From e1a26e4528eb7b9c2f462562c8265cf963f34dfb Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Thu, 20 Jun 2024 03:56:45 +0200
Subject: weiter

---
 much.tex | 150 ++++++++++++++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 109 insertions(+), 41 deletions(-)

(limited to 'much.tex')

diff --git a/much.tex b/much.tex
index a7f9bbf..249dc73 100644
--- a/much.tex
+++ b/much.tex
@@ -43,7 +43,7 @@ Poincaré covariance
 
 \begin{definition}{Von Neumann Algebra of Local Observables}{}
   \begin{equation*}
-    \localalg{\spacetimeregion{O}} = \braces{b(\varphi(f)) \mid \text{$b$ bounded}, f \in \realschwartz{M}, \supp f \subset \spacetimeregion{O}}''
+    \localalg{\spacetimeregion{O}} = \Set{b(\varphi(f)) \mid \text{$b$ bounded}, f \in \realschwartz{M}, \supp f \subset \spacetimeregion{O}}''
   \end{equation*}
 \end{definition}
 
@@ -272,17 +272,17 @@ Thus, modular theory
 
 \section{The Geometric Action of the Modular Operator Associated With a Wedge Domain}
 
-\begin{definition}{Right and Left Wedge, General Wedges}{}
+\begin{definition}{Right and Left Wedge, General Wedges}{wedge}
   The \emph{right wedge}\index{wedge!right}\nomenclature[WR]{$\rightwedge$}{right wedge}
   and \emph{left wedge}\index{wedge!left}\nomenclature[WL]{$\leftwedge$}{left wedge}
   in Minkowski space $M$ are the open subsets
   \begin{equation*}
-    \rightwedge \defequal \braces[\big]{x \in M \vcentcolon x^1 > \abs{x^0}} 
+    \rightwedge \defequal \Set[\big]{x \in M \given x^1 > \abs{x^0}} 
     \quad \text{and} \quad
-    \leftwedge \defequal \braces[\big]{x \in M \vcentcolon x^1 < -\abs{x^0}}.
+    \leftwedge \defequal \Set[\big]{x \in M \given x^1 < -\abs{x^0}}.
   \end{equation*}
 We say that a spacetime region $W \subset M$ is a \emph{wedge}\index{wedge}
-  if there exists an element $g$ of the Poincaré group
+  if there exists an element $g$ of the full Poincaré group
   such that $W = g \rightwedge$.
 \end{definition}
 
@@ -443,7 +443,7 @@ That this is generally true is the statement of the following Lemma.
     \frac{1}{2\pi i} \int_{\alpha}^{\beta} \bracks{R_A(\lambda + i \varepsilon)  - R_A(\lambda - i \varepsilon)} d\lambda 
     = E_A \parens[\big]{\bracks{a,b}}
   \end{equation*}
-  for all $a \in \RR \cup \braces{-\infty}$, $b \in \RR \cup \braces{\infty}$.
+  for all $a \in \RR \cup \Set{-\infty}$, $b \in \RR \cup \Set{\infty}$.
   Observe that $\rho(A) = \rho(U\! @AU^*)$ and that for each (common) regular value $\lambda$ we have
   \begin{equation*}
     R_{U\! @AU^*}(\lambda) = U R_A(\lambda) @ U^*\!.
@@ -492,6 +492,8 @@ In other words, $K_{\spacetimeregion{O}}$ is the unique selfadjoint operator suc
 \end{proposition}
 \todo{domain, proof}
 
+\bluetext{Maybe this is simpler in Rindler coordinates...}
+
 \section{Complex Lorentz Transformations}
 
 The main result of this section is \cref{proposition:main-result}.
@@ -504,9 +506,9 @@ of complex Minkowski space $M+iM \cong \CC^4$ with respect to the inner product
 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 $(z,\Lambda)$.
+discriminated by the sign of $\det \Lambda \in \Set{\pm 1}$ 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 $\ProperOrthochronousPoincareGroup$,
+Each of the two following sections deals with a subgroup $G$ of $\smash{\ProperOrthochronousPoincareGroup}$,
 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}
@@ -540,7 +542,7 @@ and we impose the so-called \emph{spectrum condition}
   \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}.
+where $\ClosedForwardCone \defequal \Set{a \in \RR^4 \given a \cdot a \ge 0, a^0 \ge 0}$ 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.\ $\supp(E) \subset \ClosedForwardCone$.
 
@@ -570,7 +572,7 @@ Observe that the set $\ClosedForwardTube$ is closed under vector addition and th
   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 $\exp(i(w+z) \cdot k) = \exp(iw \cdot k) \exp(iz \cdot k)$
+  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}
 
@@ -620,6 +622,19 @@ Observe that the set $\ClosedForwardTube$ is closed under vector addition and th
 
 \todo{Explain what it means for an operator-valued function of several complex variables to be analytic.}
 
+\begin{lemma}{}{complex-translation}
+  Let $g = (b,\Lambda)$ be a proper orthochronous Poincaré transform with $b \in \OpenForwardCone$.
+  Then, for all $z \in \OpenForwardTube$
+  \begin{equation*}
+    gz \in \OpenForwardTube \qquad
+    U(g) U(z) = U(gz).
+  \end{equation*}
+\end{lemma}
+
+\begin{proof}
+  \bluetext{Edge of the Wedge}
+\end{proof}
+
 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}
@@ -695,7 +710,7 @@ Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$,
   %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}{}{}
+\begin{proposition}{}{prp}
   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},
@@ -711,6 +726,27 @@ Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$,
   Dann folgt die Behauptung wohl mit Edge of the Wedge~\cite[Theorem 2-17]{Streater1964}}
 \end{proof}
 
+\begin{corollary}{}{}
+  Let $u$ be a covariant operator-valued tempered distribution,
+  and let $f \in \schwartz{\RR^4}$ be a test function. Then we have,
+  \begin{equation*}
+    U(z) u(f) \FockVacuum = \int dx \, f(x) \, u(d_{z+x}) \FockVacuum \qquad \forall z \in T_+.
+  \end{equation*}
+\end{corollary}
+
+\begin{proof}
+  The convolution formula \cref{proposition:vector-valued-convolution-formula} applied to the vector-valued distribution defined by $f \mapsto \alpha(f) = u(f) \FockVacuum$ yields
+  \begin{equation*}
+    (\alpha * \tilde{d}_z)(f) = \int dx \, f(x) \, \alpha(\tau_x d_z)
+  \end{equation*}
+  Using \cref{proposition:prp}, we calculate
+  \begin{equation*}
+    (\alpha * \tilde{d}_z)(f) = \alpha(d_z * f) = \alpha(f_z) = u(f_z) \FockVacuum = U(z) u(f) \FockVacuum.
+  \end{equation*}
+  It is easily seen by Fourier transformation that $\tau_x d_z = d_{x+z}$.
+  Hence, $\alpha(\tau_x d_z) = u(d_{x+z}) \FockVacuum$.
+\end{proof}
+
 \subsection{Complex Lorentz Boosts}
 
 The Lorentz boosts $\Lambda(t)$ given by
@@ -723,7 +759,6 @@ In particular, the vector-valued function $\CC \ni w \mapsto \Lambda(w) z$ is en
 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 \; \\
@@ -733,8 +768,9 @@ between the complex hyperbolic and trigonometric functions imply
   \end{pmatrix}
   \qquad \forall s \in \RR.
 \end{equation*}
-
-\begin{equation*}
+For later use, we give the action of $\Lambda(is)$ on a complex four-vector $x+iy$:
+\begin{equation}
+  \label{equation:pure-imaginary-lorentz-boost}
   \Lambda(is) (x+iy) =
   \begin{pmatrix}
     \cos(2 \pi @ s) x^0 - \sin(2 \pi @ s) y^1 \\
@@ -749,7 +785,7 @@ between the complex hyperbolic and trigonometric functions imply
     y^2 \\
     y^3 
   \end{pmatrix}
-\end{equation*}
+\end{equation}
 
 We
 
@@ -814,49 +850,75 @@ 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}} 
+    \mathcal{D}_0 = \Span \Set{\ran g(K) \given g \in \schwartz{\RR}} 
   \end{equation*}
   is a core for $V(z)$ for every $z \in \CC$.
 \end{lemma}
 
 \begin{proof}
-  xxx
+  $M_n = \bracks{-n,n}$
 \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
+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
+The main difference is that we state \cref{lemma:biso1} and \cref{lemma:biso2} as operator identities without reference to a field operator,
+and proof \cref{lemma:biso3} 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
+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{lemma}{}{biso1}
+Let $z = x + iy \in \OpenForwardTube$ with $x,y$ real, and suppose $y^2=y^3=0$.
   \begin{enumerate}
-    \item If $x \in \rightwedge$, then for all $s \in [0,1/4]$ 
+    \item If $x \in \rightwedge$, then for all $s \in [0,\tfrac{1}{4}]$ 
       \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 $s \in [0,-1/4]$.
+    \item If $x \in \leftwedge$, then the above holds for all $s \in [0,-\tfrac{1}{4}]$.
   \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
+  The result of applying $\Lambda(is)$ for some $s \in \RR$ to a complex four-vector $x+iy$
+  has been given in~\eqref{equation:pure-imaginary-lorentz-boost}.
+  To show that this vector lies in the open forward tube,
+  we need to verify that its imaginary part lies in the open forward cone $\OpenForwardCone$.
+  By definition, a real four-vector $a$ lies in $\OpenForwardCone$ if and only if
+  $a^0 > 0$ and $a \cdot a > 0$. If $a^2 = a^3 = 0$, then
+  these conditions are easily seen to be equivalent to the conditions
+  $a^0 > 0$ and $a^0 \mp a^1 > 0$ (for both sign choices).
+
+  Since the second and third components of the imaginary part of~\eqref{equation:pure-imaginary-lorentz-boost},
+  $y^2$ and $y^3$, vanish by assumption, it is sufficient to prove
+  \begin{equation}
+    \label{equation:inequalities}
+    \begin{aligned}
+      \sin(2 \pi s) x^1 + \cos(2 \pi s) y^0 &> 0 \ \text{and} \\
+      \sin(2 \pi s) \parens[\big]{x^1 \mp x^0} + \cos(2 \pi s) \parens[\big]{y^0 \mp y^1} &> 0.
+    \end{aligned}
+  \end{equation}
+  The assumption $x \in \rightwedge$ implies that
+  $x^1 > 0$ and $x^1 \mp x^0 > 0$,
+  by \cref{definition:wedge}.
+  The assumptions $y \in \OpenForwardCone$ and $y^2 = y^3 = 0$ imply that
+  $y^0 > 0$ and $y^0 \mp y^1 > 0$,
+  by the argument in the foregoing paragraph.
+  So, all we need to do to ensure~\eqref{equation:inequalities} holds,
+  is choose $s$ such that both $\sin(2 \pi s)$ and $\cos(2 \pi s)$ are nonnegative.
+  (Then, at least one of these will be positive.)
+  Clearly, this is true for all $s \in \bracks{0,\tfrac{1}{4}}$.
 
   \noindent\begin{minipage}{0.5\textwidth}
-  The vector-valued function $\CC \ni s \mapsto \Lambda(is) z$ is entire analytic.
+  \hspace{\parindent} 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 $s \in N$.
 
@@ -878,26 +940,36 @@ Let $z = x + iy \in \OpenForwardTube$ with $x,y$ real, and suppose $y^3=y^4=0$.
   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
+  By \cref{lemma:complex-translation}, $V(is) U(z) = U(\Lambda(is) z)$ if $is \in \RR$.
+  Hence, $f_1$ and $f_2$ agree in an open real neighborhood of $is$.
+  Now the Identity Principle implies $f_1 \equiv f_2$ on $N$.
+  Since this holds for all $\eta$ in $\mathcal{D}_0$,
+  which is a core for $V(is)^{**} = V(is)$,
+  we conclude that $U(z) \xi$ lies in the domain of $V(is)$,
+  and $V(is) U(z) \xi = U(\Lambda(is) z) \xi$.
+  As $\xi$ was arbitary, the proof is complete.
 \end{proof}
 
-\begin{lemma}{}{}
-  Let $x \in \rightwedge$ 
+Remember that $\mathcal{J} = \Lambda(i/2) = \diag(-1,-1,1,1)$.
+
+\begin{lemma}{}{biso2}
+  Let $x \in \rightwedge$, and let $e_0 = (1,0,0,0)$ be the forward timelike unit vector. Then
 \begin{equation*}
   \stronglim_{\varepsilon \downarrow 0} V(i/4) U(x+i \varepsilon e_0)
-  = U \parens[\big]{V(i/4)x} 
+  = U \parens[\big]{\Lambda(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
+  The first identity follows from \cref{lemma:biso2}(i)
+  and the strong continuity of $z \mapsto U(z)$ on $\ClosedForwardTube$ (\cref{proposition:analyticity-complex-translations}).
+  For the second identity, note that $\mathcal{J} x \in \leftwedge$ and
+  apply \cref{lemma:biso2}(ii), then use $\Lambda(-i/4) \mathcal{J} = \Lambda(i/4)$.
 \end{proof}
 
-\begin{lemma}{}{}
+\begin{lemma}{}{biso3}
   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
@@ -909,9 +981,8 @@ Let $z = x + iy \in \OpenForwardTube$ with $x,y$ real, and suppose $y^3=y^4=0$.
 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 $\diag(-1,-1,1,1)$.
 
-
 \begin{proof}
-  xxx
+  a
 \end{proof}
 
 \begin{equation*}
@@ -938,7 +1009,4 @@ In der Ungleichung aus~\cite{Much2022} ist $h$ eine Gauß-Funktion.
 
 coming soon\ldots
 
-\chapterbib
-\cleardoublepage
-
 % vim: syntax=mytex
-- 
cgit v1.2.3-70-g09d2

