supplying the family as chosen data with identity and composition laws), Local Commutativity (Definition 628, which now **consumes** the Axiom 2 family rather than choosing its own witnesses), local observables (Definition 629), Local Completeness (Definition 630), and Isometric Covariance (Definition 631, whose coherence condition, as in Minkowski, is stated for the Axiom 2 isotony family), bundled into a `HaagKastlerNet` in curved spacetime (Definition 632). The change to Axioms 2 and 3 has a visible consequence downstream: statements about nested regions have to factor a three-fold inclusion $$\mathbf{B}_1 \subseteq \mathbf{B}_2 \subseteq \mathbf{B}$$ inside a common containing algebra, and that factorisation is now the composition law of Axiom 2, so **the coherence hypotheses that earlier versions carried at each such site are gone**—curved isotony (Theorem 637) and the curved von Neumann net (Definition 639) are unconditional. An explicit hypothesis remains only where the abstract interface genuinely cannot supply it, namely basis-set preservation in Geometric Covariance (Theorem 643), discharged for nets arising from a concrete geometric spacetime; the coherence of the stabiliser action with the isotony embeddings is no longer a hypothesis there, being condition (3) of Axiom 5 transported along $$g \cdot \mathbf{B} = \mathbf{B}$$. +49 items, Definitions 627–675. The axioms are restated for a Lorentzian spacetime: Local Algebras (Definition 627), Isotony (Definition 628, again supplying the family as chosen data with identity and composition laws), Local Commutativity (Definition 629, which now **consumes** the Axiom 2 family rather than choosing its own witnesses), local observables (Definition 630), Local Completeness (Definition 631), and Isometric Covariance (Definition 632, whose coherence condition, as in Minkowski, is stated for the Axiom 2 isotony family), bundled into a `HaagKastlerNet` in curved spacetime (Definition 633). The change to Axioms 2 and 3 has a visible consequence downstream: statements about nested regions have to factor a three-fold inclusion $$\mathbf{B}_1 \subseteq \mathbf{B}_2 \subseteq \mathbf{B}$$ inside a common containing algebra, and that factorisation is now the composition law of Axiom 2, so **the coherence hypotheses that earlier versions carried at each such site are gone**—curved isotony (Theorem 638) and the curved von Neumann net (Definition 640) are unconditional. An explicit hypothesis remains only where the abstract interface genuinely cannot supply it, namely basis-set preservation in Geometric Covariance (Theorem 644), discharged for nets arising from a concrete geometric spacetime; the coherence of the stabiliser action with the isotony embeddings is no longer a hypothesis there, being condition (3) of Axiom 5 transported along $$g \cdot \mathbf{B} = \mathbf{B}$$. -- *§10.7.1 Einstein Causality in Curved Spacetime (p. 329, item 633).* The operator form of local commutativity, expressed in a representation of a common containing local algebra rather than of a global quasilocal algebra (Theorem 633). -- *§10.7.2 Local von Neumann Algebras in Curved Spacetime (p. 329, items 634–645).* The Minkowski development of §10.6.5 mirrored relative to a containing region: the local von Neumann algebra of a subregion and its bundled registration (Definitions 634–635), Microcausality (Theorem 636), unconditional Isotony (Theorem 637), the bundled form (Theorem 638), the net as an order-preserving map on the poset of subregions (Definition 639), Statistical Independence in set-level and bundled forms (Theorems 640–641), additive-free locality (Theorem 642), and Geometric Covariance via the stabiliser GNS representation (Theorem 643) with orbit-invariance of factoriality (Theorem 644) and the upgrade to a \*-algebra isomorphism (Theorem 645). -- *§10.7.3 Relative Commutants of Nested Local Algebras in Curved Spacetime (p. 332, items 646–654).* The curved counterpart of §10.6.6: the relative commutant of a nested pair inside a containing region (Definition 646), its containment in the larger algebra, commutation with the smaller, and containment of the centre (Theorems 647–649), irreducible inclusions (Definition 650) and their consequences (Theorems 651–652), plus the abelian/centre facts specialised to curved local algebras (Theorems 653–654). -- *§10.7.4 Purity of States on Local Algebras in Curved Spacetime (p. 333, items 655–659).* Each local algebra is a unital C\*-algebra with its own state space, so the abstract purity characterisations are registered per region: Pure $$\iff$$ Extreme Point (Theorem 655), Pure $$\iff$$ Irreducible GNS (Theorem 656), the GNS representation of a pure state generates a factor and indeed all of $$\mathcal{B}(H)$$ (Theorem 657), the Irreducible Dichotomy for a curved local algebra (Theorem 658), and GNS covariance for curved local algebras (Theorem 659). -- *§10.7.5 Covariant States in Curved Spacetime (p. 334, items 660–661).* Covariant families of local states in curved spacetime (Definition 660) and their composition law (Lemma 661). -- *§10.7.6 The Stabilizer GNS Unitary in Curved Spacetime (p. 334, items 662–668).* Since no quasilocal algebra exists on a generic Lorentzian spacetime, the covariance action restricts to the stabiliser subgroup $$\mathrm{Stab}(\mathbf{B})$$ of a region, giving automorphisms of the single local algebra $$\mathfrak{U}(\mathbf{B})$$ (Definition 662) that form a genuine group action (Lemma 663). A stabiliser-invariant state carries a unitary GNS representation of $$\mathrm{Stab}(\mathbf{B})$$ (Theorem 664), strongly continuous when the matrix coefficients are continuous (Theorem 665); a state both stabiliser-invariant and pure yields an irreducible covariant representation (Theorem 666); purity is invariant under the stabiliser action (Theorem 667), and GNS data transports along it (Theorem 668). -- *§10.7.7 KMS States for a Killing Flow (p. 336, items 669–674).* Killing flows are identified as one-parameter subgroups of the stabiliser of a region, inducing a one-parameter automorphism group on $$\mathfrak{U}(\mathbf{B})$$ (Definition 669, Lemma 670). KMS states for a Killing flow (Definition 671) are the precise algebraic sense in which the Hartle–Hawking and Gibbons–Hawking states are thermal. Such a state at positive inverse temperature automatically carries a strongly continuous one-parameter unitary group on its GNS Hilbert space implementing the flow, yielding the curved-spacetime thermal representation—the analogue of the Minkowski vacuum representation (Theorem 672); these states form a convex set (Theorem 673), and the corresponding ground state is recorded alongside them (Definition 674). +- *§10.7.1 Einstein Causality in Curved Spacetime (p. 331, item 634).* The operator form of local commutativity, expressed in a representation of a common containing local algebra rather than of a global quasilocal algebra (Theorem 634). +- *§10.7.2 Local von Neumann Algebras in Curved Spacetime (p. 331, items 635–646).* The Minkowski development of §10.6.5 mirrored relative to a containing region: the local von Neumann algebra of a subregion and its bundled registration (Definitions 635–636), Microcausality (Theorem 637), unconditional Isotony (Theorem 638), the bundled form (Theorem 639), the net as an order-preserving map on the poset of subregions (Definition 640), Statistical Independence in set-level and bundled forms (Theorems 641–642), additive-free locality (Theorem 643), and Geometric Covariance via the stabiliser GNS representation (Theorem 644) with orbit-invariance of factoriality (Theorem 645) and the upgrade to a \*-algebra isomorphism (Theorem 646). +- *§10.7.3 Relative Commutants of Nested Local Algebras in Curved Spacetime (p. 333, items 647–655).* The curved counterpart of §10.6.6: the relative commutant of a nested pair inside a containing region (Definition 647), its containment in the larger algebra, commutation with the smaller, and containment of the centre (Theorems 648–650), irreducible inclusions (Definition 651) and their consequences (Theorems 652–653), plus the abelian/centre facts specialised to curved local algebras (Theorems 654–655). +- *§10.7.4 Purity of States on Local Algebras in Curved Spacetime (p. 334, items 656–660).* Each local algebra is a unital C\*-algebra with its own state space, so the abstract purity characterisations are registered per region: Pure $$\iff$$ Extreme Point (Theorem 656), Pure $$\iff$$ Irreducible GNS (Theorem 657), the GNS representation of a pure state generates a factor and indeed all of $$\mathcal{B}(H)$$ (Theorem 658), the Irreducible Dichotomy for a curved local algebra (Theorem 659), and GNS covariance for curved local algebras (Theorem 660). +- *§10.7.5 Covariant States in Curved Spacetime (p. 335, items 661–662).* Covariant families of local states in curved spacetime (Definition 661) and their composition law (Lemma 662). +- *§10.7.6 The Stabilizer GNS Unitary in Curved Spacetime (p. 335, items 663–669).* Since no quasilocal algebra exists on a generic Lorentzian spacetime, the covariance action restricts to the stabiliser subgroup $$\mathrm{Stab}(\mathbf{B})$$ of a region, giving automorphisms of the single local algebra $$\mathfrak{U}(\mathbf{B})$$ (Definition 663) that form a genuine group action (Lemma 664). A stabiliser-invariant state carries a unitary GNS representation of $$\mathrm{Stab}(\mathbf{B})$$ (Theorem 665), strongly continuous when the matrix coefficients are continuous (Theorem 666); a state both stabiliser-invariant and pure yields an irreducible covariant representation (Theorem 667); purity is invariant under the stabiliser action (Theorem 668), and GNS data transports along it (Theorem 669). +- *§10.7.7 KMS States for a Killing Flow (p. 337, items 670–675).* Killing flows are identified as one-parameter subgroups of the stabiliser of a region, inducing a one-parameter automorphism group on $$\mathfrak{U}(\mathbf{B})$$ (Definition 670, Lemma 671). KMS states for a Killing flow (Definition 672) are the precise algebraic sense in which the Hartle–Hawking and Gibbons–Hawking states are thermal. Such a state at positive inverse temperature automatically carries a strongly continuous one-parameter unitary group on its GNS Hilbert space implementing the flow, yielding the curved-spacetime thermal representation—the analogue of the Minkowski vacuum representation (Theorem 673); these states form a convex set (Theorem 674), and the corresponding ground state is recorded alongside them (Definition 675). **Where the Lean lives:** `Physicslib4/AQFT/HaagKastlerCurved/` (`LocalVonNeumann`, `StabilizerAction`, `StabilizerKMS`, `Purity`, `GeometricCovariance`, …) @@ -26,78 +26,78 @@ usemathjax: true Every entry links to its node in the web blueprint and names the principal Lean declaration. -### §10.7 The axioms (p. 327) - -- [**Definition 626**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-algebras-in-curved-spacetime) — Axiom 1: Local Algebras · `Physicslib4.AQFT.HaagKastlerCurved.LocalNet` -- [**Definition 627**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isotony-in-curved-spacetime) — Axiom 2: Isotony · `Physicslib4.AQFT.HaagKastlerCurved.Isotony` -- [**Definition 628**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-commutativity-in-curved-spacetime) — Axiom 3: Local Commutativity · `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` -- [**Definition 629**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-observable) — Local Observable · `Physicslib4.AQFT.HaagKastlerCurved.IsLocalObservable` -- [**Definition 630**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-completeness-in-curved-spacetime) — Axiom 4: Local Completeness · `Physicslib4.AQFT.HaagKastlerCurved.LocalAlgebra` -- [**Definition 631**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isometric-covariance-in-curved-spacetime) — Axiom 5: Isometric Covariance · `Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance` *(implemented via the explicitly orientation-preserving subgroup — see [Formalisation status](./#formalisation-status))* -- [**Definition 632**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:haag-kastler-net-in-curved-spacetime) — Haag–Kastler Net in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` - -### §10.7.1 Einstein Causality in Curved Spacetime (p. 329) - -- [**Theorem 633**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:einstein-causality-in-curved-spacetime) — Einstein Causality in a Representation (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.einstein_causality` (+1 more) - -### §10.7.2 Local von Neumann Algebras in Curved Spacetime (p. 329) - -- [**Definition 634**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-in-curved-spacetime) — Local von Neumann Algebra in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localOperators` (+1 more) -- [**Definition 635**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-algebra-in-curved-spacetime) — $$R(\mathbf{B}')$$ as a von Neumann Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra` -- [**Theorem 636**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-microcausality-in-curved-spacetime) — Microcausality at the von Neumann Level (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_subset_centralizer` -- [**Theorem 637**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-isotony-in-curved-spacetime) — Isotony of the von Neumann Net (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_mono` -- [**Theorem 638**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-bundled-order-in-curved-spacetime) — Bundled von Neumann Microcausality and Isotony (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant` (+1 more) -- [**Definition 639**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-net-in-curved-spacetime) — The Net of von Neumann Algebras in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.vonNeumannNet` -- [**Theorem 640**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-in-curved-spacetime) — Statistical Independence (Schlieder Property) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_separating` (+1 more) -- [**Theorem 641**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-bundled-in-curved-spacetime) — Statistical Independence, bundled (curved spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_separating` -- [**Theorem 642**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:additive-free-locality-in-curved-spacetime) — Additive-Free Locality via the Spacelike Complement (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant_of_subset_spacelikeComplement_geometric` -- [**Theorem 643**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-geometric-covariance-in-curved-spacetime) — Geometric Covariance of the von Neumann Net (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumann` (+2 more) -- [**Theorem 644**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-factor-orbit-in-curved-spacetime) — Orbit-Invariance of Factoriality (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_isFactor_smul` -- [**Theorem 645**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-covariance-iso-in-curved-spacetime) — Geometric Covariance as a von Neumann Algebra Isomorphism (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannEquiv` - -### §10.7.3 Relative Commutants of Nested Local Algebras in Curved Spacetime (p. 332) - -- [**Definition 646**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:relative-commutant-in-curved-spacetime) — Relative Commutant of a Nested Pair (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant` -- [**Theorem 647**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-le-right-in-curved-spacetime) — Relative Commutant Lies in the Larger Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_le_right` -- [**Theorem 648**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-coe-commutant-in-curved-spacetime) — Relative Commutant Commutes with the Smaller Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_coe_subset_commutant` -- [**Theorem 649**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-center-in-curved-spacetime) — Relative Commutant Contains the Centre (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.center_le_relativeCommutant` -- [**Definition 650**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-inclusion-in-curved-spacetime) — Irreducible Inclusion (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsIrreducibleInclusion` -- [**Theorem 651**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-inclusion-factor-in-curved-spacetime) — An Irreducible Inclusion has Factor Ambient (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_of_isIrreducibleInclusion` -- [**Theorem 652**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:self-inclusion-factor-in-curved-spacetime) — Self-Inclusion is Irreducible iff Factor (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor` -- [**Theorem 653**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:abelian-local-von-neumann-in-curved-spacetime) — Abelian Local von Neumann Algebras (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_isAbelian` (+1 more) -- [**Theorem 654**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:center-duality-local-von-neumann-in-curved-spacetime) — Centre Duality for Local von Neumann Algebras (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_eq_commutant` (+1 more) - -### §10.7.4 Purity of States on Local Algebras in Curved Spacetime (p. 333) - -- [**Theorem 655**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme-in-curved-spacetime) — Pure $$\iff$$ Extreme Point on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.pure_iff_extreme` -- [**Theorem 656**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible-in-curved-spacetime) — Pure $$\iff$$ Irreducible GNS on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_pure_iff_irreducible` -- [**Theorem 657**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-factor-generates-in-curved-spacetime) — Pure GNS on a Local Algebra is a Factor Generating $$\mathcal{B}(H)$$ · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_factor_of_isPure` (+1 more) -- [**Theorem 658**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-dichotomy-in-curved-spacetime) — The Irreducible Dichotomy for a Curved Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.areDisjoint_or_unitaryEquiv_of_isIrreducible` -- [**Theorem 659**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-local-in-curved-spacetime) — GNS covariance for curved local algebras · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_covEquiv` (+2 more) - -### §10.7.5 Covariant States in Curved Spacetime (p. 334) - -- [**Definition 660**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-state-family-in-curved-spacetime) — Covariant Family of Local States in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsCovariantFamily` -- [**Lemma 661**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-state-family-compose-in-curved-spacetime) — Composition of Covariance in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsCovariantFamily.comp` - -### §10.7.6 The Stabiliser GNS Unitary in Curved Spacetime (p. 334) - -- [**Definition 662**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:stabilizer-action-in-curved-spacetime) — Stabiliser Action on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut` -- [**Lemma 663**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:stabilizer-action-laws-in-curved-spacetime) — The Stabiliser Action is a Group Action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut_one` (+1 more) -- [**Theorem 664**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-unitary-stabilizer-in-curved-spacetime) — GNS Unitary Representation of the Stabiliser · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer` -- [**Theorem 665**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-unitary-stabilizer-strongly-continuous-in-curved-spacetime) — Strongly Continuous Stabiliser GNS Unitary · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer_strongContinuous` -- [**Theorem 666**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-covariant-representation-in-curved-spacetime) — Irreducible Covariant Representation of a Pure Invariant State (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_irreducible_covariant_stabilizer` -- [**Theorem 667**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:purity-covariance-invariant-in-curved-spacetime) — Purity is Invariant under the Stabiliser Action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isPure_precomp_stabAut_iff` -- [**Theorem 668**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-stabilizer-action) — GNS covariance along the stabiliser action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_stabAut` (+2 more) - -### §10.7.7 KMS States for a Killing Flow (p. 336) - -- [**Definition 669**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:flow-aut-in-curved-spacetime) — Killing-Flow Automorphism Family · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.flowAut` -- [**Lemma 670**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:one-parameter-aut-flow-in-curved-spacetime) — A Killing Flow Induces a One-Parameter Group · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isOneParameterAut_flowAut` -- [**Definition 671**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state-for-flow-in-curved-spacetime) — KMS State for a Killing Flow · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow` -- [**Theorem 672**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-thermal-representation-in-curved-spacetime) — The Killing-Flow KMS Thermal Representation · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow.exists_gns_unitary_strongContinuous` -- [**Theorem 673**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex-for-flow-in-curved-spacetime) — Convexity of the Killing-Flow KMS States · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow.convexCombo` -- [**Definition 674**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:ground-state-for-flow-in-curved-spacetime) — Ground State for a Killing Flow · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsGroundStateForFlow` (+2 more) +### §10.7 The axioms (p. 328) + +- [**Definition 627**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-algebras-in-curved-spacetime) — Axiom 1: Local Algebras · `Physicslib4.AQFT.HaagKastlerCurved.LocalNet` +- [**Definition 628**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isotony-in-curved-spacetime) — Axiom 2: Isotony · `Physicslib4.AQFT.HaagKastlerCurved.Isotony` +- [**Definition 629**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-commutativity-in-curved-spacetime) — Axiom 3: Local Commutativity · `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` +- [**Definition 630**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-observable) — Local Observable · `Physicslib4.AQFT.HaagKastlerCurved.IsLocalObservable` +- [**Definition 631**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-completeness-in-curved-spacetime) — Axiom 4: Local Completeness · `Physicslib4.AQFT.HaagKastlerCurved.LocalAlgebra` +- [**Definition 632**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isometric-covariance-in-curved-spacetime) — Axiom 5: Isometric Covariance · `Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance` *(implemented via the explicitly orientation-preserving subgroup — see [Formalisation status](./#formalisation-status))* +- [**Definition 633**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:haag-kastler-net-in-curved-spacetime) — Haag–Kastler Net in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` + +### §10.7.1 Einstein Causality in Curved Spacetime (p. 331) + +- [**Theorem 634**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:einstein-causality-in-curved-spacetime) — Einstein Causality in a Representation (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.einstein_causality` (+1 more) + +### §10.7.2 Local von Neumann Algebras in Curved Spacetime (p. 331) + +- [**Definition 635**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-in-curved-spacetime) — Local von Neumann Algebra in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localOperators` (+1 more) +- [**Definition 636**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-algebra-in-curved-spacetime) — $$R(\mathbf{B}')$$ as a von Neumann Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra` +- [**Theorem 637**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-microcausality-in-curved-spacetime) — Microcausality at the von Neumann Level (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_subset_centralizer` +- [**Theorem 638**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-isotony-in-curved-spacetime) — Isotony of the von Neumann Net (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_mono` +- [**Theorem 639**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-bundled-order-in-curved-spacetime) — Bundled von Neumann Microcausality and Isotony (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant` (+1 more) +- [**Definition 640**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-net-in-curved-spacetime) — The Net of von Neumann Algebras in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.vonNeumannNet` +- [**Theorem 641**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-in-curved-spacetime) — Statistical Independence (Schlieder Property) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_separating` (+1 more) +- [**Theorem 642**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-bundled-in-curved-spacetime) — Statistical Independence, bundled (curved spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_separating` +- [**Theorem 643**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:additive-free-locality-in-curved-spacetime) — Additive-Free Locality via the Spacelike Complement (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant_of_subset_spacelikeComplement_geometric` +- [**Theorem 644**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-geometric-covariance-in-curved-spacetime) — Geometric Covariance of the von Neumann Net (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumann` (+2 more) +- [**Theorem 645**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-factor-orbit-in-curved-spacetime) — Orbit-Invariance of Factoriality (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_isFactor_smul` +- [**Theorem 646**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-covariance-iso-in-curved-spacetime) — Geometric Covariance as a von Neumann Algebra Isomorphism (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannEquiv` + +### §10.7.3 Relative Commutants of Nested Local Algebras in Curved Spacetime (p. 333) + +- [**Definition 647**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:relative-commutant-in-curved-spacetime) — Relative Commutant of a Nested Pair (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant` +- [**Theorem 648**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-le-right-in-curved-spacetime) — Relative Commutant Lies in the Larger Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_le_right` +- [**Theorem 649**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-coe-commutant-in-curved-spacetime) — Relative Commutant Commutes with the Smaller Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_coe_subset_commutant` +- [**Theorem 650**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-center-in-curved-spacetime) — Relative Commutant Contains the Centre (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.center_le_relativeCommutant` +- [**Definition 651**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-inclusion-in-curved-spacetime) — Irreducible Inclusion (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsIrreducibleInclusion` +- [**Theorem 652**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-inclusion-factor-in-curved-spacetime) — An Irreducible Inclusion has Factor Ambient (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_of_isIrreducibleInclusion` +- [**Theorem 653**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:self-inclusion-factor-in-curved-spacetime) — Self-Inclusion is Irreducible iff Factor (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor` +- [**Theorem 654**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:abelian-local-von-neumann-in-curved-spacetime) — Abelian Local von Neumann Algebras (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_isAbelian` (+1 more) +- [**Theorem 655**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:center-duality-local-von-neumann-in-curved-spacetime) — Centre Duality for Local von Neumann Algebras (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_eq_commutant` (+1 more) + +### §10.7.4 Purity of States on Local Algebras in Curved Spacetime (p. 334) + +- [**Theorem 656**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme-in-curved-spacetime) — Pure $$\iff$$ Extreme Point on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.pure_iff_extreme` +- [**Theorem 657**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible-in-curved-spacetime) — Pure $$\iff$$ Irreducible GNS on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_pure_iff_irreducible` +- [**Theorem 658**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-factor-generates-in-curved-spacetime) — Pure GNS on a Local Algebra is a Factor Generating $$\mathcal{B}(H)$$ · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_factor_of_isPure` (+1 more) +- [**Theorem 659**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-dichotomy-in-curved-spacetime) — The Irreducible Dichotomy for a Curved Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.areDisjoint_or_unitaryEquiv_of_isIrreducible` +- [**Theorem 660**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-local-in-curved-spacetime) — GNS covariance for curved local algebras · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_covEquiv` (+2 more) + +### §10.7.5 Covariant States in Curved Spacetime (p. 335) + +- [**Definition 661**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-state-family-in-curved-spacetime) — Covariant Family of Local States in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsCovariantFamily` +- [**Lemma 662**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-state-family-compose-in-curved-spacetime) — Composition of Covariance in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsCovariantFamily.comp` + +### §10.7.6 The Stabiliser GNS Unitary in Curved Spacetime (p. 336) + +- [**Definition 663**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:stabilizer-action-in-curved-spacetime) — Stabiliser Action on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut` +- [**Lemma 664**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:stabilizer-action-laws-in-curved-spacetime) — The Stabiliser Action is a Group Action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut_one` (+1 more) +- [**Theorem 665**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-unitary-stabilizer-in-curved-spacetime) — GNS Unitary Representation of the Stabiliser · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer` +- [**Theorem 666**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-unitary-stabilizer-strongly-continuous-in-curved-spacetime) — Strongly Continuous Stabiliser GNS Unitary · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer_strongContinuous` +- [**Theorem 667**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-covariant-representation-in-curved-spacetime) — Irreducible Covariant Representation of a Pure Invariant State (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_irreducible_covariant_stabilizer` +- [**Theorem 668**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:purity-covariance-invariant-in-curved-spacetime) — Purity is Invariant under the Stabiliser Action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isPure_precomp_stabAut_iff` +- [**Theorem 669**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-stabilizer-action) — GNS covariance along the stabiliser action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_stabAut` (+2 more) + +### §10.7.7 KMS States for a Killing Flow (p. 337) + +- [**Definition 670**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:flow-aut-in-curved-spacetime) — Killing-Flow Automorphism Family · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.flowAut` +- [**Lemma 671**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:one-parameter-aut-flow-in-curved-spacetime) — A Killing Flow Induces a One-Parameter Group · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isOneParameterAut_flowAut` +- [**Definition 672**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state-for-flow-in-curved-spacetime) — KMS State for a Killing Flow · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow` +- [**Theorem 673**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-thermal-representation-in-curved-spacetime) — The Killing-Flow KMS Thermal Representation · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow.exists_gns_unitary_strongContinuous` +- [**Theorem 674**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex-for-flow-in-curved-spacetime) — Convexity of the Killing-Flow KMS States · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow.convexCombo` +- [**Definition 675**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:ground-state-for-flow-in-curved-spacetime) — Ground State for a Killing Flow · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsGroundStateForFlow` (+2 more) [Home](./) · [§10.1 GNS](sec10-1-gns.html) · [§10.2 Spectral](sec10-2-spectral.html) · [§10.3 Unbounded](sec10-3-unbounded.html) · [§10.4 Stone](sec10-4-stone.html) · [§10.5 Spacetime](sec10-5-spacetime.html) · [§10.6 Minkowski](sec10-6-haag-kastler.html) · **§10.7 Curved** · [§10.8 Covariance](sec10-8-general-covariance.html) · [Axioms](axioms.html) · [Guide](guide.html) diff --git a/home_page/sec10-8-general-covariance.md b/home_page/sec10-8-general-covariance.md index f0c19a4..b0ed868 100644 --- a/home_page/sec10-8-general-covariance.md +++ b/home_page/sec10-8-general-covariance.md @@ -1,5 +1,5 @@ --- -title: "§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 337–339)" +title: "§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 338–339)" usemathjax: true --- @@ -7,9 +7,9 @@ usemathjax: true ← [§10.7 Curved](sec10-7-curved.html) -# §10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 337–339) +# §10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 338–339) -Two definitions, and the newest structural addition to the blueprint. The gauge group of general relativity is the full diffeomorphism group of $$M$$, so the physical content of a spacetime is its diffeomorphism-equivalence class and not the pair $$(M, g)$$; the hole argument shows that treating a relabelling as physical would destroy determinism, and Leibniz equivalence resolves it by declaring diffeomorphic models to represent the same physical situation. Accordingly, an equivalence of Haag–Kastler nets is defined (Definition 675) as a chosen family of unital \*-isomorphisms $$\Theta_\mathbf{B} : \mathfrak{U}_1(\mathbf{B}) \to \mathfrak{U}_2(e(\mathbf{B}))$$ along a basis-set-preserving bijection $$e$$ of carriers, natural with respect to the isotony embeddings. The carriers are related by data rather than by a type equality, deliberately: an equality of carrier types cannot be transported along and would force every comparison through a cast. A net theory is then a section assigning a net to every Lorentzian spacetime, and it is generally covariant when the nets over $$L$$ and over its pullback $$\psi^* L$$ are equivalent along $$\psi$$ (Definition 676). In Lean the nets are indexed over the identity-component bridge `toAbstractIdentityComponent`, so Axiom 5 of each net asks only for isometries connected to the identity, as in the blueprint. Four points are worth carrying away: this is a **postulate, not a theorem**—nothing forces the two nets to be isomorphic, and it says nothing about backgrounds that are not diffeomorphism-related; the morphism is specified geometrically rather than causally, because a purely causal morphism would admit the dilations (Theorems 422–423) and thereby demand a scale covariance that is false for a massive theory; the relabelling must be $$\psi$$ and not the identity, since a basis set of one metric is in general not a basis set of the other; and general covariance is a property of the section $$L \mapsto \mathfrak{U}_L$$, not a sixth field of the net structure, so no restriction to diffeomorphisms connected to the identity is needed here. +Two definitions, and the newest structural addition to the blueprint. The gauge group of general relativity is the full diffeomorphism group of $$M$$, so the physical content of a spacetime is its diffeomorphism-equivalence class and not the pair $$(M, g)$$; the hole argument shows that treating a relabelling as physical would destroy determinism, and Leibniz equivalence resolves it by declaring diffeomorphic models to represent the same physical situation. Accordingly, an equivalence of Haag–Kastler nets is defined (Definition 676) as a chosen family of unital \*-isomorphisms $$\Theta_\mathbf{B} : \mathfrak{U}_1(\mathbf{B}) \to \mathfrak{U}_2(e(\mathbf{B}))$$ along a basis-set-preserving bijection $$e$$ of carriers, natural with respect to the isotony embeddings. The carriers are related by data rather than by a type equality, deliberately: an equality of carrier types cannot be transported along and would force every comparison through a cast. A net theory is then a section assigning a net to every Lorentzian spacetime, and it is generally covariant when the nets over $$L$$ and over its pullback $$\psi^* L$$ are equivalent along $$\psi$$ (Definition 677). In Lean the nets are indexed over the identity-component bridge `toAbstractIdentityComponent`, so Axiom 5 of each net asks only for isometries connected to the identity, as in the blueprint. Four points are worth carrying away: this is a **postulate, not a theorem**—nothing forces the two nets to be isomorphic, and it says nothing about backgrounds that are not diffeomorphism-related; the morphism is specified geometrically rather than causally, because a purely causal morphism would admit the dilations (Theorems 422–423) and thereby demand a scale covariance that is false for a massive theory; the relabelling must be $$\psi$$ and not the identity, since a basis set of one metric is in general not a basis set of the other; and general covariance is a property of the section $$L \mapsto \mathfrak{U}_L$$, not a sixth field of the net structure, so no restriction to diffeomorphisms connected to the identity is needed here. **Where the Lean lives:** `Physicslib4/AQFT/HaagKastlerCurved/GeneralCovariance.lean` @@ -19,10 +19,10 @@ Two definitions, and the newest structural addition to the blueprint. The gauge Every entry links to its node in the web blueprint and names the principal Lean declaration. -### §10.8 General Covariance: Nets on Pullback-Related Metrics (p. 337) +### §10.8 General Covariance: Nets on Pullback-Related Metrics (p. 338) -- [**Definition 675**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:net-equivalence-in-curved-spacetime) — Equivalence of Haag–Kastler Nets · `Physicslib4.AQFT.HaagKastlerCurved.NetEquivalence` -- [**Definition 676**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:general-covariance-in-curved-spacetime) — General Covariance · `Physicslib4.AQFT.HaagKastlerCurved.NetTheory` (+4 more) +- [**Definition 676**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:net-equivalence-in-curved-spacetime) — Equivalence of Haag–Kastler Nets · `Physicslib4.AQFT.HaagKastlerCurved.NetEquivalence` +- [**Definition 677**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:general-covariance-in-curved-spacetime) — General Covariance · `Physicslib4.AQFT.HaagKastlerCurved.NetTheory` (+4 more) [Home](./) · [§10.1 GNS](sec10-1-gns.html) · [§10.2 Spectral](sec10-2-spectral.html) · [§10.3 Unbounded](sec10-3-unbounded.html) · [§10.4 Stone](sec10-4-stone.html) · [§10.5 Spacetime](sec10-5-spacetime.html) · [§10.6 Minkowski](sec10-6-haag-kastler.html) · [§10.7 Curved](sec10-7-curved.html) · **§10.8 Covariance** · [Axioms](axioms.html) · [Guide](guide.html) diff --git a/notes/agents/audits/audit-axiom3-common-diamond-2026-10-08.md b/notes/agents/audits/audit-axiom3-common-diamond-2026-10-08.md new file mode 100644 index 0000000..76db980 --- /dev/null +++ b/notes/agents/audits/audit-axiom3-common-diamond-2026-10-08.md @@ -0,0 +1,204 @@ +# Audit: Axiom 3 restated through a common containing diamond + +Branch `numina/axiom3-common-diamond` vs base `15cdea7` (tip of `numina/aqft-in-lean`). +Scope: the change described in `notes/agents/plan-axiom3-common-diamond-2026-10-08.md` +(D1-D5, approved 2026-10-08). Read-only audit; nothing in the repo was changed. + +## Verdict + +**Pass.** The Lean restatement of `LocalCommutativity` matches the approved plan and the +blueprint, binder-for-binder parallel to the curved axiom. The protected-definition change +(the one thing CLAUDE.md flags as sensitive) is exactly D1(a) as specified. The two theorem +signatures the task asked me to confirm are unchanged, and the new iff's right side is a +byte-for-byte match of the old definition body. No `sorry`, no non-standard axioms (checked +with the tool), no new `nolint`, no `Restriction:` markers added or needed. One real gap: +**the verification receipt is stale** — it does not cover the current tree, so "lake build" +and "lake lint" have not been confirmed to pass since Stage 3's proofs were written. That +should be closed with a fresh `lean_guard.py verify` before this merges into +`numina/aqft-in-lean`. + +## 1. Changes to defs/structures/classes/instances/fields under `Physicslib4/` + +`git diff 15cdea7 -- Physicslib4` touches four files, one declaration body, and otherwise only +docstrings: + +- **`Physicslib4.AQFT.HaagKastler.LocalCommutativity`** (def, protected) — + `LocalCommutativity.lean`. + - Old: `∃ Q : QuasilocalAlgebra U i, ∀ ⦃B₁ B₂⦄ hB₁ hB₂, IsCompletelySpacelike B₁ B₂ → ∀ a b, Commute (Q.ι hB₁ a) (Q.ι hB₂ b)` — commutation in *some* quasilocal algebra. + - New: `∀ ⦃B₁ B₂ B⦄ hB₁ hB₂ hB, IsCompletelySpacelike B₁ B₂ → (h₁ : B₁ ⊆ B) → (h₂ : B₂ ⊆ B) → ∀ a b, Commute (i.map hB₁ hB h₁ a) (i.map hB₂ hB h₂ b)` — commutation of the isotony images in every common containing basis set `B`. + - Matches plan D1(a) exactly, and is binder-for-binder parallel to + `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` (verified by reading both + definitions side by side; only the Minkowski-specific `IsAlexandrovBasisSet` / + `Spacetime.IsCompletelySpacelike StandardMinkowskiSpacetime …` replace the curved `M.IsBasisSet` + / `M.IsCompletelySpacelike`). Approved. +- **`HaagKastlerNet` structure** (`Net.lean`) — the `localCommutativity` field's *type* + (`LocalCommutativity U isotony`) is untouched; only its field docstring is reworded to describe + the new statement. No structure-shape change. +- **`QuasilocalAlgebra` structure** (`QuasilocalAlgebra.lean`) — unchanged except for one comment + on the `carrier` field explaining why a free universe isn't used, updated to name the new lemma + it would otherwise have complicated. No field added, removed or retyped. +- No other def/structure/class/instance/abbrev changed under `Physicslib4/`. + +Everything here is covered by the approved plan; nothing looks unapproved. + +Note on the guard output: `lean_guard.py check --base 15cdea7` also lists `.claude/CLAUDE.md` +and `.claude/lean-policy.json` as "changes [to] a protected file". That is an artefact, not a +real finding: `.claude/` is untracked in this checkout (`git ls-files .claude/` is empty; it +shows only as `?? .claude/` in `git status`), so the guard is comparing "absent at base" against +"present now" for files that were never committed on this branch — nothing in the project's +actual history edited those files. Flagging for awareness, not action. + +## 2. Theorems: statement changes, removals, renames + +- **`LocalCommutativity.commute_ι`** (blueprint `lmm:local-commutativity-any-quasilocal`): + signature confirmed **unchanged**: + ``` + theorem LocalCommutativity.commute_ι {U : LocalNet} {i : Isotony U} + (h : LocalCommutativity U i) (Q : QuasilocalAlgebra U i) + ⦃B₁ B₂ : Set StandardMinkowskiSpacetime.Carrier⦄ + (hB₁ : IsAlexandrovBasisSet B₁) (hB₂ : IsAlexandrovBasisSet B₂) + (hs : Spacetime.IsCompletelySpacelike StandardMinkowskiSpacetime + standardMinkowskiTimeOrientation B₁ B₂) + (a : U.algebra B₁) (b : U.algebra B₂) : + Commute (Q.ι hB₁ a) (Q.ι hB₂ b) + ``` + Only the proof body changed (it now goes through directedness plus one use of the bridge, + instead of unpacking the old existential witness). +- **`HaagKastlerNet.localCommutativity` field type**: confirmed **unchanged** + (`LocalCommutativity U isotony`), see §1. +- **New `localCommutativity_iff_exists_commute_ι`** (blueprint + `lmm:local-commutativity-iff-quasilocal`). Its right-hand side was checked character-for-character + against the old `LocalCommutativity` body (both quoted above in §1) and is **identical**: + `∃ Q : QuasilocalAlgebra U i, ∀ ⦃B₁ B₂⦄ hB₁ hB₂, IsCompletelySpacelike … → ∀ a b, Commute (Q.ι hB₁ a) (Q.ι hB₂ b)`. + So the old axiom is preserved verbatim as the right side of a new iff, exactly as the plan + specified for D3. +- **`QuasilocalAlgebra.commute_ι_iff_commute_map`**: unchanged (plan said so; confirmed by diff — + no lines touched). +- **`trivialLocalNet_localCommutativity`** (`Net.lean`): statement unchanged + (`LocalCommutativity trivialLocalNet trivialLocalNetIsotony`); only the proof was rewritten + (`mul_comm` directly instead of building `trivialQuasilocalAlgebra`), matching the plan. +- **`commute_ι_of_spacelike`** (`Net.lean`): statement and proof both unchanged; only its + docstring was reworded (it no longer says Axiom 3 "only asserts this in *some* quasilocal + algebra"). +- No theorem was removed or renamed. + +## 3. sorry / non-standard axioms / Restriction markers + +- Grep for `sorry`, `admit`, `native_decide`, `debug.skipKernelTC`, `implemented_by` across + `Physicslib4/` (excluding `scratch_*`) finds no occurrences in code; the only hits are two + prose mentions of the word "sorry" in doc comments explaining that other structures are + *sorry-free*. +- `#print axioms` (via `lake env lean` on a throwaway file, tool-verified) on the five relevant + theorems — `LocalCommutativity.commute_ι`, `localCommutativity_iff_exists_commute_ι`, + `QuasilocalAlgebra.commute_ι_iff_commute_map`, `trivialLocalNet_localCommutativity`, + `HaagKastlerNet.einstein_causality` — and on the two lemmas they depend on, + `exists_quasilocalAlgebra` and `Spacetime.alexandrovBasis_directed`, all report only + `[propext, Classical.choice, Quot.sound]`. No `sorryAx`, no other axiom. +- No `Restriction:` marker was added by this change, and none is needed: the restated axiom is + unconditional in Minkowski space (a containing diamond always exists, by + `alexandrovBasis_directed`), so there is no deliberate narrowing to flag. +- `nolint`: no new uses. The only two `nolint`s in the repository + (`Spectral/BorelClasses.lean:279`, `Analysis/CStarCompletion.lean:84`) are untouched by this + branch (confirmed: `git diff 15cdea7` on those two files is empty) and out of scope. + +## 4. Blueprint consistency + +Checked `blueprint/src/sections/sec10/haag-kastler-axioms.tex` for the five named nodes: + +- **`def:local-commutativity`** — `\lean{Physicslib4.AQFT.HaagKastler.LocalCommutativity}` + exists and the Lean name resolves. Statement: "for any Alexandrov topology basis element `B` + such that `B1, B2 ⊆ B` the algebras … commute in … `U(B)`" — matches the new Lean definition + quantifier-for-quantifier (universal in `B`, isotony images, commute in `U(B)`). It states + "such a `B` always exists … so the condition is never vacuous" and that the axiom "has the same + shape as its curved counterpart", both of which I independently verified by reading the curved + definition (§1). It also correctly points to `lmm:local-commutativity-iff-quasilocal` for the + quasilocal form and no longer `\uses{def:quasilocal-algebra}`. Match. +- **`def:quasilocal-algebra`** — `\lean{Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra}`; now + carries the `ι`/cocone-condition prose that used to live partly in the Axiom 3 node. Matches + the unchanged `QuasilocalAlgebra` structure (carrier, `ι`, `ι_injective`, cocone field, + `dense_range`). Match. +- **`lmm:local-commutativity-any-quasilocal`** — `\lean{LocalCommutativity.commute_ι, + QuasilocalAlgebra.commute_ι_iff_commute_map}`; premised on "Suppose the net satisfies Axiom 3" + and concludes commutation of the images in *any* quasilocal algebra `U'`. Matches + `commute_ι`'s signature exactly (§2). +- **`lmm:local-commutativity-iff-quasilocal`** (new) — `\lean{localCommutativity_iff_exists_commute_ι}`; + states the iff with the quasilocal-algebra form on the right, "and in that case this holds in + every quasilocal algebra of the net." Matches the Lean iff and its proof shape (`⇒` via + `exists_quasilocalAlgebra` + `commute_ι`; `⇐` via the bridge), confirmed by reading both proof + and blueprint proof text side by side. +- **`thrm:einstein-causality`** — unchanged statement; the surrounding prose (line 790) now reads + "Local commutativity … is an algebraic statement about the local algebras, which in the + quasilocal algebra takes the form of `lmm:local-commutativity-any-quasilocal`", replacing the + old "about the quasilocal algebra" wording — exactly the plan's Stage 1 item 5 edit, and its + proof's `\uses` now includes `lmm:local-commutativity-any-quasilocal`. Match. + +**Remaining "quasilocal algebra" prose, checked for staleness:** +- `blueprint/src/sections/sec9/axiom-3-local-commutativity.tex` (curved chapter): the diff adds + one sentence — "This is exactly where curved spacetime parts ways with Minkowski spacetime: + there, any two basis elements lie in a common, larger basis element …, so isotony always + places their algebras in a common algebra, and Axiom 3 can be stated in it." This is accurate + and matches plan item D5/Stage-1-item-6; the surrounding sentences about "no basis element `B` + contains both" in curved spacetime are still true and were correctly left alone. +- `blueprint/src/sections/sec5/quasilocal-algebra.tex`: fully rewritten per plan D4(a), with the + two approved refinements from the progress log ("nonempty nested regions", the common-upper-bound + parenthetical) present. The final `\begin{quote}…\end{quote}` now reads "commute in + `U(B)` for every basis element `B` containing both `B1` and `B2`" — the quasilocal-algebra + reading is kept as an explicitly-flagged equivalent reading in the following remark, not as the + primary statement. No stale "commute in the quasilocal algebra" claim remains as the main + statement. +- `EinsteinCausality.lean` module docstring: now says Axiom 3 "asserts that the local algebras … + commute in any local algebra containing both; by `commute_ι` they then commute inside the + quasilocal algebra" — consistent, no stale claim. +- I did not find any remaining prose (grepped Chapter 10 and the touched Lean docstrings) that + still asserts Axiom 3 itself lives in the quasilocal algebra, as opposed to merely having an + equivalent form there. + +The `def:local-commutativity` node still carries `\leanok` (the plan's Stage 1 item 8 called for +stripping it in the text edit and re-recording it once Lean matched; since Stage 2/3 are already +done and the Lean does match, `\leanok` being present now is the correct final state, not a +leftover). + +## 5. Integrity: verify receipt + +- `.git/lean-kit/verify-receipt.json` records `lake build` and `lake lint` both passing, but at + `2026-10-07T10:32:35+00:00` — **before** the base commit `15cdea7` (dated `22:45:12` the same + day) and well before any commit on this branch. +- I recomputed the guard's digest over the current index and worktree (read-only, via the kit's + own `digest_of`/`index_entries`/`worktree_entries` helpers) and both differ from the receipt's + digest. **The receipt does not cover the current tree.** Concretely: `lake build` / `lake lint` + have not been confirmed to pass since Stage 3's proofs (`commute_ι`, the new iff, + `trivialLocalNet_localCommutativity`) were written, as far as the recorded receipt shows. The + plan's progress log claims "build and lint clean" for Stage 3, but no fresh receipt backs that + claim. +- No `.git/lean-kit/approvals.json` file exists in this checkout, so there is no + `lean_guard.py approve` record for the protected `LocalCommutativity` change; the only recorded + approval is the user's sign-off text in the plan file itself ("approved 2026-10-08... D1-D5"). + That is consistent with how the plan describes itself, but it means the guard's own approval + ledger has nothing to check against. +- Recommendation (not an action I can take): run `python3 ~/.claude/lean-kit/lean_guard.py verify` + on this tree before merging, so the receipt actually covers what is being merged. + +## Summary of classifications + +Using the statement-audit vocabulary for the four blueprint/Lean pairs explicitly in scope: + +- **Match**: `def:local-commutativity`, `def:quasilocal-algebra`, + `lmm:local-commutativity-any-quasilocal`, `lmm:local-commutativity-iff-quasilocal`, + `thrm:einstein-causality` — 5 of 5. +- **Notation only / Lean assumes more / Lean concludes less / different objects / edge cases + differ**: none found. +- **Can't determine**: none. + +Non-matches: none to order. + +## What I verified with a tool vs. by reading + +- Tool-verified: the `git diff`/`git log` content for every claim in §1-§2 and §4; the guard's + `check` output; the stale-receipt finding (recomputed the digest with the kit's own code); the + axiom lists for five theorems plus two dependencies via `lake env lean` + `#print axioms`; the + `sorry`/`nolint`/`Restriction:` greps. +- Read-only judgement (no mechanical check beyond reading): the mathematical equivalence of the + blueprint prose to the Lean statements (quantifier order, which variables are universal vs. + existential); whether the curved and Minkowski definitions are "the same shape" (read both + side by side); whether the Chapter 5/9 prose changes match the plan's approved wording + (read the plan's proposed text against the actual diff). diff --git a/notes/agents/plan-axiom3-common-diamond-2026-10-08.md b/notes/agents/plan-axiom3-common-diamond-2026-10-08.md index 7d4d95e..9726a46 100644 --- a/notes/agents/plan-axiom3-common-diamond-2026-10-08.md +++ b/notes/agents/plan-axiom3-common-diamond-2026-10-08.md @@ -294,3 +294,14 @@ Statements are sorry'd, then the formalizer-reviewer runs. exists_quasilocalAlgebra + commute_ι; bridge backwards), trivialLocalNet_localCommutativity (mul_comm in ℂ). No sorry; build and lint clean. trivialQuasilocalAlgebra still used by QuasilocalIntertwiner.lean. +- 2026-10-08, Stage 4 done: axioms standard (propext, Classical.choice, Quot.sound) for the three + new/re-proved theorems; checkdecls clean; lean-kit verify receipt refreshed. lean-auditor PASS + (`notes/agents/audits/audit-axiom3-common-diamond-2026-10-08.md`): one protected change (the + approved `LocalCommutativity` body), field type and `commute_ι` signature unchanged, iff right + side identical to the old body, blueprint/Lean agree on the five nodes. +- 2026-10-08, Stage 5 done: PDF (342 pp., no errors, no overfull lines) and web rebuilt. Home page: + items renumbered (scripts updated for the absolute blueprint URLs), new entry Lemma 487, + counts 662 / 253 lemmas / 512 results and proofs / 1,146 Lean names, section page ranges set + from the TOC, nine page references corrected, Axiom 3 prose in the §10.6 overview, the guide's + Chapter 5 summary and a new note on the axioms page. Jekyll: 11 pages, no broken links, all + 665 anchors present in the web blueprint. Merged into numina/aqft-in-lean (not pushed). From c0e1c247b8d7b25df5a6c68a4d643e461c95afc8 Mon Sep 17 00:00:00 2001 From: KellyJDavis Date: Thu, 8 Oct 2026 08:42:08 +0200 Subject: [PATCH 6/6] Remove web.paux and ignore it in future --- .gitignore | 1 + blueprint/src/web.paux | Bin 154089 -> 0 bytes 2 files changed, 1 insertion(+) delete mode 100644 blueprint/src/web.paux diff --git a/.gitignore b/.gitignore index 520ba87..a8d10c2 100644 --- a/.gitignore +++ b/.gitignore @@ -8,3 +8,4 @@ # name contains a space (`... .lean thy`). 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