weiterHEADmaster

11 files changed, 392 insertions, 106 deletions

diff --git a/commutatortheorem.tex b/commutatortheorem.tex
index 9f6384f..fa8cc59 100644
--- a/commutatortheorem.tex
+++ b/commutatortheorem.tex
@@ -16,6 +16,4 @@
 \cite{ReedSimon2} 
 \cite{Nelson1972} 
 
-\chapterbib
-
 %vim: syntax=mytex
diff --git a/convolution.tex b/convolution.tex
index d3fb8bc..eee6e16 100644
--- a/convolution.tex
+++ b/convolution.tex
@@ -1,27 +1,32 @@
 \chapter{A Convolution Formula for Vector-Valued Tempered Distributions}
 \label{chapter:convolution}
-
 \blockcquote{Bisognano1975}{%
   The extension to vector-valued tempered distributions is trivial.
 }
-Recall that the class $\schwartz{\RR^n}$ of complex-valued Schwartz functions on $\RR^n$
+Recall that the class $\SchwartzFunctions{\RR^n}$ of complex-valued Schwartz functions on $\RR^n$
 is closed under convolution, a operation that assigns to functions $f$ and $g$ a third one, $f * g$,
 given by
 \begin{equation*}
   (f*g)(x) = \int f(x-y) g(y) \, dy
   \qquad x \in \RR^n.
 \end{equation*}
-
-\begin{definition}{Convolution of a Distribution with a Test Function}{}
-  Let $u \in \tempdistrib{\RR^n}$ be tempered distribution and
-  let $f \in \schwartz{\RR^n}$ be a Schwartz test function.
+\begin{definition}{Convolution of a Distribution with a Test Function}{convolution-distribution-test-function}
+  Let $u \in \TemperedDistributions{\RR^n}$ be a tempered distribution and
+  let $f \in \SchwartzFunctions{\RR^n}$ be a Schwartz test function.
   Then the \emph{convolution} of $u$ with $f$ is
-  the tempered distribution $u * f \in \tempdistrib{\RR^n}$ defined by
+  the tempered distribution $u * f \in \TemperedDistributions{\RR^n}$ defined by
   \begin{equation*}
-    (u * f)(g) \defequal u(\tilde{f} * g) \qquad g \in \schwartz{\RR^n},
+    (u * f)(g) \defequal u(\tilde{f} * g) \qquad g \in \SchwartzFunctions{\RR^n},
   \end{equation*}
   where $\tilde{f}(x) = f(-x)$ for all $x \in \RR^n$.
 \end{definition}
+The motivation and justification for this definition is provided by the adjoint identity
+\begin{equation*}
+  \int (h * f)(x) \, g(x) \, dx =
+  \int h(x) \, (\tilde{f} * g)(x) \, dx
+\end{equation*}
+holding for all $f,g,h \in \SchwartzFunctions{\RR^n}$.
+
 It is well-known that the convolution can be expressed by the integral
 \begin{equation*}
   (u * f)(g) = \int u(\tau_x \tilde{f}@@) g(x) \, dx
@@ -30,34 +35,269 @@ emphasizing its character of a smoothing operation.
 The purpose of this appendix is to state and prove
 a vector-valued version of this formula.
 
-Let $X$ be a complex Banach space.
-Denote by $C^{\infty}(\RR^n,X)$  the vector space of all functions $f : \RR^n \to X$
+We proceed to develop a generalization of the Bochner integral
+for functions valued in a separable Fréchet space,
+as this will facilitate our proof of the convolution formula.
+
+We consider a $\sigma$-finite measure space $(X,\SigmaAlgebra{A},\mu)$,
+a separable Fréchet space $Y$ (over $\CC$) and the task is
+to define the integral of functions $f \vcentcolon X \to Y$.
+Recall that a measure space is said to be \emph{$\sigma$-finite}
+if it can be exhausted by a countable number of measurable subsets of finite measure.
+By \emph{Fréchet space} we mean a complete Hausdorff locally convex (topological vector) space
+which possesses countable neighborhood bases.
+We will make use of a countable family $P@@$ of seminorms that generates the topology of $@@Y$.
+A topological space is called \emph{separable} if it contains a countable dense subset.
+
+A function $f \vcentcolon X \to Y$ will be called \emph{simple}
+if it is of the form $\sum_{i=1}^n \chi_{A_i} y_i$
+where $n \in \NN$, $A_i \in \SigmaAlgebra{A}$ with $\mu(A_i) < \infty$, and $y_i \in Y$.
+Naturally, the \emph{integral} of $f$ is defined to be the vector $\int f = \sum_{i=1}^n \mu(A_i) y_i \in Y$.
+We say that a function $f \vcentcolon X \to Y$ is \emph{strongly measurable}
+if it is the $\mu$-almost everywhere pointwise limit of simple functions.
+
+\begin{definition}{Generalized Bochner Integral}{}
+  Suppose $(X,\SigmaAlgebra{A},\mu)$ is a $\sigma$-finite measure space,
+  and $Y@@$ is a separable Fréchet space
+  whose topology is generated by a family $P@@$ of seminorms.
+A strongly measurable function $f \vcentcolon X \to Y$ is called \emph{(generalized Bochner) integrable}
+if there exists a sequence $(f_n)$ of simple functions such that
+\begin{equation}
+  \label{equation:bochner-integrable}
+  \lim_{n \to \infty} \int_X p \circ (f_n - f) \, d\mu = 0
+  \qquad \forall p \in P.
+\end{equation}
+In this case, the \emph{(generalized Bochner) integral} of $f$ is defined by
+\begin{equation}
+  \label{equation:bochner-integral}
+  \int_X f \ d\mu \defequal
+  \lim_{n \to \infty} \int_X f_n \, d\mu.
+\end{equation}
+\end{definition}
+This definition needs justification.
+First, for the integral in~\eqref{equation:bochner-integrable} to be meaningful,
+the functions $p \circ (f_n - f)$ must be $\mu$-measurable.
+Since $f$ is strongly measurable, there exists simple functions $s_k$ such that $f (x) = \lim_{k \to \infty} s_k(x)$ for almost all $x \in X$.
+The continuity of $p$ implies that $p \circ (f_n - f)$ is the almost everywhere limit simple scalar functions, namely $p \circ (f_n - s_k)$,
+and as such must be measurable.
+%We will defer this question for now.
+Second, we have to verify that the limit in~\eqref{equation:bochner-integral} exists
+and is independent of the particular sequence $(f_n)$.
+Remember that the sets $U_{F,\epsilon} = \braces{y \in Y \vcentcolon p(y) < \epsilon \forall p \in F}$,
+where $F \subset P$ is finite and $\epsilon > 0$,
+form a neighborhood basis for $0 \in Y$.
+Consider any such $U_{F,\epsilon}$.
+Then, for all $p \in F$ and $m,n \in \NN$
+\begin{equation*}
+  p \parens*{\smallint f_n - \smallint f_m}
+  %= p \parens*{\smallint (f_n - f_m)}
+  \le \smallint p \circ (f_n - f_m)
+  \le \smallint p \circ (f - f_n) +  \smallint p \circ (f - f_m).
+\end{equation*}
+By~\eqref{equation:bochner-integrable} there exists $N_p \in \NN$ such that
+$p \parens*{\int f_n - \int f_m} < \epsilon$ for all $m,n \ge N_p$.
+If we set $N = \max \braces{N_p \vcentcolon p \in F}$, then $\int f_n - \int f_m \in U_{F,\epsilon}$ for all $m,n \ge N$.
+This shows that $(\int f_n)$ is a Cauchy sequence in the topological vector space $Y$.
+Now the existence of a limit point follows from the completeness of $Y$.
+It is unique because the topology is Hausdorff.
+
+
+\begin{theorem}{Generalized Bochner Integrability Criterion}{generalized-bochner}
+  Suppose $X$ is a $\sigma$-finite measure space,
+  and $Y@@$ is a separable Fréchet space
+  whose topology is generated by a countable family $P@@$ of seminorms.
+  A function $f \vcentcolon X \to Y@@$ is generalized Bochner integrable if and only if it is strongly measurable and
+  \begin{equation*}
+    \int_X p \circ f \ d\mu < \infty
+    \qquad \forall p \in P.
+  \end{equation*}
+\end{theorem}
+
+\begin{proof}
+  Since $X$ is $\sigma$-finite,
+  $X = \bigcup_{m=1}^{\infty} X_m$ with $\mu(X_m) < \infty$ and $X_m \subset X_{m+1}$.
+  Clearly, $f$ is the pointwise limit of
+  the functions $f_m = f \chi_{X_m}$, as $m \to \infty$.
+  Let $(p_i)_{i \in \NN}$ be an enumeration of the countable family $P$ of seminorms
+  generating the locally convex topology on $Y$.
+  Since $Y$ is separable,
+  there is a dense sequence $(y_j)_{j \in \NN}$ of vectors in $Y$.
+  For $n,j \in \NN$ let
+  \begin{gather*}
+    C_{nj} = y_j + U_{\braces{p_1, \ldots, p_n},1/n}
+    = \braces[\big]{y \in Y \vcentcolon p_i(y - y_j) \le \tfrac{1}{n} \forall i=1,\ldots,n} \\
+    B_{nj} = f^{-1} C_{nj} \qquad
+    A_{nj} = B_{nj} \setminus \bigcup_{k=1}^{j-1} B_{nk}
+  \end{gather*}
+  Observe that for each fixed $n$ the sets $C_{nj}$ cover $Y$,
+  the sets $B_{nj}$ cover $X$ and
+  the sets $A_{nj}$ partition $X$.
+  Moreover, the sets $B_{nj}$, and consequently $A_{nj}$, are $\mu$-measurable
+  because the functions $x \mapsto p_i \parens[\big]{f(x) - y_j}$ are $\mu$-measurable.
+  Then, the functions
+  \begin{equation*}
+    f_{mn} = \sum_{j=1}^{\infty} \chi_{X_m \cap A_{nj}} y_j
+  \end{equation*}
+  satisfy $p_i(f(x) - f_{mn}(x)) \le \frac{1}{n}$ for all $x \in X$ when $i \le n$.
+  Hence, $p_i \circ f_{mn} \le p_i \circ f + \frac{1}{n}$.
+  Since $f_{mn}$ is supported in $X_m$, a set of finite measure, and $\int p_i \circ f < \infty$,
+  we conclude $\int p_i \circ f_{mn} < \infty$ for all $i \le n$.
+  For each $(m,n) \in \NN^2$ choose $J(m,n)$ so large that
+  \begin{equation*}
+    \int_{\bigcup_{j=J(m,n)+1}^{\infty} X_m \cap A_{nj}} p_i \circ f_{mn} < \frac{\mu(X_m)}{n}
+    \qquad \forall i=1,\ldots,n.
+  \end{equation*}
+  The functions
+  \begin{equation*}
+    s_{mn} = \sum_{j=1}^{J(m,n)} \chi_{X_m \cap A_{nj}} y_j
+  \end{equation*}
+  are simple and satisfy
+  \begin{equation*}
+    \int p_i \circ (f_m - s_{mn})
+    \le \int p_i \circ (f_m - f_{mn}) + \int p_i \circ (f_{mn} - s_{mn})
+    < \frac{2\mu(X_m)}{n}
+  \end{equation*}
+  for $n \ge i$.
+  It follows that
+  \begin{equation*}
+    \lim_{n \to \infty} \int p_i \circ (f_m - s_{mn}) = 0
+    \qquad \forall i \in \NN.
+  \end{equation*}
+  For each $m \in \NN$ choose $N(m)$ so large that
+  \begin{equation*}
+    \int p_i \circ (f_m - s_{mN(m)}) < \frac{1}{m} 
+    \qquad \forall i=1,\ldots,m.
+  \end{equation*}
+  and therefore
+  \begin{equation*}
+    \int p_i \circ (f - s_{m N(m)})
+    \le \frac{1}{m} + \int p_i \circ (f - f_m) 
+  \end{equation*}
+  by the triangle inequality.
+  %This implies
+  %\begin{equation*}
+    %\lim_{n \to \infty} \int p_i \circ (f_m - s_{mN(m)}) = 0
+    %\qquad \forall i \in \NN.
+  %\end{equation*}
+  For each $i \in \NN$ the increasing sequence $(p_i \circ f_{m})_m$ of positive real-valued measurable functions
+  converges pointwise to the function $p_i \circ f$,
+  which is by hypothesis is integrable.
+  By Dominated Convergence, $\int p_i \circ (f-f_m) \to  0$, as $m \to \infty$.
+  \begin{equation*}
+    \lim_{m \to \infty} \int p_i \circ (f - s_{m N(m)}) = 0
+    \qquad \forall i \in \NN.
+  \end{equation*}
+  This proves that $f$ is generalized Bochner integrable.
+\end{proof}
+
+\begin{theorem}{}{integral-commutes-with-operator}
+  Suppose $X$ is a $\sigma$-finite measure space.
+  Let $Y$ and $Z$ be separable Fréchet spaces,
+  and let $T \vcentcolon Y \to Z$ be a continuous linear operator.
+  If $f \vcentcolon X \to Y$ is generalized Bochner integrable,
+  then $T \circ f \vcentcolon X \to Z$ is generalized Bochner integrable, and
+  \begin{equation*}
+    \int T \circ f =
+    T \! \int \! f.
+  \end{equation*}
+\end{theorem}
+
+\begin{proof}
+  Clearly, the composition $T \circ f$ is strongly measurable
+  because $T$ is continuous and $f$ is strongly measurable.
+  Suppose that the locally convex topologies on $Y$ and $Z$
+  are generated by the seminorm families $P$ and $Q$, respectively.
+  If $q \in Q$, then the fact that $T$ is continuous and linear implies that
+  there exists a finite subset $F \subset P$ and a constant $M \ge 0$
+  such that $q \circ T \le M \max_{p \in F} p$.
+  If $(f_n)$ is a sequence of simple functions such that $\int p \circ (f - f_n) \to 0$, 
+  then $\int q \circ T \circ (f-f_n) \to 0$.
+  This shows that $T \circ f$ is generalized Bochner integrable, and
+  \begin{equation*}
+    \int T \circ f = \lim_{n \to \infty} \int T \circ f_n
+    = T \lim_{n \to \infty} \int f_n = T \int f.\qedhere
+  \end{equation*}
+  %By \cref{theorem:generalized-bochner},
+  %it follows that $\int q \circ T \circ f < \infty$,
+\end{proof}
+
+We now return to tempered distributions.
+Denote by $\TestFunctions{\RR^n}$  the vector space of all functions $f \vcentcolon \RR^n \to \CC$
 such that the derivatives $\partial^{\alpha} f$ exist and are continuous for all multi-indices $\alpha \in \NN^n$.
-We define the space $\schwartz{\RR^n,X}$ of \emph{$X$-valued Schwartz functions} to be the vector space
+Recall that the space $\SchwartzFunctions{\RR^n}$ of \emph{Schwartz functions} is defined to be the vector space
 \begin{equation*}
-  \schwartz{\RR^n,X} \defequal \braces{f \in C^{\infty}(\RR^n,X) \vcentcolon \norm{f}_{\alpha,\beta} < \infty \forall \alpha,\beta \in \NN^n}
+  \SchwartzFunctions{\RR^n,X} \defequal \braces{f \in \TestFunctions{\RR^n} \vcentcolon \norm{f}_{\alpha,\beta} < \infty \ \forall \alpha,\beta \in \NN^n}
 \end{equation*}
 equipped with the locally convex topology induced by the family of seminorms
 \begin{equation*}
-  \norm{f}_{\alpha,\beta} = \sup_{x \in \RR^n} \abs{x^{\alpha}} \norm{\partial^{\beta} f(x)}_X.
+  \norm{f}_{\alpha,\beta} = \sup_{x \in \RR^n} \abs{x^{\alpha}} \abs{\partial^{\beta} f(x)}.
 \end{equation*}
-We define the space $\tempdistrib{\RR^n,X}$ of \emph{$X$-valued tempered distributions} to be the vector space
+It is well known that the Schwartz space is a separable Fréchet space.
+Now let $X$ be any separable Fréchet space.
+We define the space $\TemperedDistributions{\RR^n\!,X}$ of \emph{$X$-valued tempered distributions} to be the vector space
 \begin{equation*}
-  \tempdistrib{\RR^n,X} \defequal \BoundedLinearOperators[\big]{\schwartz{\RR^n},X}.
+  \TemperedDistributions{\RR^n\!,X} \defequal \ContinousLinearOperators[\big]{\SchwartzFunctions{\RR^n},X}.
 \end{equation*}
+of all continuous linear operators $\SchwartzFunctions{\RR^n} \to X$
 equipped with the bounded convergence topology.
+The convolution of a $X$-valued tempered distribution $v$ with a Schwartz function $f$
+is defined in the same way as in \cref{definition:convolution-distribution-test-function}, that is by
+  \begin{equation*}
+    (v * f)(g) \defequal v(\tilde{f} * g) \qquad g \in \SchwartzFunctions{\RR^n}.
+  \end{equation*}
 
-\begin{proposition}{Vector-Valued Convolution Formula}{}
-  Let $v \in \tempdistrib{\RR^n,X}$ be tempered distribution with values in a Banach space $X$, and
-  let $f \in \schwartz{\RR^n}$ be a Schwartz test function. Then one has
+\begin{proposition}{Vector-Valued Convolution Formula}{vector-valued-convolution-formula}
+  Let $v \in \TemperedDistributions{\RR^n\!,X}$ be a tempered distribution with values in a separable Fréchet space $X$, and
+  let $f \in \SchwartzFunctions{\RR^n}$ be a Schwartz test function. Then one has
   \begin{equation*}
-    (v * f)(g) = \int v(\tau_x \tilde{f}@@) g(x) \, dx \qquad g \in \schwartz{\RR^n}.
+    (v * f)(g) = \int v(\tau_x \tilde{f}@@) g(x) \, dx \qquad g \in \SchwartzFunctions{\RR^n}.
   \end{equation*}
 \end{proposition}
 
-Der Beweis ist in Arbeit ;)
+\begin{proof}
+  We fix a Schwartz function $g$, and consider the finite measure $\mu = \abs{g} \lambda$ on $\RR^n$,
+  where $\lambda(x) = dx$ is the Lebesgue measure.
+  We show that the mapping $x \mapsto \tau_x \tilde{f}$ is a generalized Bochner $\mu$-integrable function $\RR^n \to \SchwartzFunctions{\RR^n}$
+  using \cref{theorem:generalized-bochner}.
+  For all $\alpha,\beta \in \NN^n$ we see by substituting $x+y$ for $y$ that
+  \begin{equation*}
+    \norm{\tau_x \tilde{f}}_{\alpha,\beta} =
+    \sup_{y} \abs{y^{\alpha} \partial^{\beta} (\tau_x \tilde{f})(y)} =
+    \sup_{y} \abs{(x+y)^{\alpha} \partial^{\beta} \tilde{f}(y)}.
+  \end{equation*}
+  There exists constants $c_{\gamma \delta}$ with
+  $\abs{(x+y)^{\alpha}} \le \sum_{\gamma + \delta = \alpha} c_{\gamma \delta} \abs{x^{\gamma} y^{\delta}}$,
+  and it follows that
+  \begin{equation*}
+    \int \norm{\tau_x \tilde{f}}_{\alpha,\beta} \, d \mu(x) 
+    \le \sum_{\gamma + \delta = \alpha} c_{\gamma \delta} \norm{\tilde{f}}_{\delta,\beta} \int \abs{x^{\gamma}} g(x) \, dx < \infty 
+  \end{equation*}
+  because $g$ is Schwartz class.
+  Hence, $x \mapsto \tau_x \tilde{f}$ defines an integrable function.
+  
+  The mapping $v \vcentcolon \SchwartzFunctions{\RR^n} \to X$ is linear and continuous by definition.
+  By \cref{theorem:integral-commutes-with-operator},
+  the composite mapping $x \mapsto v(\tau_x \tilde{f})$ is a $\mu$-integrable function $\RR^n \to X$, and
+  \begin{equation}
+    \label{equation:general-bochner-appears}
+    \int v(\tau_x \tilde{f}) \, d\mu(x) = v \parens[\bigg]{\int \tau_x \tilde{f} \, d\mu(x)}
+  \end{equation}
+  For every fixed $y \in \RR^4$ the evaluation mapping $\ev_{\! @@y} \vcentcolon \SchwartzFunctions{\RR^4} \to \CC$, $h \mapsto h(y)$, clearly is continuous.
+  A second invocation of \cref{theorem:integral-commutes-with-operator} delivers
+  \begin{equation*}
+    \ev_{\! @@y} \parens[\bigg]{\int \tau_x \tilde{f} \, d\mu(x)} =
+    \int \ev_{\! @@y}(\tau_x \tilde{f}) \, d\mu(x) =
+    \int \tilde{f}(y-x) g(x) \, dx =
+    (\tilde{f} * g)(y) 
+  \end{equation*}
+  and the proof is complete.
+\end{proof}
+
+Let us point out that even in the special case that $X$ is a Banach space
+the integral on the right hand side of~\eqref{equation:general-bochner-appears}
+only has meaning as a generalized Bochner integral,
+since the integrand takes values in $\SchwartzFunctions{\RR^n}$,
+which is not a Banach space.
+We could not have performed this step with the ordinary Bochner integral.
 
 %\nomenclature[B]{$\BoundedLinearOperators{X,Y}$}{bounded linear operators from $X$ to $Y$\nomnorefpage}
-
-\chapterbib
-\cleardoublepage
diff --git a/distributions.tex b/distributions.tex
index fad319b..6bd290d 100644
--- a/distributions.tex
+++ b/distributions.tex
@@ -24,7 +24,3 @@ by
 \nocite{Friedlander1999}
 \nocite{Hoskins2005}
 \nocite{Hoskins2009}
-
-
-\chapterbib
-\cleardoublepage
diff --git a/fewstereveson.tex b/fewstereveson.tex
index cb1918b..27da7f2 100644
--- a/fewstereveson.tex
+++ b/fewstereveson.tex
@@ -5,7 +5,4 @@
 
 Goal: make~\cite{Fewster1998} rigorous based on the previous chapter
 
-\chapterbib
-\cleardoublepage
-
 % vim: syntax=mytex
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
diff --git a/preamble.tex b/preamble.tex
index 8479b31..fb67d0a 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -16,16 +16,15 @@
 %\usepackage{graphicx}
 \usepackage{tikz}
 \usepackage{tcolorbox}
-%\usepackage{wrapfig}
 \usepackage[style=ext-alphabetic]{biblatex}
 \usepackage[intoc,refpage]{nomencl}
 \usepackage{makeidx}
-\usepackage{idxlayout}
 \usepackage{hyperref}
 \usepackage{bookmark}
 \usepackage{hypdestopt}
 \usepackage[capitalize,nameinlink]{cleveref}
 %\usepackage{refcheck}
+%\usepackage{nag}
 
 % ---------- fontspec
 \setfontfamily\fausansoffice{FAUSansOffice}
@@ -88,8 +87,10 @@
 \DeclareMathOperator{\dom}{dom}
 \DeclareMathOperator{\ran}{ran}
 \DeclareMathOperator{\Span}{span}
+\DeclareMathOperator{\ev}{ev}
 % extend amsmath's proof environment
-\NewDocumentEnvironment{myproof}{Ob}{\IfNoValueTF{#1}{\begin{proof}}{\begin{proof}[\proofname\ of \Cref{#1}]}}{\end{proof}}
+\newenvironment{myproof}[1]{\proof[\proofname\ of \Cref{#1}]}{\endproof}
+%\NewDocumentEnvironment{myproof}{Ob}{\IfNoValueTF{#1}{\begin{proof}}{\begin{proof}[\proofname\ of \Cref{#1}]}}{\end{proof}}
 
 % ---------- mathtools
 \DeclarePairedDelimiter\abs{\lvert}{\rvert}
@@ -99,6 +100,8 @@
 \DeclarePairedDelimiter\braces{\lbrace}{\rbrace}
 \DeclarePairedDelimiter\angles{\langle}{\rangle}
 % TODO set macro with proper spacing
+\newcommand\given{\vcentcolon}
+\DeclarePairedDelimiterX\Set[1]{\lbrace}{\rbrace}{#1}
 \DeclarePairedDelimiter\bra{\lvert}{\rangle}
 \DeclarePairedDelimiter\ket{\langle}{\rvert}
 \DeclarePairedDelimiterX\innerp[2]{\langle}{\rangle}{#1,#2}
@@ -111,7 +114,7 @@
 \usetikzlibrary{arrows.meta}
 
 % ---------- tcolorbox
-\tcbuselibrary{skins,theorems,breakable} % add breakable library?
+\tcbuselibrary{skins,theorems,breakable}
 \tcbset{%
   beforeafter skip balanced=0.4\baselineskip,
   mythmstyle/.style={%
@@ -151,16 +154,10 @@
 \newenunciation{example}{gray}
 \newenunciation{remark}{gray}
 
-%\theoremstyle{definition}
-%\newtheorem{defin}
-%\renewenvironment{definition}[2]{\begin{defin}[#1]\label{#2}}{\end{defin}}
-
 % ---------- biblatex
 \addbibresource[glob]{bib/*.bib}
 \ExecuteBibliographyOptions{%
   refsegment=chapter,
-  sorting=none,
-  defernumbers,
   giveninits,
   backref,
 }
@@ -170,6 +167,7 @@
   \bookmark[level=section,italic,dest=refbm:\arabic{refsegment}]{#1}
 }
 \newcommand*{\chapterbib}{\printbibliography[segment=\therefsegment,heading=chapterbib]}
+\AddToHook{include/end}{\chapterbib}
 \DefineBibliographyStrings{english}{%
   backrefpage={ref.\ on \pno}, %TODO use \addspace ?
   backrefpages={ref.\ on \ppno}
@@ -275,7 +273,7 @@
 % --------------
 \newcommand*{\hilb}[1]{\mathcal{#1}}
 \newcommand*{\Hilb}[1]{\mathcal{#1}}
-% algebraic dircet sum
+% algebraic direct sum
 \newcommand{\AlgebraicDirectSum}[1]{\sideset{}{_{\!\ts{alg}}}\bigoplus#1}
 % Hilbert space direct sum
 %\newcommand{\AlgebraicDirectSum}[1]{\sideset{}{_{\!\ts{Hilb}}}\bigoplus #1}
@@ -283,10 +281,14 @@
 % Test functions and distributions
 % --------------
 \newcommand*{\testfun}[1]{\mathcal{D}(#1)}
-\newcommand*{\distrib}[1]{\mathcal{D}'(#1)}
-\newcommand*{\schwartz}[1]{\mathcal{S}(#1)}
+\newcommand*{\TestFunctions}[2][]{\mathcal{D}\parens[#1]{#2}}
+\newcommand*{\distrib}[1]{\mathcal{D}'(#1)} %todo replace by command below
+\newcommand*{\Distributions}[2][]{\mathcal{D}\parens[#1]{#2}}
+\newcommand*{\schwartz}[1]{\mathcal{S}(#1)} %todo replace by command below
+\newcommand*{\SchwartzFunctions}[2][]{\mathcal{S}\parens[#1]{#2}}
 \newcommand*{\realschwartz}[1]{\mathcal{S}_{\RR}(#1)}
 \newcommand*{\tempdistrib}[1]{\mathcal{S}'(#1)}
+\newcommand*{\TemperedDistributions}[2][]{\mathcal{S}'\parens[#1]{#2}}
 \newcommand*{\tempdistribnoarg}{\mathcal{S}'}
 
 % Fock spaces
@@ -314,7 +316,7 @@
 
 % Standard Subspaces
 % ------------------
-% real scalarproduct
+% real scalar product
 \DeclarePairedDelimiterXPP\realscalarp[2]{\Re}{\langle}{\rangle}{}{#1,#2}
 \DeclarePairedDelimiterXPP\symplecticp[2]{\Im}{\langle}{\rangle}{}{#1,#2}
 % symplectic complement
@@ -340,8 +342,9 @@
 \newcommand*{\vNa}[1]{\mathcal{#1}}
 \newcommand*{\localalg}[1]{\vNa{R}(#1)}
 
-% Measure Theroy
-\newcommand*{\BorelSigmaAlgebra}[2][]{\mathfrak{B}\parens[#1]{#2}}
+% Measure Theory
+\newcommand*{\SigmaAlgebra}[1]{\mathfrak{#1}}
+\newcommand*{\BorelSigmaAlgebra}[2][]{\SigmaAlgebra{B}\parens[#1]{#2}}
 
 % Lorentz and Poincaré groups, subgroups and connected components
 \newcommand*{\LorentzGroup}{\mathcal{L}}
@@ -378,6 +381,7 @@
 
 % Functional Analysis
 \newcommand*{\BoundedLinearOperators}[2][]{B\parens[#1]{#2}}
+\newcommand*{\ContinousLinearOperators}[2][]{L\parens[#1]{#2}}
 \DeclareMathOperator*{\stronglim}{s-lim}
 \DeclareMathOperator*{\weaklim}{w-lim}
 
diff --git a/sampleappendix.tex b/sampleappendix.tex
index 09d68f4..6cdf6cb 100644
--- a/sampleappendix.tex
+++ b/sampleappendix.tex
@@ -15,5 +15,3 @@ $x \equiv y$
 \nocite{*}
 
 \cref{lemma:xxx}
-
-\chapterbib
diff --git a/samplesection.tex b/samplesection.tex
index 35f1c38..205fa6f 100644
--- a/samplesection.tex
+++ b/samplesection.tex
@@ -6,5 +6,3 @@ Just some \index{sample text}sample text.
 \section{Subsection}
 
 \section{Another Subsection}
-
-\chapterbib
diff --git a/second.tex b/second.tex
index 241d4c1..d7e1a38 100644
--- a/second.tex
+++ b/second.tex
@@ -4,6 +4,3 @@
   First quantization is a mystery, but second quantization is a functor.
 }
 Just more text.
-
-\chapterbib
-\cleardoublepage
diff --git a/standard.tex b/standard.tex
index 93ccdcc..86bd1ab 100644
--- a/standard.tex
+++ b/standard.tex
@@ -19,6 +19,3 @@ test
 \begin{definition}{cyclic, separating}{}
   test
 \end{definition}
-
-\chapterbib
-\cleardoublepage
diff --git a/stresstensor.tex b/stresstensor.tex
index 79c0930..70a79ea 100644
--- a/stresstensor.tex
+++ b/stresstensor.tex
@@ -1087,10 +1087,6 @@ where
   and $\psi_n \equiv 0$ for $n \ne 2$.
 \end{proposition}
 
-\begin{equation*}
-  \energydensity(f) \Omega = ?
-\end{equation*}
-
 \section{Essential Selfadjointness of Renormalized Products}
 
 \begin{lemma}{H-Bounds for the Renormalized Product}{}
@@ -1148,7 +1144,4 @@ where
   \end{equation*}
 \end{theorem}
 
-\chapterbib
-\cleardoublepage
-
 % vim: syntax=mytex