From c66c3bc73d5d627ec7051e9ada6316c98ae072e0 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 29 May 2024 14:04:59 +0200
Subject: weiter

---
 much.tex | 546 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 494 insertions(+), 52 deletions(-)

(limited to 'much.tex')

diff --git a/much.tex b/much.tex
index 58aeab3..a7f9bbf 100644
--- a/much.tex
+++ b/much.tex
@@ -1,23 +1,22 @@
-\chapter{A quantum energy inequality involving local modular data}
-
+\chapter{A Quantum Energy Inequality Involving Local Modular Data}
 
 \cite{Much2022}
 
 \begin{equation*}
   \innerp{\psi}{\energydensity(f)\psi} \ge
-  - \epsilon - \norm{\smash[b]{\Delta}_{\smash[t]{\sharp}}^{-1/2} \ft{g}_{\lambda}(K_{\raisebox{5pt}{\footnotesize$\sharp$}}) \energydensity(f) \fockvaccum}
+  - \epsilon - \norm{\smash[b]{\Delta}_{\smash[t]{\sharp}}^{-1/2} \ft{g}_{\lambda}(K_{\raisebox{5pt}{\footnotesize$\sharp$}}) \energydensity(f) \FockVacuum}
 \end{equation*}
 
-
 \section{Misc}
 
-\todo{Put this somwhere else.}
+\todo{Put this somewhere else.}
 
 A \emph{Lorentz transform} is a linear automorphism of Minkowski spacetime
 which preserves the Lorentz bilinear form.
 Lorentz transforms are usually represented by (real) $4 \times 4$ matrices,
 with respect to the standard basis.
-the \emph{Lorentz group} $\FullLorentzGroup$.
+
+The \emph{Lorentz group} $\FullLorentzGroup$.
 \begin{equation*}
   \FullPoincareGroup = \RR^4 \ltimes \FullLorentzGroup
 \end{equation*}
@@ -44,7 +43,7 @@ Poincaré covariance
 
 \begin{definition}{Von Neumann Algebra of Local Observables}{}
   \begin{equation*}
-    \localalg{\spacetimeregion{O}} = \braces{b(\varphi(f)) \mid b, f \in \realschwartz{M}, \supp f \subset \spacetimeregion{O}}''
+    \localalg{\spacetimeregion{O}} = \braces{b(\varphi(f)) \mid \text{$b$ bounded}, f \in \realschwartz{M}, \supp f \subset \spacetimeregion{O}}''
   \end{equation*}
 \end{definition}
 
@@ -52,10 +51,10 @@ Poincaré covariance
 \index{modular!theory}
 
 If $\hilb{H}$ is a Hilbert space
-we shall denote the $C^*$-algebra of all bounded linear operators on $\hilb{H}$ by $B(\hilb{H})$.
+we shall denote the $C^*$-algebra of all bounded linear operators on $\hilb{H}$ by $\BoundedLinearOperators{\hilb{H}}$.
 
 \begin{definition}{Cyclic and Separating Vectors}{}
-  Suppose $\hilb{H}$ is a Hilbert space and $\mathcal{A}$ is a $C^*$-subalgebra of $B(\hilb{H})$.
+  Suppose $\hilb{H}$ is a Hilbert space and $\mathcal{A}$ is a $C^*$-subalgebra of $\BoundedLinearOperators{\hilb{H}}$.
   A vector $\Omega \in \hilb{H}$ is called
   \begin{itemize}
     \item \emph{cyclic}\index{cyclic vector} for $\mathcal{A}$ if the vector set $\mathcal{A} \Omega$ is dense in $\hilb{H}$.
@@ -66,19 +65,37 @@ Occasionally, a vector that is both cyclic and separating is called \emph{standa
 
 Recall that the commutant of a set $\mathcal{S} \subset B(\hilb{H})$ of operators
 is defined as the set of all operators $T \in B(\hilb{H})$ which commute with all operators $S$ in $\mathcal{S}$.
-We shall denote the commutant of $\mathcal{S}$ by $\mathcal{S}'$.\nomenclature{$\mathcal{A}'$}{commutant of $\mathcal{A}$}
+We shall denote the commutant of $\mathcal{S}$ by $\mathcal{S}'$.\nomenclature[A]{$\mathcal{A}'$}{commutant of $\mathcal{A}$}
 
 \begin{proposition}{}{cyclic-separating}
-  \begin{enumerate}[label=(\roman*),nosep,leftmargin=*,widest=ii]
-    \item A vector is cyclic for $\mathcal{A}$ if and only if it is separating for $\mathcal{A}'$.
-    \item If $\vNa{M}$ is a von Neumann algebra, then a vector is cyclic and separating for $\vNa{M}$
-  if and only if it is cyclic and separating for $\vNa{M}'$.
+  Let $\hilb{H}$ be a Hilbert space and $\mathcal{A}$ be a $C^*$-subalgebra of $\BoundedLinearOperators{\hilb{H}}$.
+  \begin{enumerate}
+    \item \label{item:first} A vector is cyclic for $\mathcal{A}$ if and only if it is separating for $\mathcal{A}'$.
+    \item \label{item:second} If $\mathcal{A}$ is a von Neumann algebra, then a vector is cyclic and separating for $\mathcal{A}$
+  if and only if it is cyclic and separating for $\mathcal{A}'$.
   \end{enumerate}
 \end{proposition}
 
 \begin{proof}
-  \todo{xxx}
-  The second assertion directly follows from the first and the fact that $\vNa{M}'' = \vNa{M}$.
+  First, suppose that $\Omega \in \hilb{H}$ is cyclic for $\mathcal{A}$.
+  If $A'$ is an element of $\mathcal{A}'$ with $A' \Omega = 0$,
+  then $A' A \Omega = A A' \Omega = 0$ for all $A \in \mathcal{A}$.
+  This means that $A'$ vanishes on the dense subspace $\mathcal{A} \Omega$,
+  and thus on all of $\hilb{H}$, i.e.\ $A'=0$.
+  This proves that $\Omega$ is separating for $\mathcal{A}'$.
+
+  Conversely, suppose that $\Omega$ is separating for $\mathcal{A}'$.
+  We have to show that the closed subspace $\overline{\mathcal{A}\Omega}$ is all of $\hilb{H}$.
+  Let $P$ be the orthogonal projection onto $\overline{\mathcal{A}\Omega}$.
+  Clearly, any element $A$ of $\mathcal{A}$ maps $\overline{\mathcal{A}\Omega}$ into itself.
+  Thus, $PAP=AP$, and the same holds for $A^*$, that is, $PA^*P=A^*P$.
+  Taking the adjoint of the second equation, we get $PAP=PA$. Hence, $P \in \mathcal{A}'$.
+  Now, $P \Omega = \Omega = I \Omega$, where $I$ is the identity operator on $\hilb{H}$ which obviously also belongs to $\mathcal{A}'$,
+  and the assumption that $\Omega$ is separating for $\mathcal{A'}$ implies $P=I$.
+  Consequently, $\overline{\mathcal{A} \Omega} = \hilb{H}$.
+
+  Statement~\ref{item:second} directly follows from~\ref{item:first} and
+  the fact that $\mathcal{A}'' = \mathcal{A}$ by the Double Commutant Theorem.
 \end{proof}
 
 If $\Omega$ is separating for $\mathcal{A}$,
@@ -87,7 +104,7 @@ with a unique $A \in \mathcal{A}$.
 This allows us to define an (anti-linear) operator $S_0$ in $\hilb{H}$ with domain $\mathcal{A}\Omega$ by
 \begin{equation}
   \label{equation:definition-s0}
-  \quad S_0 A\Omega \defequal S_0 A^*\Omega \qquad A \in \mathcal{A}.
+  \quad S_0 A\Omega \defequal A^*\Omega \qquad A \in \mathcal{A}.
 \end{equation}
 The operator $S_0$ is densely defined if and only if $\Omega$ is cyclic for $\mathcal{A}$.
 Since the $*$-operation on $\mathcal{A}$ is involutive,
@@ -100,12 +117,12 @@ the range of $S_0$ coincides with its domain.
 \begin{proof}
   By \cref{proposition:cyclic-separating},
   $\Omega$ is also cyclic and separating for the commutant $\vNa{A}'$.
-  Hence we may, analogously to $S_0$,
+  Hence we may, in analogy to $S_0$,
   define another anti-linear operator $F_0$ in $\hilb{H}$ with dense domain $\mathcal{A}' \Omega$ by
   \begin{equation*}
-    \quad F_0 B\Omega \defequal F_0 B^*\Omega \qquad B \in \mathcal{A'}.
+    \quad F_0 B\Omega \defequal B^*\Omega \qquad B \in \mathcal{A'}.
   \end{equation*}
-  By definition of $S_0$ and $F_0$ we have for every $A \in \mathcal{A}$ and $B \in \mathcal{A}'$
+  By definitions of $S_0$ and $F_0$, we have for every $A \in \mathcal{A}$ and $B \in \mathcal{A}'$
   \begin{equation*}
     \innerp{S_0 A \Omega}{B \Omega} = 
     \innerp{\Omega}{AB \Omega} = 
@@ -124,10 +141,10 @@ the range of $S_0$ coincides with its domain.
   Suppose $\Omega$ is a cyclic and separating vector for a von Neumann algebra $\mathcal{A}$.
   The closure $S = \operatorclosure{S_0}$
   of the operator $S_0$ defined on $\mathcal{A}\Omega$ by
-  $S_0 A\Omega = S_0 A^*\Omega$
+  $S_0 A\Omega = A^*\Omega$
   for $A \in \mathcal{A}$
   is called the
-  \emph{Tomita operator}\index{Tomita operator}\index{operator!Tomita}\nomenclature{$S$}{Tomita operator}
+  \emph{Tomita operator}\index{Tomita operator}\index{operator!Tomita}\nomenclature[S]{$S$}{Tomita operator}
   for the pair $(\mathcal{A},\Omega)$.
 \end{definition}
 
@@ -135,7 +152,7 @@ It is a well-known fact that closed operators can be decomposed
 in a similar fashion to the polar coordinate representation $z = e^{i\arg z} \abs{z}$
 of a complex number.
 We state the theorem in its somewhat uncommon variant for anti-linear operators,
-as this is our only use case.
+as this will be our only use case.
 
 \begin{theorem}{Polar Decomposition for Anti-Linear Closed Operators}{polar-decomposition}
   \index{polar decomposition}
@@ -156,7 +173,7 @@ as this is our only use case.
 \end{theorem}
 
 Proofs of this statement are contained in~\cite{ReedSimon1} and~\cite{Schmüdgen2012}.
-When we speak of \emph{the} polar composition we tacitly assume that the additional conditions
+When we speak of \emph{the} polar composition of an operator we tacitly assume that the additional conditions
 ensuring uniqueness are satisfied.
 
 Now we are able to introduce the fundamental objects of modular theory.
@@ -170,7 +187,7 @@ Now we are able to introduce the fundamental objects of modular theory.
   \end{equation*}
   be its polar decomposition.
   The anti-unitary operator $J$ is called
-  \emph{modular conjugation}\index{modular!conjugation}\nomenclature{$J$}{modular conjugation}.
+  \emph{modular conjugation}\index{modular!conjugation}\nomenclature[J]{$J$}{modular conjugation}.
   The positive selfadjoint operator $\Delta$ is called
   \emph{modular operator}\index{modular!operator}\index{operator!modular}\nomenclature{$\Delta$}{modular operator}.
   The pair $(J,\Delta)$ is said to be the \emph{modular data}\index{modular!data}\index{modular!objects} associated to
@@ -187,8 +204,6 @@ Now we are able to introduce the fundamental objects of modular theory.
 
 The modular group is a strongly continuous one-parameter unitary group on $\hilb{H}$.
 
-\newpage
-
 \begin{proposition}{}{modular-data-unitary}
   Suppose $\vNa{M}$ is a von Neumann algebra acting on a Hilbert space $\hilb{H}$.
   Let $U$ be a unitary operator on $\hilb{H}$.
@@ -202,11 +217,11 @@ The modular group is a strongly continuous one-parameter unitary group on $\hilb
 \begin{proof}
   To prove the first assertion,
   consider any $A \in (U\vNa{M}U^*)''$.
-  By the double commutant theorem,
+  By the Double Commutant Theorem~\cite[Theorem 18.6]{Zhu1993},
   it suffices to show that $A \in U\vNa{M}U^*$.
   As $\vNa{M}$ is a von Neumann algebra,
   this is equivalent to $U^*\! AU \in \vNa{M}''$,
-  again by the double commutant theorem.
+  again by the Double Commutant Theorem.
   Let $B \in \vNa{M}'$.
   It is easy to check that $UBU^* \in (U\vNa{M}U^*)'$.
   By assumption, $A$ lies in the commutant of $(U\vNa{M}U^*)'$.
@@ -221,15 +236,15 @@ The modular group is a strongly continuous one-parameter unitary group on $\hilb
   Now $A=0$ follows from the assumption that $\Omega$ is separating for $\vNa{M}$.
   We have shown that the mapping $UAU^*U\Omega = UA\Omega$ from $U\vNa{M}U^* \to \hilb{H}$ is injective.
 
-  Let $S = \overline{S_0}$ be the Tomita operator associated to $(\vNa{M},\Omega)$,
-  and let $S' = \overline{S'_0}$ be the Tomita operator associated to $(U\vNa{M}U^*,U\Omega)$.
+  Let $S = \operatorclosure{S_0}$ be the Tomita operator associated to $(\vNa{M},\Omega)$,
+  and let $S' = \operatorclosure{S'_0}$ be the Tomita operator associated to $(U\vNa{M}U^*,U\Omega)$.
   Then we have
   \begin{equation*}
     (S'_0 U) A \Omega =
     S'_0 (U A U^*)  U \Omega =
     (U A^* U^*)  U \Omega =
     U A^* \Omega =
-    U S_0 A \Omega
+    (U S_0) A \Omega
   \end{equation*}
   for all $A \in \vNa{M}$. Consequently, $S'_0 = U S_0 U^*$ as operators with domain $U\vNa{M}\Omega$.
   Taking the closure, we obtain $S' = U S U^*$.
@@ -243,12 +258,12 @@ The modular group is a strongly continuous one-parameter unitary group on $\hilb
   associated to the pair $(U\vNa{M}U^*,U\Omega)$.
 \end{proof}
 
-\newpage
-
 Finally, let us outline how modular theory enters into algebraic quantum field theory.
 
-\begin{theorem}{Reeh-Schlieder Theorem}{reeh-schlieder}
-  \todo{spell it out}
+\begin{theorem}{Reeh--Schlieder Theorem}{reeh-schlieder}
+  Let $\spacetimeregion{O}$ be any open spacetime region.
+  Then the vacuum vector $\Omega$ is cyclic for $\localalg{\spacetimeregion{O}}$.
+  If $\spacetimeregion{O}'$ is non-empty, then $\Omega$ is also separating for $\localalg{\spacetimeregion{O}}$.
 \end{theorem}
 
 By Reeh-Schlieder (\cref{theorem:reeh-schlieder}), the vacuum $\Omega$ is cyclic and separating for $\localalg{\spacetimeregion{O}}$.
@@ -286,7 +301,7 @@ since they are transformed into each other by space inversion.
 \end{proof}
 
 In the standard representation of the Lorentz group, the boost (or velocity transformation) along the $x^1$-axis
-with rapidity $2 \pi t$ is given by the matrix\footnote{
+with rapidity $2 \pi t$ is given by the matrix\footnote{%
   This matrix depends on the choice of metric signature.
   Ours is $(+,-,-,-)$.
   For $(-,+,+,+)$, use
@@ -300,21 +315,22 @@ with rapidity $2 \pi t$ is given by the matrix\footnote{
   \end{equation*}
   }
 
-\begin{equation*}
+\begin{equation}
+  \label{equation:lorentz-boost}
   \Lambda(t) = \begin{pmatrix}
     \cosh(2 \pi @ t) & \sinh(2 \pi @ t) & \; 0 \; & \; 0 \; \\
     \sinh(2 \pi @ t) & \cosh(2 \pi @ t) & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & 1 \\
   \end{pmatrix}
-\end{equation*}
+\end{equation}
 
 The following proposition shows that $t \mapsto \Lambda(t)$ is
 a one-parameter subgroup of the stabilizer group of the right wedge
 with respect to the action of the Lorentz group on subsets of Minkowski space.
 
 \begin{proposition}{}{}
-  \begin{enumerate}[label=(\roman*),nosep,leftmargin=*,widest=ii]
+  \begin{enumerate}
     \item $\Lambda(s + t) = \Lambda(s) \Lambda(t)$ for all $s,t \in \RR$.
     \item $\Lambda(t) \rightwedge = \rightwedge$ for all $t \in \RR$.
   \end{enumerate}
@@ -349,7 +365,9 @@ with respect to the action of the Lorentz group on subsets of Minkowski space.
   and~\eqref{equation:image-right-wedge}. Now it follows from $\Lambda(-t) = \Lambda(t)^{-1}$ that in fact $\Lambda(t) x = y$.
 \end{proof}
 
-\begin{theorem}{Bisognano-Wichmann Theorem \textmd{\cite{Bisognano1975}}}{}
+It can easily be seen that $\rightwedge' = \leftwedge$, and so \cref{theorem:reeh-schlieder} applies.
+
+\begin{theorem}{Bisognano--Wichmann Theorem \textmd{\cite{Bisognano1975}}}{bisognano-wichmann}
   For the theory of a free scalar field in Minkowski spacetime,
   let $\spacetimeregion{O} \mapsto \localalg{\spacetimeregion{O}}$ be the net of von Neumann algebras of local observables.
   If $(J,\Delta)$ is the modular data associated to the algebra $\localalg{\rightwedge}$ of the right wedge and the vacuum $\Omega$, then
@@ -400,7 +418,7 @@ That this is generally true is the statement of the following Lemma.
   for all Borel functions $f : \RR \to \CC$.
 \end{lemma}
 
-\question{Ist diese Aussage korrekt? Ist mein Beweis richtig? Geht der auch einfacher?}
+\todo{write down the simpler proof}
 
 \begin{proof}
   For each regular value $\lambda \in \rho(A)$ let
@@ -453,14 +471,14 @@ That this is generally true is the statement of the following Lemma.
 \end{equation*}
 
 
-Recall that Stones Theorem \todo{add reference} states that
+Recall that Stone's Theorem \todo{add reference} states that
 every strongly continuous one-parameter unitary group
 is of the form $t \mapsto e^{itK}$ with a uniquely determined
 selfadjoint operator $K$, which is called \emph{infinitesimal generator} of the group.
 
 \begin{definition}{Modular Hamiltonian}{}
   The infinitesimal generator of the modular group associated to a spacetime region $\spacetimeregion{O}$ is called the
-  \emph{modular Hamiltonian}\index{modular!Hamiltonian}\nomenclature{$K_{\spacetimeregion{O}}$}{modular Hamiltonian for $\spacetimeregion{O}$}
+  \emph{modular Hamiltonian}\index{modular!Hamiltonian}\nomenclature[KO]{$K_{\spacetimeregion{O}}$}{modular Hamiltonian for $\spacetimeregion{O}$}
   for said region, and denoted $K_{\spacetimeregion{O}}$.
 \end{definition}
 
@@ -472,29 +490,453 @@ In other words, $K_{\spacetimeregion{O}}$ is the unique selfadjoint operator suc
     A\psi(p) = - \frac{2\pi}{i} \parens[\big]{\partial_0 \psi(p) \, p^1 + \partial_1 \psi(p) \, p^0}
   \end{equation*}
 \end{proposition}
+\todo{domain, proof}
 
 \section{Complex Lorentz Transformations}
 
+The main result of this section is \cref{proposition:main-result}.
+
+By definition, the \emph{complex Lorentz group}\index{Lorentz group!complex}\nomenclature[LC]{$\ComplexLorentzGroup$}{complex Lorentz group} $\ComplexLorentzGroup$ is the isometry group
+of complex Minkowski space $M+iM \cong \CC^4$ with respect to the inner product
+\begin{equation*}
+  \innerp{z_1}{z_2} = \innerp{x_1}{x_2} - \innerp{y_1}{y_2} + i \parens[\big]{\innerp{x_1}{y_2} + \innerp{x_2}{y_1}}.
+\end{equation*}
+The \emph{complex Poincaré group}\index{Poincaré group!complex}\nomenclature[PC]{$\ComplexPoincareGroup$}{complex Poincaré group} is the semidirect product $\ComplexPoincareGroup \defequal \CC^4 \ltimes \ComplexLorentzGroup$.
+The action of $\ComplexPoincareGroup$ on $M+iM$ is defined in the obvious way.
+The complex Poincaré group has just two connected components, the subgroup $\ProperComplexPoincareGroup$ and the subset $\ImproperComplexPoincareTransformations$,
+differentiated by the sign of $\det \Lambda \in \braces{\pm 1}$ for its elements $(z,\Lambda)$.
+The (real) proper orthochronous Poincaré group $\ProperOrthochronousPoincareGroup$ is a subgroup of $\ProperComplexPoincareGroup$.
+Each of the two following sections deals with a subgroup $G$ of $\ProperOrthochronousPoincareGroup$,
+and the possibility of extending a unitary representation of $G$ to a larger set within $\ProperComplexPoincareGroup$.
+
 \subsection{Analytic Continuation of the Space-Time Translation Group}
 
+%\todo{a short intro}
+
+Let $a \mapsto U(a)$ be a strongly continuous unitary representation of the additive group of $\RR^4$ (on some separable Hilbert space).
+By a generalization of Stone's Theorem~\cite[Theorem VIII.12]{ReedSimon1},
+there exists a unique projection-valued measure $E$ on $\RR^4$ such that
+\begin{equation}
+  \label{equation:spectral-resolution-translation}
+  U(a) = \int_{\RR^4} \exp(ia \cdot k) \, dE(k) \qquad a \in \RR^4.
+\end{equation}
+Then one can define a vector $P$ of unbounded selfadjoint operators
+\begin{equation*}
+  P_i = \int_{\RR^4} k_i \, dE(k) \qquad i=0,\ldots,3
+\end{equation*}
+which have a common dense domain $D$ and satisfy
+\begin{equation*}
+  a \cdot P = \int_{\RR^4} a \cdot k \, dE(k) \qquad a \in \RR^4.
+\end{equation*}
+We are specifically interested in the representation
+\begin{equation*}
+  U(a) \defequal U(a,I) \qquad a \in \RR^4
+\end{equation*}
+obtained by restricting the unitary representation of the Poincaré group $\RestrictedPoincareGroup$ on Fock space to the subgroup of spacetime translations.
+In this case the vector operator $P$ carries the physical meaning of energy-momentum,
+and we impose the so-called \emph{spectrum condition}
+\begin{equation*}
+  \langle a \cdot P \rangle_{\psi} \ge 0 \qquad
+  \forall \psi \in D \;
+  \forall a \in \ClosedForwardCone,
+\end{equation*}
+where $\ClosedForwardCone \defequal \braces{a \in \RR^4 \vcentcolon a \cdot a \ge 0, a^0 \ge 0}$ is the \emph{closed forward cone}\index{cone!closed forward}\nomenclature[V]{$\ClosedForwardCone$}{closed forward cone}.
+It can be shown \cite{Uhlmann1961} that the spectrum condition is equivalent to the statement that
+the support of the spectral measure is contained in the closed forward cone, i.e.\ $\supp(E) \subset \ClosedForwardCone$.
+
+Spectral calculus allows us to extend $a \mapsto U(a)$ to complex arguments $z \in \CC^4$ by simply replacing $a$ with $z$ in the spectral resolution~\eqref{equation:spectral-resolution-translation} of $U(a)$. 
+However, one obtains, in general, an unbounded operator.
+It is a consequence of the spectrum condition that $U(z)$ is bounded whenever $z$ lies in the \emph{closed forward tube}\index{tube!closed}\nomenclature[T]{$\ClosedForwardTube$}{closed forward tube} $\ClosedForwardTube \defequal \RR^4 + i\ClosedForwardCone$.
+Observe that the set $\ClosedForwardTube$ is closed under vector addition and thus forms a commutative monoid; it is not a group.
+
+\begin{proposition}{}{}
+  For every $z \in \ClosedForwardTube$ the operator
+  \nomenclature[U]{$U(z)$}{complex translation}
+  \begin{equation}
+    \label{equation:definition-complex-translation}
+    U(z) \defequal \int_{\ClosedForwardCone} \exp(iz \cdot k) \, dE(k)
+  \end{equation}
+  is bounded.
+  Moreover, $U(w+z) = U(w) U(z)$ for all $w,z \in \ClosedForwardTube$.
+\end{proposition}
+
+\begin{proof}
+  By a general property of spectral integrals~\cite[Proposition 4.18]{Schmüdgen2012},
+  the operator $U(z)$ is bounded if (and only if)
+  the function $f(k) = \exp(iz \cdot k)$ is bounded $E$-almost everywhere.
+  In view of the fact that $E$ is supported in the closed forward cone $\ClosedForwardCone$,
+  it is sufficient to show that $f$ is bounded on $\ClosedForwardCone$.
+  %the $E$-essential supremum of the function
+  Since $z$ lies in the closed forward tube, $z=x+iy$ with $x \in \RR^4$ and $y \in \ClosedForwardCone$.
+  Now $\abs{f(k)} = \exp(-y \cdot k)$, and on $\ClosedForwardCone$ this is bounded by $1$ because $y \cdot k \ge 0$ for all $k \in \ClosedForwardCone$.
+
+  The identity $U(w+z) = U(w) U(z)$ follows from $\exp(i(w+z) \cdot k) = \exp(iw \cdot k) \exp(iz \cdot k)$
+  and the boundedness of the operators, see~\cite[Proposition 4.16(iii) and (v)]{Schmüdgen2012}.
+\end{proof}
+
+\begin{proposition}{}{}
+  If $(b,\Lambda) \in \RestrictedPoincareGroup$ and $z \in \ClosedForwardTube$, then $\Lambda z \in \ClosedForwardTube$ and
+  \begin{equation*}
+    U(b,\Lambda) U(z) U(b,\Lambda)^* = U(\Lambda z).
+  \end{equation*}
+\end{proposition}
+
+\begin{proof}
+  We exploit the uniqueness of the projection-valued measure $E$ satisfying~\eqref{equation:spectral-resolution-translation}.
+  Since $\Lambda$ acts continuously on $\RR^4$, $\Lambda^{-1} S$ is a Borel set whenever $S \subset \RR^4$ is, and
+  \begin{equation*}
+    F(S) \defequal U(b,\Lambda) E(\Lambda^{-1} S) U(b,\Lambda)^*
+    \qquad S \in \BorelSigmaAlgebra{\RR^4}
+  \end{equation*}
+  is a well-defined projection-valued measure on $\RR^4$.
+  By the Transformation Formula for Spectral Integrals~\cite[Proposition 4.24]{Schmüdgen2012}, we have
+  \begin{align}
+    \int_{\RR^4} \exp(iz \cdot k) \, dF(k)
+    &= U(b,\Lambda) \bracks[\bigg]{\,\int_{\RR^4} \!\exp(iz \cdot \Lambda k) \, dE(k)} U(b,\Lambda)^* \nonumber\\
+    &= U(b,\Lambda) \bracks[\bigg]{\,\int_{\RR^4} \!\exp(i \Lambda^{-1} z \cdot k) \, dE(k)} U(b,\Lambda)^* \nonumber\\
+    \label{equation:F-integral}
+    &= U(b,\Lambda) U(\Lambda^{-1} z)  U(b,\Lambda)^*
+  \end{align}
+  for all $z \in \ClosedForwardTube$.
+  In particular, for $z=a \in \RR^4$ the last term equals
+  \begin{equation*}
+    U(b,\Lambda) U(\Lambda^{-1} a)  U(b,\Lambda)^* = U(a) \qquad a \in \RR^4,
+  \end{equation*}
+  which can be seen by applying $U$ to the identity
+  \begin{equation*}
+    (b,\Lambda) (\Lambda^{-1} a, I) (b,\Lambda)^{-1} = (a,I).
+  \end{equation*}
+  Hence, $U(a) = \int \exp(ia \cdot k) dF(k)$ for all $a \in \RR^4$. We conclude $E = F$.
+  Now~\eqref{equation:F-integral} asserts that $U(z) = U(b,\Lambda) U(\Lambda^{-1} z) U(b,\Lambda)$ for all $z \in \ClosedForwardTube$
+  and a substitution of $z$ by $\Lambda z$ yields the desired identity.
+\end{proof}
+
+%\begin{proposition}{Analyticity of the Complex Translation Monoid}{analyticity-complex-translations}
+  \begin{proposition}{\textmd{\cite[Theorem 4]{Uhlmann1961}}}{analyticity-complex-translations}
+    The operator-valued map $z \mapsto U(z)$ given by~\eqref{equation:definition-complex-translation} is
+  strongly continuous on $\ClosedForwardTube$ and
+  analytic on $\OpenForwardTube$.
+\end{proposition}
+
+\todo{Explain what it means for an operator-valued function of several complex variables to be analytic.}
+
+Next we consider an operator-valued tempered distribution $u$ that is \emph{covariant}
+in the sense that it obeys the relativistic transformation law
+\begin{equation}
+  \label{equation:covariance-distribution}
+  U(g) u(f) U(g)^* = u(f_g) \qquad g \in \RestrictedPoincareGroup, f \in \schwartz{\RR^4},
+\end{equation}
+where $f_g(x) = f(g^{-1} x)$ for all $x \in M$.
+In particular, if $g=(a,I)$ is the translation by a vector $a \in \RR^4$,
+then~\eqref{equation:covariance-distribution} and the invariance of the vacuum vector $\FockVacuum$ imply
+\begin{equation}
+  \label{equation:real-translation-law}
+  U(a) u(f) \FockVacuum = u(f_a) \FockVacuum \qquad \forall a \in \RR^4.
+\end{equation}
+We would like to extend this law to complex translation vectors,
+but translating a function defined on $\RR^4$ by a complex vector is not a sensible operation.
+Nevertheless, we have $\FT{f_a}(p) = \exp(ia \cdot p) \ft{f}(p)$ in Fourier space,
+and $\exp(iz \cdot p) \ft{f}(p)$ is a well defined function of $p \in \RR^4$ even when $z \in \CC^4$.
+The obvious idea would be to define $f_z$ as the inverse Fourier transform of this function.
+This does not work because $\exp(iz \cdot p) \ft{f}(p)$ is generally not in the Schwartz class.
+However, thanks to the spectrum condition we may modify this function outside of the closed forward cone.
+
+\begin{lemma}{}{depends-only-on-restriction}
+  Let $u$ be a covariant operator-valued tempered distribution.
+  Then the vector $u(f) \FockVacuum$, where $f \in \schwartz{\RR^4}$,
+  depends only on the restriction of $\ft{f}$ to $\ClosedForwardCone$.
+\end{lemma}
+
+\begin{proof}
+  We consider a Schwartz function $g \in \schwartz{\RR^4}$ and
+  the operator $G = \int g(k) dE(k)$,
+  where $E$ is the unique projection-valued measure on $\RR^4$ such that
+  $U(a) = \int \exp(ik \cdot a) dE(k)$ for all $a \in \RR^4$.
+  Let $g(k) = (2 \pi)^{-2} \int \ift{g}(a) \exp(ia \cdot k) da$ be the Fourier decomposition of $g$.
+  \begin{multline*}
+    \hspace{1cm} (2 \pi)^2 G = \int_{\ClosedForwardCone} \!\int_{\RR} \ift{g}(a) \exp(ia \cdot k) \, da \, dE(k) = \\
+    = \int_{\RR} \ift{g}(a) \!\int_{\ClosedForwardCone} \exp(ia \cdot k) \, dE(k) \ da
+    = \int_{\RR} \ift{g}(a) U(a) \, da \hspace{1cm}
+  \end{multline*}
+  \question{Darf ich hier wirklich die Integrationsreihenfolge vertauschen?}
+
+  Recall that the Fourier transform of $u$ is defined by $\ft{u}(f) = u(\ft{f}@@)$ for $f \in \schwartz{\RR^4}$.
+  We obtain the action of the translation group on $\ft{u}(\ft{f}@@)\FockVacuum$ by definition chasing and~\eqref{equation:real-translation-law}:
+  \begin{equation*}
+    U(a) \ft{u}(\ft{f}@@)\FockVacuum
+    = U(a) u(f)\FockVacuum
+    = u(f_a)\FockVacuum
+    = \ft{u}(\ft{f}_a)\FockVacuum
+    = \ft{u}(\ft{f}e_a)\FockVacuum
+  \end{equation*}
+  Here $e_a$ stands for the function $e_a(p) = \exp(ia \cdot p)$.
+  \begin{equation*}
+    G @\ft{u}(\ft{f}@@)\FockVacuum
+    = \int \ift{g}(a) @\ft{u}(\ft{f}e_a)\FockVacuum \, da
+    = \ft{u} \parens[\bigg]{\ft{f} \int \ift{g}(a) e_a da} \FockVacuum 
+    = \ft{u}(\ft{f} g) \FockVacuum
+  \end{equation*}
+  The second identity is due to the continuity of the vector-valued map $f \mapsto u(f) \FockVacuum$.
+  If the support of $g$ does not intersect the support of $E$, i.e.\ the closed forward cone, then $G=0$.
+  Thus, $\ft{u}(\ft{f} g) \FockVacuum = 0$.
+  This proves that $u(f_1) \FockVacuum = u(f_2) \FockVacuum$ when $\supp(\ft{f_1} - \ft{f_2}) \subset \ClosedForwardCone$.
+\end{proof}
+
+This fact inspires the definition
+\begin{equation*}
+  f_z \defequal d_z * f \qquad z \in \ClosedForwardTube
+\end{equation*}
+where $d_z$ is any Schwartz function on $\RR^4$ such that $\FT{d_z}(p) = \exp(iz \cdot p)$ for all $p \in \ClosedForwardCone$.
+Such a function does exist \todo{elaborate, smooth cutoff}. Then $f_z$ will be Schwartz class as well.
+Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$, by~\cref{lemma:depends-only-on-restriction}.
+
+%\begin{lemma}{}{}
+  %For every $z \in \ClosedForwardTube$ there exists a Schwartz function $d_z \in \schwartz{\RR^4}$
+  %such that $\ft{e_z} \in \schwartz{\RR^4}$ and $\ft{e_z}(p) = \exp(iz \cdot p)$ for $p \in \ClosedForwardCone$.
+%\end{lemma}
+
+\begin{proposition}{}{}
+  Let $u$ be a covariant operator-valued tempered distribution,
+  and let $f \in \schwartz{\RR^4}$ be a test function. Then we have,
+  in generalization of~\eqref{equation:real-translation-law},
+  \begin{equation*}
+    U(z) u(f) \FockVacuum = u(f_z) \FockVacuum \qquad \forall z \in T_+.
+  \end{equation*}
+\end{proposition}
+
+\begin{proof}
+  By \cref{proposition:analyticity-complex-translations},
+  the function $z \mapsto U(z) u(f) \FockVacuum$ is analytic on the open forward tube.
+  \todo{Zeige, dass $z \mapsto u(f_z) \FockVacuum$ ebenfalls analytisch ist.
+  Dann folgt die Behauptung wohl mit Edge of the Wedge~\cite[Theorem 2-17]{Streater1964}}
+\end{proof}
+
 \subsection{Complex Lorentz Boosts}
 
+The Lorentz boosts $\Lambda(t)$ given by
+the matrices~\eqref{equation:lorentz-boost} in standard representation
+have a natural interpretation for complex parameters,
+since the hyperbolic functions $\cosh$ and $\sinh$ extend analytically to the whole complex plane.
+In view of Lemma xxx it follows immediately that the matrix-valued function $\CC \ni w \mapsto \Lambda(w)$ is entire analytic.
+In particular, the vector-valued function $\CC \ni w \mapsto \Lambda(w) z$ is entire analytic for every fixed vector $z \in \CC^4$.
+
+We are particularly interested in the case of a purely imaginary parameter.
+The relations $\cosh iz = \cos z$ and $\sinh iz = i \sin z$
+between the complex hyperbolic and trigonometric functions imply
+
+\begin{equation*}
+  \Lambda(is) = \begin{pmatrix}
+    \phantom{i}\cos(2 \pi @ s) & i\sin(2 \pi @ s) & \; 0 \; & \; 0 \; \\
+    i\sin(2 \pi @ s) & \phantom{i}\cos(2 \pi @ s) & 0 & 0 \\
+    0 & 0 & 1 & 0 \\
+    0 & 0 & 0 & 1 \\
+  \end{pmatrix}
+  \qquad \forall s \in \RR.
+\end{equation*}
+
+\begin{equation*}
+  \Lambda(is) (x+iy) =
+  \begin{pmatrix}
+    \cos(2 \pi @ s) x^0 - \sin(2 \pi @ s) y^1 \\
+    \cos(2 \pi @ s) x^1 - \sin(2 \pi @ s) y^0 \\
+    x^2 \\
+    x^3 
+  \end{pmatrix}
+  +i
+  \begin{pmatrix}
+    \sin(2 \pi @ s) x^1 + \cos(2 \pi @ s) y^0 \\
+    \sin(2 \pi @ s) x^0 + \cos(2 \pi @ s) y^1 \\
+    y^2 \\
+    y^3 
+  \end{pmatrix}
+\end{equation*}
+
+We
+
+\begin{equation*}
+  \mathcal{J} \defequal \Lambda(i/2) = \begin{pmatrix}
+    -1 & 0 & \; 0 \; & \; 0 \; \\
+    0 & -1 & 0 & 0 \\
+    0 & 0 & 1 & 0 \\
+    0 & 0 & 0 & 1 \\
+  \end{pmatrix}
+\end{equation*}
+
+\begin{equation*}
+  \mathcal{J}_{\pm} \defequal \Lambda(\pm i/4) = \begin{pmatrix}
+    0 & \pm i & \; 0 \; & \; 0 \; \\
+    \pm i & 0 & 0 & 0 \\
+    0 & 0 & 1 & 0 \\
+    0 & 0 & 0 & 1 \\
+  \end{pmatrix}
+\end{equation*}
+
+We now turn to the unitary representation of (real) Lorentz boosts
+\begin{equation*}
+  V(t) \defequal U \parens[\big]{0,\Lambda(t)} \qquad t \in \RR
+\end{equation*}
+on Fock space and aim for an analytic extension similar to the previous section.
+By Stone's theorem theorem there exists a unique selfadjoint operator $K$ such that
+\begin{equation*}
+  V(t) = \exp(itK) = \int_{\RR} \exp(it \lambda) \,dE_K(\lambda),
+\end{equation*}
+where $E_K$ is the spectral measure on $\RR$ associated to $K$.
+Now we define \emph{complex Lorentz boosts} to be the operators
+\nomenclature[V]{$V(z)$}{complex Lorentz boost}
+\begin{equation*}
+  V(z) \defequal \int_{\RR} \exp(iz \lambda) \,dE_K(\lambda) \qquad z \in \CC.
+\end{equation*}
+In contrast to the previous section, we
+
+
+\begin{lemma}{}{}
+  Suppose $A$ is a selfadjoint unbounded operator on some Hilbert space $\hilb{H}$.
+  For each complex number $z$ define the closed normal operator $V(z) = e^{izA}$ by means of functional calculus.
+  Let $g \in \schwartz{\RR}$ be a Schwartz function.
+  \begin{enumerate}
+    \item $V(z) V(w) = V(z + w)$ for all $z,w \in \CC$.
+    \item The operator $g(A)$ is bounded, and its range is contained in the domain of $V(z)$ for all $z \in \CC$.
+    \item The operator $V(z) g(A)$ is bounded for all $z \in \CC$, and has spectral resolution
+      \begin{equation*}
+        V(z) g(A) = \int e^{iz \lambda} g(\lambda) dE_A(\lambda).
+      \end{equation*}
+    \item The function $z \mapsto V(z) g(A)$ is entire analytic.
+  \end{enumerate}
+\end{lemma}
+
+Remember that a \emph{core}\index{operator!core for an} for a closed densely defined unbounded operator $T$
+is, by definition, a linear subspace $\mathcal{D}_0$ of its domain $\Domain{T}$ such that
+the closure of the restriction of $T$ to $\mathcal{D}_0$ coincides with $T$.
+%symbolically $\overline{T \vert \mathcal{D}_0} = T$.
+Each core of $T$ is necessarily a dense subspace of $\Domain{T}$,
+but a dense subspace of $\Domain{T}$ need not be a core for $T$.
+
+\begin{lemma}{A Common Core for All Complex Lorentz Boosts}{common-core-for-complex-lorentz-boots}
+  Adopt the notation of the foregoing lemma. The linear subspace
+  \begin{equation*}
+    \mathcal{D}_0 = \Span \braces{\ran g(K) \vcentcolon g \in \schwartz{\RR}} 
+  \end{equation*}
+  is a core for $V(z)$ for every $z \in \CC$.
+\end{lemma}
+
+\begin{proof}
+  xxx
+\end{proof}
+
+\subsection{Application to the Energy Density}
+
+\bluetext{Achtung: Dieser Abschnitt ist noch roh, lückenhaft und enthält inkonsistente Notation und falsche Aussagen.}
+The following three Lemmas are variations of the arguments
+brought forward by~\citeauthor{Bisognano1975} in their proof of \cref{theorem:bisognano-wichmann}.
+The main difference is that we state xxx and xxx as operator identities without reference to a field operator,
+and proof xxx for arbitrary Lorentz-covariant operator-valued distributions
+rather than products of field operators.
+This generalization is necessary for the application to the energy density.
+In addition, we provide in \cref{chapter:convolution} a complete proof of the convolution formula for vector-valued distributions.
+
+Roughly speaking, the following Lemma asserts that a translation by a complex vector
+followed by a suitable imaginary boost is again a complex translation.
+
+\begin{lemma}{}{}
+Let $z = x + iy \in \OpenForwardTube$ with $x,y$ real, and suppose $y^3=y^4=0$.
+  \begin{enumerate}
+    \item If $x \in \rightwedge$, then for all $s \in [0,1/4]$ 
+      \begin{equation*}
+        \Lambda(is) z \in \OpenForwardTube, \qquad
+        \ran U(z) \subset \dom V(is), \qquad 
+  V(is) U(z) = U \parens[\big]{\Lambda(is) z}.
+      \end{equation*}
+    \item If $x \in \leftwedge$, then the above holds for all $s \in [0,-1/4]$.
+  \end{enumerate}
+\end{lemma}
+\nomenclature[dom]{$\dom T$}{domain of the operator $T$\nomnorefpage}
+\nomenclature[ran]{$\ran T$}{range of the operator $T$\nomnorefpage}
+
+\begin{proof}
+  xxx
+
+  \noindent\begin{minipage}{0.5\textwidth}
+  The vector-valued function $\CC \ni s \mapsto \Lambda(is) z$ is entire analytic.
+  In particular it is continuous, and we have shown that it maps the compact subset $[0,1/4]$ into the open set $\OpenForwardTube$.
+  This implies that there exists a connected open neighborhood $N \subset \CC$ of $[0,1/4]$ such that $\Lambda(is) z \in \OpenForwardTube$ for all $s \in N$.
+
+  \end{minipage} \hfill
+  \begin{minipage}{0.45\textwidth}
+    \begin{center}
+    \begin{tikzpicture}[baseline=10]
+      \draw[->] (-1,0) -- (3,0) node[right] {\footnotesize$\Real s$};
+      \draw[->] (0,-1) -- (0,2) node[left] {\footnotesize$\Imag s$};
+      \draw[faunat,thick,{Parenthesis[]}-{Parenthesis[]}] (0,-0.5) -- (0,0.5);
+      \draw[thick,{Bracket[]}-{Bracket[]}] (-0.3pt,0) -- (2,0);
+      \draw (1,1) node {$N$};
+      \draw (2,0) node[above] {\footnotesize$\tfrac{1}{4}$};
+      \draw plot [smooth cycle] coordinates {(2.5,0) (2,1) (1,0.7) (0.3,1) (-0.5,0.7) (-0.3,-0.7) (1.6,-0.7)};
+    \end{tikzpicture}
+    \end{center}
+  \end{minipage}
+
+  Let $\xi \in \hilb{F}$ be arbitrary, and let $\eta$ be in the common dense domain $\mathcal{D}_0$ of the operators $V(is)$ from \cref{lemma:common-core-for-complex-lorentz-boots}.
+  Then the function $f_1(s) = \innerp{V(is)^* \eta}{U(z) \xi}$ is well-defined, and entire analytic by Lemma xxx.
+  The function $f_2(s) = \innerp{\eta}{U(\Lambda(is) z) \xi}$ is analytic on $N$, by \cref{proposition:analyticity-complex-translations}.
+  By Lemma xxx, $f_1$ and $f_2$ agree in an open real neighborhood $is$.
+  Since $N$ is an open neighborhood of $0$, there is an $\epsilon >0$ such that $i(-\epsilon,\epsilon) \subset N$.
+  It follows that $f_1 \equiv f_2$ on $N$.
+  core \ldots
+\end{proof}
+
 \begin{lemma}{}{}
-  Suppose $A$ is a selfadjoint operator on some Hilbert space $\hilb{H}$.
-  For all complex numbers $z$ define a closed normal operator $V(z) = e^{izA}$ by means of functional calculus.
-  Let $g$ be a xxx function. Then the range of the bounded operator $g(A)$ is contained in the domain of $V(z)$ for all $z$, and
+  Let $x \in \rightwedge$ 
+\begin{equation*}
+  \stronglim_{\varepsilon \downarrow 0} V(i/4) U(x+i \varepsilon e_0)
+  = U \parens[\big]{V(i/4)x} 
+  = \stronglim_{\varepsilon \downarrow 0} V(-i/4) U(\mathcal{J}x+i \varepsilon e_0)
+\end{equation*}
+\end{lemma}
+
+\begin{proof}
+  xxx
+\end{proof}
+
+\begin{lemma}{}{}
+  Suppose that $u$ is a covariant operator-valued tempered distribution.
+  Let $f \in \schwartz{M}$ with $\supp f \subset \rightwedge$, and
+  let $g \in \schwartz{M}$ be arbitrary. Then
   \begin{equation*}
-    V(z) g(A) = \int e^{iz \lambda} g(\lambda) dE_A(\lambda).
+    V(i/2) g(K) u(f) \FockVacuum = g(K) u(f_{\mathcal{J}}) \FockVacuum
   \end{equation*}
 \end{lemma}
 
-\subsection{A Convolution Theorem for Vector-Valued Tempered Distributions}
+Here, $K$ is the infinitesimal generator of the group $t \mapsto V(t)$ of real Lorentz boosts,
+$\FockVacuum$ is the Fock vacuum, and $\mathcal{J}$ is the Lorentz transformation given by the diagonal matrix $\diag(-1,-1,1,1)$.
 
-\blockcquote{Bisognano1975}{%
-  The extension to vector-valued tempered distributions is trivial.
-}
 
+\begin{proof}
+  xxx
+\end{proof}
+
+\begin{equation*}
+  \Delta^{-1/2} g(K) \energydensity(f) \FockVacuum = g(K) \energydensity(f^J) \FockVacuum
+\end{equation*}
+
+Die Anwendung auf die Energiedichte $\energydensity$:
+
+\begin{proposition}{}{main-result}
+  Suppose $W \subset M$ is any wedge domain, with associated modular operator $\Delta_W$ and modular Hamiltonian $K_W$.
+  Let $f \in \schwartz{M}$ with $\supp f \subset W$, and
+  let $h \in \schwartz{M}$ be arbitrary. Then
+  \begin{equation*}
+    \norm{\Delta_W^{-1/2} h(K_W) \energydensity(f) \FockVacuum}
+    = \norm{h(K) \energydensity(f_{\mathcal{J}g}) \FockVacuum},
+  \end{equation*}
+  where $K$ is the modular Hamiltonian of the right wedge $\rightwedge$,
+  and $g$ is any element of $\RestrictedPoincareGroup$ such that $W = g \rightwedge$,
+  and $\mathcal{J} = \diag(-1,-1,1,1)$.
+\end{proposition}
+In der Ungleichung aus~\cite{Much2022} ist $h$ eine Gauß-Funktion.
+
+\section{Calculating Gaussians of the Modular Hamiltonian}
 
+coming soon\ldots
 
 \chapterbib
 \cleardoublepage
-- 
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