weiter

11 files changed, 1205 insertions, 75 deletions

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@@ -1,38 +1,47 @@
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diff --git a/analytic2.tex b/analytic2.tex
new file mode 100644
index 0000000..e0dc68f
--- /dev/null
+++ b/analytic2.tex
@@ -0,0 +1,135 @@
+\chapter{Analytic Vectors}
+
+\info{Dies ist nur ein Relikt meines Studiums analytischer Vektoren. Wird wieder entfernt, falls nicht benötigt.}
+
+\begin{definition}{Analytic Vector for an Operator}{analytic-vector-operator}
+  Let $A : D(A) \to \hilb{H}$ be an unbounded linear operator in a complex Hilbert space $\hilb{H}$.
+  A vector $x \in \hilb{H}$ is said to be an \emph{analytic vector} for $A$,
+  if $x$ lies in the domain of the power $A^n$ for all $n \in \NN$, and the power series
+  \begin{equation*}\tag{power-series-analytic-vector}
+    \sum_{n=0}^{\infty} \frac{A^n x}{n!} \, z^n
+  \end{equation*}
+  has a nonzero radius of convergece.
+  If the power series converges for all $z \in \CC$,
+  we say that $x$ is an \emph{entire analytic vector} for $A$.
+\end{definition}
+Note that, if $x$ is analytic for $A$, then the power series
+\begin{equation*}
+  \sum_{n=0}^{\infty} \frac{\norm{A^n x}}{n!} \, z^n
+\end{equation*}
+converges for all complex $z$ with $\abs{z} < t$,
+that is, in the open disc with radius $t$ centered in the origin of the complex plane.
+This is a well-known consequence of the convergence behavior of power series.
+
+\begin{definition}{Analyticity of Vector-Valued Functions}{}
+  Let $G \subset \CC$ be open and let $\hilb{H}$ be a Hilbert space.
+  A function $f : G \to \hilb{H}$ is called
+  \begin{itemize}
+    \item \emph{strongly analytic} at $a \in G$, if the limit
+      \begin{equation*}
+        \lim_{z \to a} \frac{f(z) - f(a)}{z-a}
+      \end{equation*}
+      exists in norm.
+    \item \emph{weakly analytic} in $a \in G$, if for each $w \in \CC$ the scalar-valued function
+      \begin{equation*}
+        G \longrightarrow \CC, \quad z \longmapsto \innerp{w}{f(z)}
+      \end{equation*}
+      is analytic in $a$.
+  \end{itemize}
+\end{definition}
+
+\begin{lemma}{Equivalence of Weak and Strong Analyticity}{}
+  Let $G \subset \CC$ be open.
+  Then a Banach space-valued function is strongly analytic on $G$ if and only if it is weakly analytic on $G$.
+\end{lemma}
+\begin{myproof}
+  Let $X$ be a Banach space and suppose that the function $f : G \to X$ is weakly analytic.
+  By definition, for each $g \in X'$ the scalar valued function $g \circ f : G \to \CC$ is analytic on $G$.
+  Consider a point $a \in G$.
+  Since $G$ is open, there exists a circular contour $\gamma$ around $a$ such that $\gamma$ and its interior lie wholly inside of $G$.
+  By Cauchy’s Integral Formula we have
+  \begin{equation*}
+    g(f(z)) = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(f(w))}{w-z} \, dw
+  \end{equation*}
+  for any $z$ in the interior of $\gamma$.
+  Writing
+  \begin{equation*}
+    Q(z) = \frac{f(z) - f(a)}{z - a}
+  \end{equation*}
+  for the difference quotient, we get
+  \begin{equation*}
+    g(Q(z)) = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(f(w))}{(w-z)(w-a)} \, dw
+  \end{equation*}
+  and
+  \begin{equation*}
+    g \parens*{\frac{Q(z) - Q(z')}{z - z'}} = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(f(w))}{(w-z)(w-z')(w-a)} \, dw
+  \end{equation*}
+  for all $z,z'$ in the interior of $\gamma$.
+  The family of vectors $f(w) \in X$, indexed by complex numbers $w$ on the contour $\gamma$, can be viewed as a family of bounded linear functionals $C(f(w)) : X' \to \CC$
+  via the canonical embedding $C : X \to X''$ of $X$ into its bidual. For every fixed $g \in X'$ the set of values $C(f(w))(g) = g(f(w))$ is bounded, because the function $g \circ f$ is continous and the contour is compact.
+  In other words, the family of functionals $C(f(w))$, $w \in \gamma$, is pointwise bounded.
+  The Uniform Boundedness Theorem implies that there exists a constant $M > 0$ such that $\abs{g(f(w))} \le M \norm{g}$ for all $w$ on $\gamma$ and all $g \in X'$.
+  \begin{equation*}
+    \abs*{g \parens*{\frac{Q(z) - Q(z')}{z - z'}}} \le \frac{M}{2 \pi} \norm{g} \int_{\gamma} \frac{dw}{\abs{w-z}\abs{w-z'}\abs{w-a}}
+  \end{equation*}
+  If we restict $z,z'$ to a neighbourhood $N$ of $a$ that stays away from $\gamma$, then the integral on the right hand side is
+  bounded by a constant independent of $z$ and $z'$.
+  Absorbing all constants into $M' > 0$ we obtain
+  \begin{equation*}
+    \abs{g(Q(z) - Q(z'))} \le M' \norm{g} \abs{z-z'} \quad \forall z,z' \in N.
+  \end{equation*}
+  \begin{equation*}
+    \norm{Q(z) - Q(z')} = \sup_{\substack{g \in X'\\ \norm{g} \le 1}} \abs{g(Q(z) - Q(z'))} \le M' \norm{z - z'}.
+  \end{equation*}
+  Hence, the limit of $Q(z)$ for $z \to a$ exists by completeness of $X$.
+\end{myproof}
+
+\begin{definition}{Analytic Vector for an Unitary Group}{analytic-vector-unitary-group}
+  Let $\sigma : \RR \to U(\hilb{H})$ be a strongly continuous one-parameter unitary group on a complex Hilbert space $\hilb{H}$.
+  A vector $x \in \hilb{H}$ is said to be an \emph{analytic vector} for $\sigma$, if there exist
+  \begin{itemize}
+    \item a number $\lambda > 0$, defining a strip $I_{\lambda} = \braces{z : \abs{\Im z} < 1}$, and
+    \item a vector-valued function $f : I_{\lambda} \to \hilb{H}$,
+  \end{itemize}
+  with the properties that
+  \begin{itemize}
+    \item $f(t) = \sigma_t(x)$ for all $t \in \RR$,
+    \item $f$ is weakly analytic on $I_{\lambda}$.
+  \end{itemize}
+  In this case we write $f(z) = \sigma_z(x)$ for $z \in I_{\lambda}$.
+\end{definition}
+
+\begin{proposition}{}{}
+  Let $\sigma : \RR \to U(\hilb{H})$ be a strongly continuous one-parameter unitary group on a complex Hilbert space $\hilb{H}$
+  and let $A$ be its infinitesimal generator.
+  Then a vector $x \in \hilb{H}$ is analytic for $\sigma$ if and only if it is analytic for $A$.
+\end{proposition}
+
+\begin{myproof}
+  First, suppose that $x$ is an analytic vector for $\sigma$.
+  Then, there exist a number $\lambda > 0$ and a function $f : I_{\lambda} \to X$ as in \cref{definition:analytic-vector-unitary-group}.
+  In particular, $f$ is (strongly) analytic on the strip $I_{\lambda}$, which contains the disk $\braces{z : \abs{z} \le r}$ when $r \le \lambda$.
+  Hence we have Cauchy estimates
+  \begin{equation*}
+    \norm{f^{(n)}(0)} \le \frac{n!}{r^n} M \quad \forall n \in \NN,
+    \quad \text{where} \ M = \sup_{\abs{z} = r} \norm{f(z)}.
+  \end{equation*}
+  For real $t$ we have $f(t) = \sigma_t(x) = \exp(itA) x$ and
+  the mapping $t \mapsto f(t)$ is strongly differentiable with derivatives
+  $f^{(n)}(0) = (iA)^n x$.
+  This implies that the power series
+  \begin{equation*}
+    \sum_{n=0}^{\infty} \frac{\norm{A^n x}}{n!} t^n \le M \sum_{n=0}^{\infty} \frac{t^n}{r^n} 
+  \end{equation*}
+  is convergent for $t \le \lambda$ by majorization. Hernce $x$ is an analytic vector for the operator $A$.
+
+  Coversely, suppose that $x$ is analytic for the generator $A$ of $\sigma$.
+  Then, by \cref{definition:analytic-vector-operator}, $x$ lies in the domains of all powers $A^n$, $n \in \NN$, and the power series
+  \begin{equation*}
+    \sum_{n=0}^{\infty} \frac{\norm{A^n x}}{n!} z^n
+  \end{equation*}
+  has a positive radius of convergence $t>0$.
+\end{myproof}
+
+\chapterbib
+\cleardoublepage
diff --git a/bib/articles.bib b/bib/articles.bib
index 2368b40..60d4d4f 100644
--- a/bib/articles.bib
+++ b/bib/articles.bib
@@ -1,8 +1,8 @@
 @article{Epstein1965,
   author       = {Epstein, H. and Glaser, V. and Jaffe, A.},
   title        = {Nonpositivity of the Energy Density in Quantized Field Theories},
-  publisher    = {Società Italiana di Fisica},
   journaltitle = {Il Nuovo Cimento},
+  publisher    = {Società Italiana di Fisica},
   volume       = {36},
   issue        = {3},
   date         = {1965},
diff --git a/bib/misc.bib b/bib/misc.bib
new file mode 100644
index 0000000..b32be08
--- /dev/null
+++ b/bib/misc.bib
@@ -0,0 +1,28 @@
+@book{ReedSimon1,
+   maintitle =    {Methods of Modern Mathematical Physics},
+   title =     {Functional Analysis},
+   author =    {Michael Reed and Barry Simon},
+   publisher = {Academic Press},
+   date =      {1980},
+   edition = {Revised and Enlarged Edition},
+   volume =    {1},
+}
+@book{ReedSimon2,
+   maintitle =    {Methods of Modern Mathematical Physics},
+   title =     {Fourier Analysis, Self-Adjointness},
+   author =    {Michael Reed and Barry Simon},
+   publisher = {Academic Press},
+   date =      {1975},
+   volume =    {2},
+}
+@article{Nelson1972,
+ title     = {Time-Ordered Operator Products of Sharp-Time Quadratic Forms},
+ author    = {Edward Nelson},
+ publisher = {Elsevier Science},
+ journal   = {Journal of Functional Analysis},
+ issn      = {0022-1236,1096-0783},
+ date      = {1972},
+ volume    = {11},
+ number     = {2},
+ pages     = {211--219},
+}
diff --git a/bib/much.bib b/bib/much.bib
new file mode 100644
index 0000000..c2f76f4
--- /dev/null
+++ b/bib/much.bib
@@ -0,0 +1,29 @@
+@misc{Much2022,
+      title={An approximate local modular quantum energy inequality in general quantum field theory}, 
+      author={Albert Much and Albert Georg Passegger and Rainer Verch},
+      year={2022},
+      eprint={2210.01145},
+      archivePrefix={arXiv},
+      primaryClass={math-ph}
+}
+@article{Bisognano1975,
+ title     = {On the duality condition for a Hermitian scalar field},
+ author    = {Joseph J. Bisognano and Eyvind H. Wichmann},
+ publisher = {American Institute of Physics},
+ journal   = {Journal of Mathematical Physics},
+ issn      = {0022-2488,1089-7658},
+ year      = {1975},
+ volume    = {16},
+ number     = {4},
+ pages     = {985-1007},
+}
+@book{Schmüdgen2012,
+   title =     {Unbounded Self-adjoint Operators on Hilbert Space},
+   author =    {Konrad Schmüdgen},
+   publisher = {Springer},
+   isbn =      {9789400747524},
+   year =      {2012},
+   series =    {Graduate Texts in Mathematics},
+   edition =   {1},
+   volume =    {265},
+}
diff --git a/bib/test.bib b/bib/test.bib
new file mode 100644
index 0000000..c6d07d6
--- /dev/null
+++ b/bib/test.bib
@@ -0,0 +1,32 @@
+@book{book:4004309,
+   title =     {Introduction to Algebraic and Constructive Quantum Field Theory},
+   author =    {John C. Baez and Irving Ezra Segal and Zhengfang Zhou},
+   publisher = {Princeton University Press},
+   isbn =      {9781400862504},
+   year =      {1992},
+   series =    {Princeton Series in Physics},
+   volume =    {},
+}
+@article{Wick1950,
+ title     = {The Evaluation of the Collision Matrix},
+ author    = {G. C. Wick},
+ publisher = {American Physical Society},
+ journal   = {Physical Review},
+ issn      = {0031-899X,1536-6065},
+ year      = {1950},
+ volume    = {80},
+ number     = {2},
+ pages     = {268--272},
+}
+@article{Fewster1998,
+ title     = {Bounds on negative energy densities in flat spacetime},
+ author    = {C. J .Fewster and S. P. Eveson},
+ publisher = {American Physical Society},
+ journal   = {Physical Review D},
+  ISSN = {1089-4918},
+ year      = {1998},
+ volume    = {58},
+  number = {8},
+  month = sep,
+ pages     = {084010},
+}
diff --git a/bib/test/distributions.bib b/bib/test/distributions.bib
new file mode 100644
index 0000000..81746dc
--- /dev/null
+++ b/bib/test/distributions.bib
@@ -0,0 +1,170 @@
+@book{Halperin1952,
+   series =    {Canadian Mathematical Congress, Lecture Series},
+   volume =    {1},
+   title =     {Introduction to the Theory of Distributions},
+   author =    {Israel Halperin},
+   note =    {Based on the lectures given by \textsc{L. Schwartz}.},
+   publisher = {University of Toronto Press},
+   year =      {1952},
+}
+@book{Lighthill1958,
+   title =     {Introduction to Fourier Analysis and Generalised Functions},
+   author =    {M. J. Lighthill},
+   publisher = {Cambridge University Press},
+   year =      {1958},
+}
+@article{Gårding1959,
+ doi       = {10.1007/bf02724838},
+ title     = {Functional Analysis},
+ author    = {L. Gårding and J. L. Lions},
+ publisher = {Società Italiana di Fisica},
+ journaltitle   = {Il Nuovo Cimento},
+ issn      = {0029-6341,1827-6121},
+ year      = {1959},
+ volume    = {14},
+ issue     = {1 Supplement},
+ pages     = {9--66},
+}
+@book{Erdélyi1962,
+   title =     {Operational Calculus and Generalized Functions},
+   author =    {Arthur Erdélyi},
+   publisher = {Holt, Rinehart and Winston},
+   year =      {1962},
+}
+@book{Schwartz1966,
+   title =     {Théorie des distributions},
+   author =    {Laurent Schwartz},
+   publisher = {Hermann},
+   year =      {1966},
+   language =      {french},
+}
+@book{Trèves1967,
+   title =     {Topological Vector Spaces, Distributions and Kernels},
+   author =    {François Trèves},
+   publisher = {Academic Press},
+   year =      {1967},
+}
+@book{Jones1982,
+   title =     {The Theory of Generalised Functions},
+   author =    {D. S. Jones},
+   publisher = {Cambridge University Press},
+   year =      {1982},
+   edition =   {2},
+}
+@book{GenFunc1,
+  maintitle    = {Generalized Functions},
+  volume       = {1},
+  title        = {Properties and Operations},
+  author       = {I. M. Gel'fand and G. E. Shilov},
+  publisher    = {Academic Press},
+  year         = {1964},
+  language     = {english},
+  origlanguage = {russian},
+  translator   = {Eugene Saletan},
+}
+@book{GenFunc2,
+  maintitle    = {Generalized Functions},
+  volume       = {2},
+  title        = {Spaces of Fundamental and Generalized Functions},
+  author       = {I. M. Gel'fand and G. E. Shilov},
+  publisher    = {Academic Press},
+  year         = {1968},
+  language     = {english},
+  origlanguage = {russian},
+  translator   = {Morris D. Friedman and Amiel Feinstein and Christian P. Peltzer},
+}
+@book{GenFunc3,
+  maintitle    = {Generalized Functions},
+  volume       = {3},
+  title        = {Theory of Differential Equations },
+  author       = {I. M. Gel'fand and G. E. Shilov},
+  publisher    = {Academic Press},
+  year         = {1967},
+  language     = {english},
+  origlanguage = {russian},
+  translator   = {Meinhard E. Mayer},
+}
+@book{GenFunc4,
+  maintitle    = {Generalized Functions},
+  volume       = {4},
+  title        = {Applications of Harmonic Analysis},
+  author       = {I. M. Gel'fand and N. Ya. Vilenkin},
+  publisher    = {Academic Press},
+  year         = {1964},
+  language     = {english},
+  origlanguage = {russian},
+  translator   = {Amiel Feinstein},
+}
+@book{GenFunc5,
+  maintitle    = {Generalized Functions},
+  volume       = {5},
+  title        = {Integral Geometry and Representation Theory},
+  author       = {I. M. Gel'fand and M. I. Graev and N. Ya. Vilenkin},
+  publisher    = {Academic Press},
+  year         = {1966},
+  language     = {english},
+  origlanguage = {russian},
+  translator   = {Eugene Saletan},
+}
+@book{GenFunc6,
+  maintitle    = {Generalized Functions},
+  volume       = {6},
+  title        = {Representation Theory and Automorphic Functions},
+  author       = {I. M. Gel'fand and M. I. Graev and I. I. Pyatetskii-Shapiro},
+  publisher    = {W. B. Saunders},
+  year         = {1969},
+  language     = {english},
+  origlanguage = {russian},
+  translator   = {K. A. Hirsch},
+}
+@book{Richards1990,
+   title =     {The Theory of Distributions},
+   subtitle =     {A Nontechnical Introduction},
+   author =    {Ian Richards and Heekyung Youn},
+   publisher = {Cambridge University Press},
+   year =      {1990},
+}
+@book{Strichartz1994,
+   title =     {A Guide to Distribution Theory and Fourier Transforms},
+   author =    {Robert Strichartz},
+   publisher = {CRC Press},
+   year =      {1994},
+   series =    {Studies in Advanced Mathematics},
+   edition =   {},
+   volume =    {},
+}
+@article{Wightman1996,
+ doi       = {10.1002/prop.2190440204},
+ title     = {How It Was Learned that Quantized Fields Are Operator-Valued Distributions},
+ author    = {A. S. Wightman},
+ publisher = {John Wiley and Sons},
+ journaltitle   = {Fortschritte der Physik},
+ issn      = {0015-8208,1521-3978},
+ year      = {1996},
+ volume    = {44},
+ issue     = {2},
+ pages     = {143--178},
+}
+@book{Friedlander1999,
+   title =     {Introduction to the Theory of Distributions},
+   author =    {F. G. Friedlander and M. Joshi},
+   publisher = {Cambridge University Press},
+   year =      {1999},
+   edition =   {2},
+}
+@book{Hoskins2005,
+   title =     {Theories of Generalised Functions},
+   title =     {Distributions, Ultradistributions and Other Generalised Functions},
+   author =    {R. F. Hoskins and J. S. Pinto},
+   publisher = {Woodhead Publishing},
+   year =      {2005},
+   edition =   {2},
+}
+@book{Hoskins2009,
+   title =     {Delta Functions},
+   subtitle =     {Introduction to Generalised Functions},
+   author =    {R. F. Hoskins},
+   publisher = {Woodhead Publishing},
+   year =      {2009},
+   edition =   {2},
+}
diff --git a/main.tex b/main.tex
index fc44017..91aaaab 100644
--- a/main.tex
+++ b/main.tex
@@ -1,5 +1,5 @@
 \input{preamble}
-\includeonly{stresstensor,fewstereveson,commutatortheorem,index}
+\includeonly{stresstensor,fewstereveson,much,commutatortheorem,analytic2,symbols,index}
 \begin{document}
 \frontmatter
 \include{titlepage}
@@ -12,10 +12,12 @@
 \include{standard}
 \include{stresstensor}
 \include{fewstereveson}
+\include{much}
 \include{samplesection}
 \appendix
 \include{sampleappendix}
 \include{commutatortheorem}
+\include{analytic2}
 \backmatter
 \include{bibliography}
 \include{symbols}
diff --git a/much.tex b/much.tex
new file mode 100644
index 0000000..58aeab3
--- /dev/null
+++ b/much.tex
@@ -0,0 +1,502 @@
+\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}
+\end{equation*}
+
+
+\section{Misc}
+
+\todo{Put this somwhere 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} $\FullLorentzGroup$.
+\begin{equation*}
+  \FullPoincareGroup = \RR^4 \ltimes \FullLorentzGroup
+\end{equation*}
+
+The relativistic transformation law for one-particle states is given by
+\begin{equation*}
+  \parens[\big]{U(a,\Lambda) \psi}(p) = e^{ia \cdot p} \psi(\Lambda^{-1} p),
+  \quad \psi \in \hilb{H}, (a,\Lambda) \in \ProperOrthochronousPoincareGroup.
+\end{equation*}
+The mapping $(a,\Lambda) \mapsto U(a,\Lambda)$ is a (irreducible) unitary representation
+of the proper orthochronous Poincaré group $\ProperOrthochronousPoincareGroup$
+on the one-particle Hilbert space.
+By applying any of the second quantization functors we obtain a representation on the multi-particle state space.
+\begin{equation*}
+  \parens[\big]{U(a,\Lambda) \psi}{}_n(p_1,\ldots,p_n) = e^{ia \cdot (p_1 + \cdots + p_n)} \psi_n(\Lambda^{-1} p_1, \ldots, \Lambda^{-1} p_n),
+\end{equation*}
+
+Poincaré covariance
+\begin{equation}
+  \label{equation:poincare-covariance-local-algebras}
+  U(g) \localalg{\spacetimeregion{O}} U(g)^* = \localalg{g\spacetimeregion{O}}
+  \qquad g \in \ProperOrthochronousPoincareGroup
+\end{equation}
+
+\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}}''
+  \end{equation*}
+\end{definition}
+
+\section{Basic Concepts of Modular Theory}
+\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 $B(\hilb{H})$.
+
+\begin{definition}{Cyclic and Separating Vectors}{}
+  Suppose $\hilb{H}$ is a Hilbert space and $\mathcal{A}$ is a $C^*$-subalgebra of $B(\hilb{H})$.
+  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 $\hilb{H}$.
+    \item \emph{separating}\index{separating vector} for $\mathcal{A}$ if the map $A \mapsto A \Omega$ from $\mathcal{A}$ into $\hilb{H}$ is injective.
+  \end{itemize}
+\end{definition}
+Occasionally, a vector that is both cyclic and separating is called \emph{standard}\index{standard vector}.
+
+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 $\mathcal{A}$}
+
+\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 $\vNa{M}'$.
+  \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+  \todo{xxx}
+  The second assertion directly follows from the first and the fact that $\vNa{M}'' = \vNa{M}$.
+\end{proof}
+
+If $\Omega$ is separating for $\mathcal{A}$,
+then every element of $\mathcal{A}\Omega$ is of the form $A\Omega$
+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}.
+\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,
+the range of $S_0$ coincides with its domain.
+
+\begin{lemma}{}{}
+  If $\Omega$ is a cyclic and separating vector for a von Neumann algebra $\mathcal{A}$,
+  then the operator $S_0$ defined by~\eqref{equation:definition-s0} is closable.
+\end{lemma}
+\begin{proof}
+  By \cref{proposition:cyclic-separating},
+  $\Omega$ is also cyclic and separating for the commutant $\vNa{A}'$.
+  Hence we may, analogously to $S_0$,
+  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'}.
+  \end{equation*}
+  By definition 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} = 
+    \innerp{\Omega}{BA \Omega} = 
+    \innerp{F_0 B\Omega}{A \Omega}.
+  \end{equation*}
+  This adjoint identity establishes that $S_0 \subset F_0^*$.
+  (The \enquote{twisted} appearance of the identity is correct,
+  since it involves anti-linear operators on both sides.)
+  The Hilbert adjoint $F_0^*$ of $F_0$ is closed.
+  Hence, we have shown that $S_0$ has a closed extension, and
+  this implies that $S_0$ is closable.
+\end{proof}
+
+\begin{definition}{Tomita operator}{}
+  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
+  $S_0 A\Omega = S_0 A^*\Omega$
+  for $A \in \mathcal{A}$
+  is called the
+  \emph{Tomita operator}\index{Tomita operator}\index{operator!Tomita}\nomenclature{$S$}{Tomita operator}
+  for the pair $(\mathcal{A},\Omega)$.
+\end{definition}
+
+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.
+
+\begin{theorem}{Polar Decomposition for Anti-Linear Closed Operators}{polar-decomposition}
+  \index{polar decomposition}
+  Let $T$ be an arbitrary closed anti-linear operator in a Hilbert space $\hilb{H}$.
+  Then there exist
+  a positive selfadjoint linear operator $\abs{T}$ and
+  a partial anti-linear isometry $U$
+  such that
+  \begin{equation*}
+    T = U \abs{T} \qquad \bracks[\big]{\text{in particular, $\Domain{T} = \Domain{\abs{T}}$}}.
+  \end{equation*}
+  The operators $U$ and $\abs{T}$ are uniquely determined given the additional conditions
+  \begin{equation*}
+    \ker\abs{T} = \ker T \qquad
+    (\ker U)^\perp = (\ker T)^\perp \qquad
+    \ran U = \overline{\ran T}.
+  \end{equation*}
+\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
+ensuring uniqueness are satisfied.
+
+Now we are able to introduce the fundamental objects of modular theory.
+
+\begin{definition}{Modular Conjugation, Modular Operator}{}
+  Suppose $\vNa{M}$ is a von Neumann algebra acting on a Hilbert space $\hilb{H}$,
+  and suppose $\Omega \in \hilb{H}$ is a cyclic and separating vector for $\vNa{M}$.
+  Let $S$ be the Tomita operator for $(\vNa{M},\Omega)$ and let
+  \begin{equation*}
+    S = J \Delta^{1/2}
+  \end{equation*}
+  be its polar decomposition.
+  The anti-unitary operator $J$ is called
+  \emph{modular conjugation}\index{modular!conjugation}\nomenclature{$J$}{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
+  the pair $(\vNa{M},\Omega)$.
+\end{definition}
+
+\todo{clarify why $J$ is anti-unitary}
+
+\begin{definition}{Modular Group}{}
+  Adopt the notation of the foregoing definition.
+  The mapping $\RR \ni t \mapsto \Delta^{it}$ is called the \emph{modular group}\index{modular!group} associated to 
+  $(\vNa{M},\Omega)$.
+\end{definition}
+
+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 $\hilb{H}$.
+  Then $U\vNa{M}U^*$ is a von Neumann algebra on $\hilb{H}$.
+  Suppose further that $\Omega \in \hilb{H}$ is a cyclic and separating vector for $\vNa{M}$.
+  Then $U \Omega$ is cyclic and separating for $U\vNa{M}U^*$.
+  Let $(J,\Delta)$ be the modular data associated to $(\vNa{M},\Omega)$.
+  Then $(UJU^*,U{\Delta}U^*)$ is the modular data associated to $(U\vNa{M}U^*,U\Omega)$.
+\end{proposition}
+
+\begin{proof}
+  To prove the first assertion,
+  consider any $A \in (U\vNa{M}U^*)''$.
+  By the double commutant theorem,
+  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.
+  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 $(U\vNa{M}U^*)'$.
+  Thus we find that $[U^*\! AU,B] = U^* [A,UBU^*] U = 0$, as desired.
+
+  The set of vectors $U\vNa{M}U^* U\Omega = U\vNa{M}\Omega$ is dense in $\hilb{H}$,
+  since it is the image of $\vNa{M} \Omega$ under the homeomorphism $U$.
+  Thus, the vector $U\Omega$ is cyclic for $U\vNa{M}U^*$.
+  Let us show that it is also separating.
+  Suppose $A$ is in $\vNa{M}$ and $UAU^*U\Omega = UA\Omega = 0$
+  Since unitaries are injective, $A\Omega = 0$.
+  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 $(U\vNa{M}U^*,U\Omega)$.
+  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
+  \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 $S' = U S U^*$.
+  We can write this as $S' = UJU^* U\Delta^{1/2} U^*$,
+  where $S = J \Delta^{1/2}$ is the polar decomposition of the Tomita operator.
+  It is straightforward to check that
+  $UJU^*$ is anti-unitary and $U\Delta^{1/2} U^*$ is positive selfadjoint,
+  and satisfy the additional condition of \cref{theorem:polar-decomposition}.
+  The uniqueness of the polar decomposition implies that
+  $UJU^*$ is the modular conjugation and $U\Delta U^*$ is the modular conjugation
+  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}
+\end{theorem}
+
+By Reeh-Schlieder (\cref{theorem:reeh-schlieder}), the vacuum $\Omega$ is cyclic and separating for $\localalg{\spacetimeregion{O}}$.
+Thus, modular theory
+
+
+\section{The Geometric Action of the Modular Operator Associated With a Wedge Domain}
+
+\begin{definition}{Right and Left Wedge, General Wedges}{}
+  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}} 
+    \quad \text{and} \quad
+    \leftwedge \defequal \braces[\big]{x \in M \vcentcolon 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
+  such that $W = g \rightwedge$.
+\end{definition}
+
+Instead of the right wedge,
+we could just as well have used the left wedge to define the notion of a general wedge,
+since they are transformed into each other by space inversion.
+
+\begin{lemma}{}{general-wedge-from-right-wedge}
+  If a spacetime region $W$ is a wedge,
+  then there exists an element $g$ of the proper orthochronous Poincaré group
+  such that $W = g \rightwedge$.
+\end{lemma}
+
+\begin{proof}
+  \todo{xxx}
+\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{
+  This matrix depends on the choice of metric signature.
+  Ours is $(+,-,-,-)$.
+  For $(-,+,+,+)$, use
+  \begin{equation*}
+  \Lambda(t) = \begin{pmatrix}
+    \phantom{-}\cosh(2 \pi @ t) & -\sinh(2 \pi @ t) & \; 0 \; & \; 0 \; \\
+    -\sinh(2 \pi @ t) & \phantom{-}\cosh(2 \pi @ t) & 0 & 0 \\
+    0 & 0 & 1 & 0 \\
+    0 & 0 & 0 & 1 \\
+  \end{pmatrix}.
+  \end{equation*}
+  }
+
+\begin{equation*}
+  \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*}
+
+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]
+    \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}
+\end{proposition}
+
+\begin{proof}
+  The first property can be verified by direct computation.
+  Let us prove the second.
+  By definition, the image $\Lambda(t) x$ of a vector $x \in M$ lies in $\rightwedge$ if and only if
+  \begin{equation*}
+    x^0 \sinh(2 \pi t) + x^1 \cosh(2 \pi t) 
+    > \abs{x^0 \cosh(2 \pi t) + x^1 \sinh(2 \pi t)},
+  \end{equation*}
+  or equivalently
+  \begin{equation*}
+    x^0 \parens[\big]{\sinh(2 \pi t) \mp \cosh(2 \pi t)}
+    + x^1 \parens[\big]{\cosh(2 \pi t) \mp \sinh(2 \pi t)} > 0
+  \end{equation*}
+  for both sign choices.
+  Using the definitions of the hyperbolic sine and cosine, this may be further simplified to $(x^1 \mp x^0) e^{2 \pi t} > 0$,
+  which holds if and only if $x^1 > \abs{x^0}$,
+  since the exponential is always positive.
+  So we have shown that
+  \begin{equation}
+    \label{equation:image-right-wedge}
+    \Lambda(t) x \in \rightwedge \iff x \in \rightwedge.
+  \end{equation}
+  This implies that $\Lambda(t)\rightwedge \subset \rightwedge$ for all $t \in \RR$.
+  Conversely, given an arbitrary vector $y \in \rightwedge$,
+  we have to find $x \in \rightwedge$ such that $\Lambda(t) x = y$.
+  Consider $x = \Lambda(-t) y$. Clearly, $x \in \rightwedge$, because of $y \in \rightwedge$
+  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}}}{}
+  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
+  \begin{equation*}
+    J = \Theta \cdot U\parens[\big]{0, R_{23}(\pi)} \qquad
+    \Delta^{it} = U\parens[\big]{0,\Lambda(t)},
+  \end{equation*}
+  where $U$ is the theory's unitary representation of the proper orthochronous Poincaré group.
+\end{theorem}
+
+\todo{give definition of $\Theta$ and $R_{23}$}
+
+Note that above statement is for the right wedge only.
+Let us investigate how the modular group changes, if we consider another wedge region.
+By \cref{lemma:general-wedge-from-right-wedge} any wedge $W$ can be obtained as $W = g\rightwedge$, where $g$ is a proper orthochronous Poincaré transformation.
+The covariance property~\eqref{equation:poincare-covariance-local-algebras} of $\vNa{R}$ implies
+\begin{equation*}
+  \localalg{W} =
+  U(g) \localalg{\rightwedge} U(g)^*.
+\end{equation*}
+The vacuum $\Omega$ is Poincaré invariant:
+\begin{equation*}
+  U(g) \Omega = \Omega.
+\end{equation*}
+We write $(J_W,\Delta_W)$ for the modular data associated to $(\localalg{W},\Omega)$.
+By \cref{proposition:modular-data-unitary}
+\begin{equation*}
+  J_W = U(g) J U(g)^* \qquad 
+  \Delta_W = U(g) \Delta U(g)^*
+\end{equation*}
+Recall that the modular group $\Delta_W^{it}$ is defined by means of functional calculus.
+This raises the following problem: given a selfadjoint operator $A$, a unitary operator $U$
+and a suitable function $f$ we want to express $f(UAU^*)$ in terms of $f(A)$, if possible.
+Note that two different functional calculi are at play here, the former is for $UAU^*$ and the latter for $A$.
+Simple functions such as polynomials suggest $f(UAU^*) = Uf(A)U^*$.
+That this is generally true is the statement of the following Lemma.
+
+
+\begin{lemma}{}{functional-calclus-unitary-trafo}
+  Suppose that $A$ is a selfadjoint operator on a Hilbert space $\hilb{H}$,
+  with spectral measure $E_A$.
+  Suppose $U$ is an unitary operator on $\hilb{H}$, and
+  let $E_{U\! @AU^*}$ denote the spectral measure of the (selfadjoint) operator $UAU^*$.
+  Then we have $U E_A U^* = E_{U\! @AU^*}$, and
+  \begin{equation*}
+    U f(A) U^* = f(U\! @@AU^*)
+  \end{equation*}
+  for all Borel functions $f : \RR \to \CC$.
+\end{lemma}
+
+\question{Ist diese Aussage korrekt? Ist mein Beweis richtig? Geht der auch einfacher?}
+
+\begin{proof}
+  For each regular value $\lambda \in \rho(A)$ let
+  \begin{equation*}
+    R_A(\lambda) = (A-\lambda)^{-1}
+  \end{equation*}
+  denote the resolvent operator of $A$.
+  This proof is based on Stone's Formula \todo{reference},
+  which relates the resolvent to the spectral projections of $A$:
+  If $E_A$ is the spectral measure of $A$ and $\alpha < \beta$ are real numbers, then
+  \begin{equation*}
+    \stronglim_{\varepsilon \downarrow 0}
+    \frac{1}{\pi i} \int_{\alpha}^{\beta} \bracks{R_A(\lambda + i \varepsilon)  - R_A(\lambda - i \varepsilon)} d\lambda 
+    = E_A \parens[\big]{\bracks{\alpha,\beta}} + E_A \parens[\big]{\parens{\alpha,\beta}}
+  \end{equation*}
+  Recall that a spectral measure is countably additive.
+  As a consequence,
+  \begin{equation*}
+    \stronglim_{\alpha \uparrow a}
+    \stronglim_{\beta \downarrow b}
+    \stronglim_{\varepsilon \downarrow 0}
+    \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 $a \in \RR \cup \braces{-\infty}$, $b \in \RR \cup \braces{\infty}$.
+  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^*\!.
+  \end{equation*}
+  Since conjugation with an unitary commutes with the strong operator limit, we obtain
+  \begin{equation*}
+    E_{U\! @AU^*} \parens[\big]{\bracks{a,b}}
+    = U E_A \parens[\big]{\bracks{a,b}} U^*
+  \end{equation*}
+  for all $a,b \in \RR$.
+  The collection $\mathcal{A}$ of all subsets $S$ of $\RR$ such that
+    $E_{U\! @AU^*} \parens[\big]{S} = U E_A \parens[\big]{S} U^*$
+    is a $\sigma$-algebra on $\RR$.
+    We have shown that all closed intervals belong to $\mathcal{A}$.
+    It is well known that the Borel-$\sigma$-algebra $\mathcal{B}$ of $\RR$
+    is generated by the closed intervals. Hence, $\mathcal{B} \subset \mathcal{A}$.
+    This shows that the spectral measures $U E_A U^*$ and $E_{U\! @AU^*}$ coincide.
+\end{proof}
+
+\begin{equation*}
+  \Delta_W^{it}
+  = U(g) \Delta^{it} U(g)^*
+  = U(g) U\parens[\big]{0,\Lambda(t)} U(g)^*
+  = U\parens[\big]{g(0,\Lambda(t))g^{-1}}
+\end{equation*}
+
+
+Recall that Stones 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 $\spacetimeregion{O}$}
+  for said region, and denoted $K_{\spacetimeregion{O}}$.
+\end{definition}
+
+In other words, $K_{\spacetimeregion{O}}$ is the unique selfadjoint operator such that $\Delta_{\spacetimeregion{O}}^{it} = e^{itK_{\spacetimeregion{O}}}$ for all $t \in \RR$.
+
+\begin{proposition}{}{}
+  The modular Hamiltonian for the right wedge is given by $d \Gamma(A)$, where
+  \begin{equation*}
+    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}
+
+\section{Complex Lorentz Transformations}
+
+\subsection{Analytic Continuation of the Space-Time Translation Group}
+
+\subsection{Complex Lorentz Boosts}
+
+\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
+  \begin{equation*}
+    V(z) g(A) = \int e^{iz \lambda} g(\lambda) dE_A(\lambda).
+  \end{equation*}
+\end{lemma}
+
+\subsection{A Convolution Theorem for Vector-Valued Tempered Distributions}
+
+\blockcquote{Bisognano1975}{%
+  The extension to vector-valued tempered distributions is trivial.
+}
+
+
+
+\chapterbib
+\cleardoublepage
+
+% vim: syntax=mytex
diff --git a/preamble.tex b/preamble.tex
index 4a88ef7..35d94ad 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -15,7 +15,7 @@
 %\usepackage{graphicx}
 \usepackage{tcolorbox}
 \usepackage[style=ext-alphabetic]{biblatex}
-\usepackage[intoc]{nomencl}
+\usepackage[intoc,refpage]{nomencl}
 \usepackage{makeidx}
 \usepackage{idxlayout}
 \usepackage{hyperref}
@@ -82,6 +82,7 @@
 \numberwithin{equation}{chapter}
 \DeclareMathOperator{\supp}{supp}
 \DeclareMathOperator{\dom}{dom}
+\DeclareMathOperator{\ran}{ran}
 % extend amsmath's proof environment
 \NewDocumentEnvironment{myproof}{Ob}{\IfNoValueTF{#1}{\begin{proof}}{\begin{proof}[\proofname\ of \Cref{#1}]}}{\end{proof}}
 
@@ -177,7 +178,8 @@
 % ---------- nomencl
 \makenomenclature
 \renewcommand*{\nomname}{List of Symbols}
-\def\pagedeclaration#1{, \hyperlink{page.#1}{page\nobreakspace#1}}
+%\def\pagedeclaration#1{, \hyperlink{page.#1}{page\nobreakspace#1}}
+\def\pagedeclaration#1{, \hyperlink{page.#1}{#1}}
 
 % ---------- makeidx
 \makeindex
@@ -250,6 +252,8 @@
 % Fourier transformation
 % ----------------------
 \newcommand*{\ft}[1]{\hat{#1}}
+\newcommand*{\wideft}[1]{\widehat{#1}}
+\newcommand*{\ift}[1]{\check{#1}}
 \newcommand*{\FT}[1]{\mathcal{F}\parens*{#1}}
 \newcommand*{\iFT}[1]{\mathcal{F}^{-1}\parens*{#1}}
 
@@ -269,6 +273,7 @@
 \newcommand*{\schwartz}[1]{\mathcal{S}(#1)}
 \newcommand*{\realschwartz}[1]{\mathcal{S}_{\RR}(#1)}
 \newcommand*{\tempdistrib}[1]{\mathcal{S}'(#1)}
+\newcommand*{\tempdistribnoarg}{\mathcal{S}'}
 
 % Fock spaces
 % -----------
@@ -289,6 +294,8 @@
 % ---------
 \newcommand*{\QF}[1]{QF(#1)}
 \newcommand{\QFequal}{\overset{\text{\scriptsize QF}}{=}}
+% operator associated to a quadratic form
+\newcommand*{\QFop}[1]{{#1}_{\mathrm{op}}}
 
 % Standard Subspaces
 % ------------------
@@ -311,3 +318,29 @@
   }}
 
 \newcommand{\defequal}{\overset{\text{\scriptsize def}}{=}}
+
+\newcommand*{\energydensity}{\varrho}
+\newcommand*{\fockvaccum}{\Omega}
+
+% Observable Algebras
+\newcommand*{\vNa}[1]{\mathcal{#1}}
+\newcommand*{\localalg}[1]{\vNa{R}(#1)}
+
+
+\newcommand*{\FullLorentzGroup}{\mathcal{L}}
+\newcommand*{\ProperOrthochronousLorentzGroup}{\FullLorentzGroup_{+}^{\uparrow}}
+\newcommand*{\FullPoincareGroup}{\mathcal{P}}
+\newcommand*{\ProperOrthochronousPoincareGroup}{\FullPoincareGroup_{+}^{\uparrow}}
+
+% spacetime domains
+\newcommand*{\spacetimeregion}[1]{\mathcal{#1}}
+\newcommand*{\rightwedge}{W_{\! R}}
+\newcommand*{\leftwedge}{W_{\! L}}
+
+\newcommand*{\todo}[1]{{\color{blue}TODO: #1}}
+\newcommand*{\question}[1]{{\color{blue}Question: #1}}
+\newcommand*{\info}[1]{{\color{blue}Info: #1}}
+
+\newcommand*{\operatorclosure}[1]{\overline{#1}}
+
+\DeclareMathOperator*{\stronglim}{s-lim}
diff --git a/stresstensor.tex b/stresstensor.tex
index a4bb6fb..4c128c2 100644
--- a/stresstensor.tex
+++ b/stresstensor.tex
@@ -35,9 +35,13 @@ as a service to the reader.
   \item Given a complex-valued function $f$ on $M$, we define its \emph{Fourier transform} $\ft{f}\,$ by
     \begin{equation}
       \label{fourier-transform}
-      \hat{f}(p) = \frac{1}{(2 \pi)^2} \int_{M} e^{i p \cdot x} f(x) \, dx
+      \ft{f}(p) \defequal \int_{M} e^{i p \cdot x} f(x) \, dx
     \end{equation}
-    whenever the integral converges. The \emph{inverse Fourier transform} is TODO
+    whenever the integral converges. The \emph{inverse Fourier transform} is defined by
+    \begin{equation*}
+      \label{inverse-fourier-transform}
+      \ift{f}(p) \defequal \frac{1}{(2 \pi)^2} \int_{M} e^{-i p \cdot x} f(x) \, dx.
+    \end{equation*}
   \item To a mathematician $\overline{\phantom{z}}$ usually means complex conjugation and ${}^*$ indicates the Hilbert adjoint of an operator,
     while a physicist may read ${}^*$ as complex conjugation and
     denotes the Hilbert adjoint with ${}^{\dagger}$.
@@ -85,9 +89,15 @@ as a service to the reader.
     \begin{equation*}
       E : \schwartz{M} \to \hilb{H}, \quad f \mapsto Ef = \left.\ft{f}\,\right\vert {X_m^+}
     \end{equation*}
+    We define a $\RR$-linear mapping $\phi$ by
+    \begin{equation*}
+      \realschwartz{M} \ni f \mapsto \varphi(f) = \Phi_{\mathrm{S}}(Ef) = \frac{1}{\sqrt{2}} \parens*{a(Ef) + a(Ef)^\dagger}
+    \end{equation*}
+    This extedns to complex valued test functions $f \in \schwartz{M}$
     \begin{equation*}
-      \realschwartz{M} \ni f \mapsto \Phi(f) = \Phi_{\mathrm{S}}(Ef) = \frac{1}{\sqrt{2}} \parens*{a(Ef) + a(Ef)^\dagger}
+      \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(E\bar{f}) + a(Ef)^\dagger}
     \end{equation*}
+    called the \emph{massive free scalar quantum field} 
   \item
     annihilation and creation operators, $f \in \schwartz{M}$, $\psi \in \BosonFock{\hilb{H}}$
     \begin{align*}
@@ -176,7 +186,7 @@ leads to
   \parens[\big]{a(p)^\dagger \psi} {}_n (k_1, \ldots, k_n)
   = \frac{1}{\sqrt{n}} \sum_{i=1}^n \delta(p - k_i) \, \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n),
 \end{equation}
-where the symmetization is necessary to obtain an expression that
+where the symmetrization is necessary to obtain an expression that
 at least has a chance of being a $n$ Boson state.
 However, it clearly is not a $L^2$ function.
 Given any state $\psi'$, we can
@@ -207,7 +217,7 @@ For completeness, we give a precise definition of quadratic form.
     q : D(q) \times D(q) \to \CC,
   \end{equation*}
   where $D(q)$ is a linear subspace of $\hilb{H}$, called the \emph{form domain}\index{form domain}\index{quadratic form!domain of a},
-  such that $q$ is conjugate linear in its first agrument
+  such that $q$ is conjugate linear in its first argument
   and linear in its second argument (i.e.\ sesquilinear).
   We say that $q$ is \emph{densely defined}
   if $D(q)$ is dense in $\hilb{H}$.
@@ -221,13 +231,13 @@ but one may obtain an operator with trivial domain.
 
 \begin{definition}{Operator Associated to a Quadratic Form}{}
   Suppose $q$ is a densely defined quadratic form on a complex Hilbert space $\hilb{H}$.
-  The linear \emph{operator associated to}\index{quadratic form!operator associated to a} $q$, denoted $q_{\mathrm{op}}$,
+  The linear \emph{operator associated to}\index{quadratic form!operator associated to a} $q$, denoted $\QFop{q}$,
   is defined on the domain
   \begin{equation*}
-    D(q_{\mathrm{op}}) = \braces{\psi \in D(q) \mid \text{the map $q(\cdot,\psi) : D(q) \to \CC$ is bounded}},
+    D(\QFop{q}) = \braces{\psi \in D(q) \mid \text{the map $q(\cdot,\psi) \vcentcolon D(q) \to \CC$ is bounded}},
   \end{equation*}
-  and maps $\psi \in D(q_{\mathrm{op}})$ to the vector $q_{\mathrm{op}}\psi$ in $\hilb{H}$ satisfying
-  $q(\psi',\psi) = \innerp{\psi'}{q_{\mathrm{op}}\psi}$,
+  and maps $\psi \in D(\QFop{q})$ to the vector $\QFop{q}\psi$ in $\hilb{H}$ satisfying
+  $q(\psi'\!,\psi) = \innerp{\psi'\!}{\QFop{q}\psi}$,
   which exists and is unique by Riesz’s Representation Theorem.
 \end{definition}
 
@@ -299,7 +309,7 @@ the $\alpha^{(0)},\alpha^{(1)}_i,\alpha^{(2)}_{j,k},\ldots$ are complex numbers,
 of which only finitely many are nonzero,
 and $e$ is a special object representing an empty product of $z$'s.
 To make this mathematically precise:
-we are speaking of the noncommutative associative algebra over $\CC$
+we are speaking of the non-commutative associative algebra over $\CC$
 freely generated by the elements of $\hilb{H}$.
 The unit of the algebra is $e$.
 
@@ -314,7 +324,7 @@ where $z,z' \in \hilb{H}$.
 \begin{definition}{Infinitesimal Weyl Algebra}{}
   Let $\hilb{H}$ be a complex Hilbert space.
   The \emph{infinitesimal Weyl algebra}\index{infinitesimal Weyl algebra} $\WeylAlg(\hilb{H})$ over $\hilb{H}$
-  is the noncommutative associative algebra over $\CC$
+  is the non-commutative associative algebra over $\CC$
   generated by the elements of $\hilb{H}$, with the relations
   \begin{equation*}
     zz' - z'z = i \Imag \innerp{z}{z'} \, e \qquad z,z' \in \hilb{H},
@@ -371,10 +381,10 @@ the formula makes sense even for $r=0$ and asserts that $\normord{e} = e$.
 The cases $r=1$ and $r=2$ read
 \begin{align*}
   \normord{z} &=
-  \frac{1}{\sqrt{2}} \parens[\big]{\weylannihilator(z) + \weylcreator(z)} = z \\
+  \frac{1}{\sqrt{2}} \parens[\big]{\weylannihilator(z) + \weylcreator(z)} = z, \\
   \normord{z_1 z_2} &= \frac{1}{2}
-  \parens[\big]{\weylannihilator(z_1) \weylannihilator(z_2) + \weylannihilator(z_1) \weylcreator(z_2)
-  + \weylcreator(z_1) \weylannihilator(z_2) + \weylcreator(z_1) \weylcreator(z_2) } 
+  \parens[\big]{ \weylannihilator(z_1) \weylannihilator(z_2) + \weylcreator(z_1) \weylannihilator(z_2)
+  + \weylcreator(z_2) \weylannihilator(z_1) + \weylcreator(z_1) \weylcreator(z_2) }.
 \end{align*}
 This suggests that the normally ordered product $\normord{z_1 \!\cdots z_r}$
 is symmetric in $z_1,\ldots,z_n$. This is in fact true, and becomes evident
@@ -470,20 +480,19 @@ and may be obtained via integration by parts.
 
 Naturally, we now define the \emph{distributional derivative} of the field by
 \begin{equation*}
-  D \varphi(f) = \varphi(D^{\dagger} f) \qquad \forall f \in \schwartz{\RR^d}
+  D \varphi(f) = \varphi(D^{\dagger} f) \qquad \forall f \in \schwartz{\RR^4}
 \end{equation*}
-As one expects, $D\varphi$ is an operator-valued tempered distribution on $M=\RR^d$.
-TODO
+As one expects, $D\varphi$ is an operator-valued tempered distribution on $M=\RR^4$.
+In terms of creation and annihilation operators we have 
 \begin{equation}
   \label{derivative-free-field}
-  D \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(ED^{\dagger}f)^{\dagger} + a(ED^{\dagger}f)}
+  D \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(ED^{\dagger}f)^{\dagger} + a(E\overline{D^{\dagger}f})}.
 \end{equation}
-
-
-The operator corresponding to $D$ in Fourier space is the multiplication operator
+In Fourier space the operator $D^\dagger$ corresponds to muliplication with the polynomial
 \begin{equation*}
-  -i \sum_{\alpha} a_{\alpha} p_0^{\alpha_0} (-p_1)^{\alpha_1} (-p_2)^{\alpha_2} (-p_3)^{\alpha_3}
+  \ft{D}(p) \defequal \sum_{\alpha} i^{\abs{\alpha}} a_{\alpha} (+p^0)^{\alpha_0} (-p^1)^{\alpha_1} (-p^2)^{\alpha_2} (-p^3)^{\alpha_3}
 \end{equation*}
+If $D=\partial^{\mu}$, then $\ft{D}(p) = i @ p_{\!\mu}$, were the potential sign is concealed by lowering the index.
 
 
 Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients.
@@ -494,8 +503,8 @@ Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex)
   \sum_{\sigma \in S_r}
   \sum_{s=0\vphantom{S}}^{r}
   \frac{1}{s!(r-s)!}
-  \prod_{i=1\vphantom{S}}^{s} a^\dagger(D^\dagger_{\sigma(i)}f)
-  \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} a(D^\dagger_{\sigma(j)}f) 
+  \prod_{i=1\vphantom{S}}^{s} a^\dagger(ED^\dagger_{\sigma(i)}f)
+  \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} a(E\overline{D^\dagger_{\sigma(j)}f}) 
 \end{gather}
 
 \section{Renormalized Products of the Free Field and~its~Derivatives}
@@ -516,7 +525,7 @@ this approach incurs significant technical difficulties.
 \begin{lemma}{Integral Representation of the Renormalized Product}{renormalized-product-integral-representation}  
   Let $\varphi$ be the free scalar quantum field with mass parameter $m > 0$.
   Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients.
-  Then, for arbitrary Schwartz functions $f \in \schwartz{M}$ and Fock states $\psi,\psi' \in \BosonFock{L^2(X_m^+,d\Omega_m)}$,
+  Then, for arbitrary Schwartz functions $f \in \realschwartz{M}$ and Fock states $\psi,\psi' \in \BosonFock{L^2(X_m^+,d\Omega_m)}$,
   we have
   \begin{equation*}
     \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} =
@@ -552,7 +561,32 @@ this approach incurs significant technical difficulties.
   \end{multline*}  
 \end{lemma}
 
-TODO(Note about the remaining dependence of $K$ on $f$.)
+Note that $K$ has a remaining dependence on $f$ via $\chi$
+even thogh the notation does not indicate this.
+This is made explicit in the alternative integral representation
+  \begin{equation}
+    \label{equation:alternative-integral-representation}
+    \begin{multlined}
+      \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} =\\
+      \hspace{1cm} \int dp_1 \!\cdots dp_r
+      \sum_{s=0}^{r} 
+      \, \ft{f}(p_1) \cdots\! \ft{f}(p_s)
+      \, \overline{\ft{f}(p_{s+1}) \cdots\! \ft{f}(p_r)}
+      \, \tilde{K}^s_{\psi'\!,\psi}(p_1,\ldots,p_r)
+    \end{multlined}
+  \end{equation}
+  where
+  \begin{multline*}
+    \tilde{K}^s_{\psi'\!,\psi}(p_1,\ldots,p_r) =
+    P_s(p_1,\ldots,p_r)
+    \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
+    \delta_{m-s}^{n-(r-s)} \\
+    \cdot \sqrt{m(m-1) \cdots (m-s+1)} \sqrt{n(n-1) \cdots (n-(r-s)+1)} \\
+    \cdot \int dk_1 \cdots dk_{m-s} 
+    \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)}
+    \ \psi_n(k_1,\ldots,k_{n-(r-s)},p_{s+1},\ldots,p_r).
+  \end{multline*}
+  This will be more convenient for xxx
 
 \begin{myproof}[lemma:renormalized-product-integral-representation]
   From equation~\eqref{equation:renormalized-product},
@@ -617,24 +651,30 @@ In particular, for squares ($r=2$) we have
 
 The following assertion is key to realizing the idea of taking the limit $f \to \delta_x$.
 
-\begin{lemma}{}{integral-kernel-h-bound}
+\begin{lemma}{H-bounds for the Integral Kernel}{integral-kernel-h-bound}
   In the setting of \cref{lemma:renormalized-product-integral-representation},
   there exist a constant $C$, and a positive integer $l$,
-  such that for arbitrary states $\psi,\psi' \in \BosonFock{\hilb{H}}$,
-  and test functions $f \in \schwartz{M}$,
+  such that for arbitrary test functions $f \in \schwartz{M}$
+  and states $\psi,\psi' \in \Domain{H^l}$,
   the function $K_{\psi'\!,\psi}$ is integrable (that is, $L^1$)
   and satisfies the $H$-bound
   \begin{equation*}
     \norm{K_{\psi'\!,\psi}}_1 \le
     C \norm{(1+H)^l \psi'} \norm{(1+H)^l \psi}.
   \end{equation*}
+  More specifically, it is sufficient to choose $l > rd + r/2$,
+  where $d$ is the highest order of differentiation occuring in $D_1, \ldots, D_r$.
 \end{lemma}
 
-The Hamilton operator $H$ acts on $n$-particle states $\psi_n$
-by multiplication with $\omega(p_1)$
-In the following proof it will we convenient to use the abbreviation
+The Hamilton operator $H$ acts on $n$-particle states $\psi_n$ as follows:
+\begin{gather*}
+  H \psi_n(p_1,\ldots,p_n) = \parens[\big]{\omega(p_1) + \cdots + \omega(p_n)} \psi_n(p_1,\ldots,p_n) \\
+  \shortintertext{where}
+  \omega(p) = \omega(\symbfit{p}) = \sqrt{m^2 + \abs{\symbfit{p}}^2} = \sqrt{m^2 + (p^1)^2 + (p^2)^2 + (p^3)^2}.
+\end{gather*}
+In the following proof it will be convenient to use the abbreviation
 \begin{equation*}
-  \omega(p_1,\ldots,p_s) = \omega(p_1) + \cdots + \omega(p_s)
+  \omega(p_1,\ldots,p_s) \defequal \omega(p_1) + \cdots + \omega(p_n).
 \end{equation*}
 
 \begin{myproof}[lemma:integral-kernel-h-bound]
@@ -778,28 +818,71 @@ In the following proof it will we convenient to use the abbreviation
     \le \frac{\omega(p_1) + \cdots + \omega(p_s)}{s} 
     \le \omega(p') \le 1 + \omega(k,p'), \nonumber\\
     \shortintertext{hence}
-    \label{equation:one-plus-omega-estimate}
+    \label{equation:one-plus-omega-estimate1}
     \parens[\big]{1+\omega(k,p')} {}^{-a}
-    \le \parens[\big]{\omega(p_1) \cdots \omega(p_s)} {}^{-a/s}.
+    \le \parens[\big]{\omega(p_1) \cdots \omega(p_s)} {}^{-a/s}, \\
+    \shortintertext{and similarly}
+    \label{equation:one-plus-omega-estimate2}
+    \parens[\big]{1+\omega(k,p)} {}^{-a}
+    \le \parens[\big]{\omega(p_1) \cdots \omega(p_{r-s})} {}^{-a/(r-s)}.
   \end{gather}
-  The estimates~\eqref{equation:polynomial-estimate} and~\eqref{equation:one-plus-omega-estimate} entail
+  The estimates~\eqref{equation:polynomial-estimate},~\eqref{equation:one-plus-omega-estimate1} and~\eqref{equation:one-plus-omega-estimate2} entail
   \begin{equation*}
     \abs{F(k,p',p)} \le C_s
     \prod_{i=1}^{s} \omega(p_i)^{d_i-a/s} 
-    \prod_{j=s+1}^{r} \omega(p_j)^{d_j-a/(r-s)} 
+    \prod_{j=s+1}^{r} \omega(p_j)^{d_j-a/(r-s)}.
   \end{equation*}
+  Since the right hand side does not depend on $k$, and the $p$-variables are separated,
+  the problem reduces to proving that
+  \begin{equation}
+    \label{equation:integral-finite}
+    \int \omega(q)^{-2b} \,d\Omega(q) < \infty
+  \end{equation}
+  for $b$ large enough.
+  Recall that $d \Omega(q) = \omega(q)^{-1} d^3 \symbfit{q}$.
+  By transformation to spherical coordinates, we find that~\eqref{equation:integral-finite}
+  is equivalent to
+  \begin{equation*}
+    \int \frac{r^2}{(m^2 + r^2)^{b+1/2}} \,dr < \infty
+  \end{equation*}
+  It it well known that this holds for $b > 1$.
+  $a > r d$ 
+
+  \begin{equation}
+    \label{equation:intermediate-result}
+    \norm{K_{\psi'\!,\psi}}_1 \le
+    \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
+    \sum_{s=0}^{r} \delta_{m-s}^{n-(r-s)} C_s
+    \underbracket{\norm{(1+H)^l \psi'_m}_2}_{a'_m}
+    \underbracket{\norm{(1+H)^l \psi_n}_2}_{a_n}
+  \end{equation}
+  We introduce auxiliary variables $a'_m, a_n$ as shown above, and for convenience set $a_n = 0$ whenever $n < 0$.
+  Using this, we rewrite the right hand side of~\eqref{equation:intermediate-result}
+  and apply the Cauchy-Schwarz Inequality for sequences as follows:
+  \begin{equation*}
+    \sum_{s=0}^{r} C_s \sum_{m = 0}^{\infty} a'_m \cdot a_{m+r-2s} \le
+    \sum_{s=0}^{r} C_s \sqrt{\sum_{m=0}^{\infty} a'^2_m \sum_{n=0}^{\infty} a^2_n}
+  \end{equation*}
+  To complete the proof, observe that
+  \begin{equation*}
+    \norm{(1+H)^l \psi'}_2 =
+    \sqrt{\sum_{m=0}^{\infty} \norm{(1+H)^l \psi'_m}_2^2} =
+    \sqrt{\sum_{m=0}^{\infty} a'^2_m},
+  \end{equation*}
+  and similar for $\psi$, by definition of the inner prouct
+  and because $((1+H)^l \psi')_m = (1+H)^l \psi'_m$ for all $m$.
 \end{myproof}
 
 \begin{lemma}{Renormalized Product at a Point}{}
   In the setting of \cref{lemma:renormalized-product-integral-representation},
-  assume that $\psi,\psi'$ are in $D^l(H)$.
+  assume that $\psi,\psi'$ are in $\Domain{H^l}$.
   Let $x$ be any point in $M$ and let $\delta_x \in \tempdistrib{M}$ be the Dirac distribution supported in $x$.
   Then the limit
   \begin{equation*}
     \lim_{f \to \delta_x}
     \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi}
   \end{equation*}
-  exists and depends continously on $x$.
+  exists and depends continuously on $x$.
 \end{lemma}
 
 \begin{proof}
@@ -813,8 +896,22 @@ In the following proof it will we convenient to use the abbreviation
   The integrand is dominated by the function $\abs{K_{\psi'\!,\psi}(p_1,\ldots,p_r)}$,
   which has finite integral as it is $L^1$
   by \cref{lemma:integral-kernel-h-bound}.
+
   Moreover, the integrand converges pointwise to $K_{\psi'\!,\psi}(p_1,\ldots,p_r)$, since $\ft{f} \to 1$ when $f \to \delta_x$.
   TODO(With of choice of FT constants, $\ft{f} \to 1/(2\pi)^2$. Change here or change def?)
+
+  Since the Fourier transformation of tempered distribution
+  is a continuous mapping $\tempdistribnoarg \to \tempdistribnoarg$,
+  we have $\ft{f} \to \FT{\delta_x}$ whenever $f \to \delta_x$ in the topology of $\tempdistribnoarg$.
+  Recall that $\ft{\delta} = 1$, and thus $\FT{\delta_x}(p) = e^{ix \cdot p}$ for all $p \in M$.
+  This shows that the integrand converges pointwise to
+  \begin{equation*}
+    \sum_{s=0}^r
+    e^{ix \cdot (p_1 + \cdots + p_s)}
+    e^{-ix \cdot (p_{s+1} + \cdots + p_r)}
+    \tilde{K}_{\psi'\!,\psi}(p_1,\ldots,p_r)
+  \end{equation*}
+
   The Dominated Convergence Theorem implies
 \end{proof}
 
@@ -847,20 +944,21 @@ In the following proof it will we convenient to use the abbreviation
 \begin{lemma}{TODO}{}  
   Let $\varphi$ be a free quantum field.
   Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients.
-  Then we have for all states $\psi,\psi' \in D^l(H)$
+  Suppose that $l$ is a positive integer large enough to satisfy the
+  Then we have for all states $\psi,\psi' \in \Domain{H^l}$
   \begin{multline*}
     \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f) \,\psi} = \\
     = \int dp_1 \!\cdots dp_r
     \sum_{s=0}^{r}
     \, \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r)
-    \, L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r)
+    \, \tilde{K}_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r)
   \end{multline*}
   where
   \begin{multline*}
-    L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) =
+    \tilde{K}_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) =
+    P_s(p_1,\ldots,p_r)
     \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
-    \delta_{m-s}^{n-(r-s)}
-    \ P_s(p_1,\ldots,p_r) \\
+    \delta_{m-s}^{n-(r-s)} \\
     \cdot \sqrt{m(m-1) \cdots (m-s+1)} \sqrt{n(n-1) \cdots (n-(r-s)+1)} \\
     \cdot \int dk_1 \cdots dk_{m-s} 
     \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)}
@@ -869,10 +967,12 @@ In the following proof it will we convenient to use the abbreviation
   and $P_s(p_1,\ldots,p_r)$ is defined as before.
 \end{lemma}
 
-%\[
-    %f(T), f\left( T \right),
-    %\int_{a}^{b} f\left( x \right) d x, \frac{1}{T},
-%\]
+\begin{proof}
+  a
+\end{proof}
+
+
+\section{Definition of the Stress Tensor}
 
 In the theory of a real scalar field $\phi$ of mass $m$,
 the Lagrangian density of the Klein-Gordon action is given by
@@ -890,7 +990,7 @@ Raising the index $\nu$ and inserting \cref{lagrangian-density} yields
 \end{equation*}
 The \emph{energy density}:
 \begin{equation*}
-  \rho = T^{00} = \frac{1}{2} \parens*{\sum_{\mu=0}^{3} (\partial^{\mu}\phi)^2  + m^2 \phi^2} 
+  \energydensity = T^{00} = \frac{1}{2} \parens*{\sum_{\mu=0}^{3} (\partial^{\mu}\phi)^2  + m^2 \phi^2} 
 \end{equation*}
 The discussion in the previous section enables us to define
 the \emph{renormalized stress-energy tensor} of a free scalar field $\varphi$ by
@@ -900,11 +1000,11 @@ the \emph{renormalized stress-energy tensor} of a free scalar field $\varphi$ by
 and this is a quadratic form.
 In particular, the energy density is
 \begin{equation*}
-  \rho  = \frac{1}{2} \sum_{\mu=0}^{3} \normord{(\partial^{\mu}\phi)^2}  + \frac{1}{2} m^2 \normord{\phi^2}
+  \energydensity  = \frac{1}{2} \sum_{\mu=0}^{3} \normord{(\partial^{\mu}\varphi)^2}  + \frac{1}{2} m^2 \normord{\varphi^2}
 \end{equation*}
 
 \begin{multline*}
-  \innerp{\psi'\!}{\rho(f) \,\psi} = \\
+  \innerp{\psi'\!}{\energydensity(f) \,\psi} = \\
   = \int dp_1 dp_2
   \parens{p_1^{\mu} p_2^{\mu} + m^2} 
   \sum_{s=0}^{r} (-1)^{s+1}
@@ -925,13 +1025,13 @@ where
 \begin{theorem}{TODO}{}  
   Let $\varphi$ be a free quantum field.
   Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients.
-  Then we have for all states $\psi,\psi' \in D^l(H)$
+  Then we have for all test functions $f \in \schwartz{M}$
   \begin{multline*}
     \normord{D_1 \varphi \cdots D_r \varphi}(f) \QFequal \int dp_1 \!\cdots dp_r \sum_{s=0}^{r}
     P_s(p_1,\ldots,p_r) \, \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r) \\
     \cdot a^\dagger(p_1) \cdots a^\dagger(p_s) a(p_{s+1}) \cdots a(p_r) 
   \end{multline*}
-  as quadratic forms on $D^l(H)$, where
+  as quadratic forms on $\Domain{H^l}$, where
   \begin{multline*}
     \quad P_s(p_1,\ldots,p_r) =
     \frac{1}{\sqrt{2^r}}
@@ -944,17 +1044,107 @@ where
 
 \begin{definition}{}{}
   \begin{multline*}
-    \rho(f) \QFequal \frac{1}{4} \int dp dp' (p \cdot p' + m^2)
+    \energydensity(f) \QFequal \frac{1}{4} \int dp dp' (p \cdot p' + m^2)
     \Big\lbrack \ft{f}(p+p') a(p) a(p') + {}\\
     + 2\ft{f}(p-p') a^\dagger(p) a(p') + \ft{f}(-p-p') a^\dagger(p) a^\dagger(p') \Big\rbrack
   \end{multline*}
 \end{definition}
 
+\begin{equation*}
+  \bar{p} := \eta p = (p^0,-\symbfit{p}) 
+\end{equation*}
+
+\begin{proposition}{}{}
+  \begin{multline*}
+    \innerp{\psi'}{\energydensity(f) \psi} =
+    \frac{1}{4} \int dp dp'
+    (\bar{p} \cdot p' + m^2)
+    \bracks[\big]{2 \ft{f}(p - p') L^1_{\psi'\!,\psi}(p,p')} \\
+    + (-\bar{p} \cdot p' + m^2)
+    \bracks[\big]{\ft{f}(- p - p') L^0_{\psi'\!,\psi}(p,p') + \ft{f}(p + p') L^2_{\psi'\!,\psi}(p,p')} 
+  \end{multline*}
+\end{proposition}
+
+\begin{proposition}{}{}
+  \begin{multline*}
+    \energydensity(f) \QFequal \frac{1}{4} \int dp dp'
+    (m^2 + \bar{p} \cdot p')
+    \bracks[\Big]{2\ft{f}(p-p') a^\dagger(p) a(p')} \\
+    + (m^2 - \bar{p} \cdot p')
+    \bracks[\Big]{\ft{f}(p+p') a(p) a(p') + \ft{f}(-p-p') a^\dagger(p) a^\dagger(p')} 
+  \end{multline*}
+\end{proposition}
+
+\begin{proposition}{}{}
+  The Fock vaccum $\fockvaccum$ lies in the domain of $\energydensity(f)\QFop{}$
+  for all test functions $f \in \schwartz{M}$
+  and $\energydensity(f)\QFop{}\fockvaccum$ is the vector $\psi$ defined by
+  \begin{equation*}
+    \psi_2(p,p') = \frac{\sqrt{2}}{4} (m^2 - \bar{p} \cdot p') \ft{f}(-p-p')
+  \end{equation*}
+  and $\psi_n \equiv 0$ for $n \ne 2$.
+\end{proposition}
+
+\begin{equation*}
+  \energydensity(f) \Omega = ?
+\end{equation*}
+
 \section{Essential Selfadjointness of Renormalized Products}
 
-a
+\begin{lemma}{H-Bounds for the Renormalized Product}{}
+  \begin{equation*}
+    \abs{\innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f)\psi}} \le
+    C \norm{(I+H)^l \psi} \norm{(I+H)^l \psi'}
+  \end{equation*}
+\end{lemma}
 
-%\nocite{*}
+\begin{proof}
+  This proof is nearly identical to that of \cref{lemma:integral-kernel-h-bound}
+  and we will only cover the differences.
+  \begin{equation*}
+    \begin{multlined}[c]
+    \abs{\innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f)\psi}} \le
+      \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{s=0}^{r}
+      \delta_{m-s}^{n-(r-s)} \\
+      \hspace{2.5cm} \cdot
+    \int \!dk \int \!dp'\! \int \!dp \, \abs*{F(k,p',p) \, G'(k,p') \, G(k,p)},
+    \end{multlined}
+  \end{equation*}
+  where
+  \begin{multline}
+    F(k,p',p) = \parens[\big]{1+\omega(k,p')} {}^{-a} \parens[\big]{1+\omega(k,p)} {}^{-a} P_s(p',p) \\
+    \cdot \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r)
+  \end{multline}
+  and $G$ and $G'$ are defined as before.
+  All we have to do, is verifying that
+  \begin{equation*}
+    \sup_k \norm{F(k,\cdot,\cdot)}_2 < \infty
+  \end{equation*}
+  for a sufficiently large integer $a$.
+  Then we obtain the desired $H$-bound with $l=a+r/2$.
+
+  Recall that the Schwartz class is preserved by Fourier transform, translation and multiplication with polynomials.
+  Moreover, it is well known that Schwartz functions are square-integrable with repect to the Lorentz invariant measure on the mass shell.
+  Hence,
+  \begin{equation*}
+    \int dp_1 \abs{\ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r)}^2
+  \end{equation*}
+  is bounded by a constant independent of $p_2,\ldots,p_r$.
+\end{proof}
+
+\section{Covariance}
+
+\begin{equation*}
+  f_g(x) \defequal f(g^{-1} x) \qquad
+  x \in M, \quad g \in \ProperOrthochronousPoincareGroup.
+\end{equation*}
+
+\begin{theorem}{Covariance}{covariance-renormalized-product}
+  \begin{equation*}
+    U(g) \,\normord{D_1 \varphi \cdots D_r \varphi}(f)\, U(g)^{-1}
+    = \normord{D_1 \varphi \cdots D_r \varphi}(f_g)
+  \end{equation*}
+\end{theorem}
 
 \chapterbib
 \cleardoublepage