weiter

3 files changed, 696 insertions, 12 deletions

diff --git a/bib/stresstensor.bib b/bib/stresstensor.bib
new file mode 100644
index 0000000..c2f835d
--- /dev/null
+++ b/bib/stresstensor.bib
@@ -0,0 +1,57 @@
+@article{Segal1969,
+ author    = {Irving Segal},
+ title     = {Nonlinear Functions of Weak Processes. I},
+ publisher = {Elsevier Science},
+ journal   = {Journal of Functional Analysis},
+ issn      = {0022-1236,1096-0783},
+ year      = {1969},
+ volume    = {4},
+ number     = {3},
+ pages     = {404--456},
+}
+@article{Segal1970,
+ title     = {Nonlinear Functions of Weak Processes. II},
+ author    = {Irving Segal},
+ publisher = {Elsevier Science},
+ journal   = {Journal of Functional Analysis},
+ issn      = {0022-1236,1096-0783},
+ year      = {1970},
+ volume    = {6},
+ number     = {1},
+ pages     = {29--75},
+}
+@article{Klein1973,
+  title = {Renormalized products of the generalized free field and its derivatives},
+  volume = {45},
+  ISSN = {0030-8730},
+  number = {1},
+  journal = {Pacific Journal of Mathematics},
+  publisher = {Mathematical Sciences Publishers},
+  author = {Abel Klein},
+  year = {1973},
+  month = mar,
+  pages = {275–-292}
+}
+@article{Baez1989,
+  title = {Wick Products of the Free Bose Field},
+  volume = {86},
+  ISSN = {0022-1236},
+  number = {2},
+  journal = {Journal of Functional Analysis},
+ publisher = {Elsevier Science},
+ author    = {John C. Baez},
+  year = {1989},
+  month = oct,
+  pages = {211–-225}
+}
+@article{Fewster1998,
+ title     = {Bounds on negative energy densities in flat spacetime},
+ author    = {C. J. Fewster and S. P. Eveson},
+ publisher = {The American Physical Society},
+ journal   = {Physical Review D},
+ issn      = {1550-7998,1089-4918},
+ year      = {1998},
+ volume    = {58},
+ number     = {8},
+ pages     = {084010},
+}
diff --git a/preamble.tex b/preamble.tex
index 95aff6c..37068a4 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -10,11 +10,11 @@
 \usepackage{selnolig}
 \usepackage{amsmath,amsthm}
 \usepackage{mathtools}
-\usepackage[colon=literal]{unicode-math}
+\usepackage[colon=literal]{unicode-math} % TODO get rid of this since it messes up math italic correction
 \usepackage{enumitem}
 %\usepackage{graphicx}
 \usepackage{tcolorbox}
-\usepackage[style=ext-numeric]{biblatex}
+\usepackage[style=ext-alphabetic]{biblatex}
 \usepackage[intoc]{nomencl}
 \usepackage{makeidx}
 \usepackage{idxlayout}
@@ -98,11 +98,12 @@
 \DeclarePairedDelimiterX\LorentzBF[2]{\lparen}{\rparen}{#1,#2}
 
 % ---------- tcolorbox
-\tcbuselibrary{skins,theorems} % add breakable library?
+\tcbuselibrary{skins,theorems,breakable} % add breakable library?
 \tcbset{%
   beforeafter skip balanced=0.4\baselineskip,
   mythmstyle/.style={%
     enhanced,
+    breakable,
     sharp corners=all,
     interior hidden,
     borderline west={3pt}{0pt}{#1},
@@ -137,6 +138,10 @@
 \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{%
@@ -170,7 +175,7 @@
 
 % ---------- nomencl
 \makenomenclature
-\renewcommand{\nomname}{List of Symbols}
+\renewcommand*{\nomname}{List of Symbols}
 \def\pagedeclaration#1{, \hyperlink{page.#1}{page\nobreakspace#1}}
 
 % ---------- makeidx
@@ -213,8 +218,8 @@
 {\catcode`\@=\active\gdef@{\mkern1mu}}
 
 % ---------- misc
-\renewcommand\phi\varphi
-\renewcommand\epsilon\varepsilon
+%\renewcommand\phi\varphi
+%\renewcommand\epsilon\varepsilon
 % number systems
 \newcommand*{\NN}{\mathbb{N}}
 \newcommand*{\ZZ}{\mathbb{Z}}
@@ -228,9 +233,9 @@
 \renewcommand{\Im}{\operatorname{Im}}
 
 % emphasis for defined terms
-\newcommand{\defn}[1]{\textbf{\textit{#1}}}
+\newcommand*{\defn}[1]{\textbf{\textit{#1}}}
 
-\newcommand{\ts}[1]{\textnormal{#1}} % textual subscript
+\newcommand*{\ts}[1]{\textnormal{#1}} % textual subscript
 
 % Fourier transformation
 % ----------------------
@@ -247,6 +252,14 @@
 % Hilbert space direct sum
 %\newcommand{\AlgebraicDirectSum}[1]{\sideset{}{_{\!\ts{Hilb}}}\bigoplus #1}
 
+% Test functions and distributions
+% --------------
+\newcommand*{\testfun}[1]{\mathcal{D}(#1)}
+\newcommand*{\distrib}[1]{\mathcal{D}'(#1)}
+\newcommand*{\schwartz}[1]{\mathcal{S}(#1)}
+\newcommand*{\realschwartz}[1]{\mathcal{S}_{\RR}(#1)}
+\newcommand*{\tempdistrib}[1]{\mathcal{S}'(#1)}
+
 % Fock spaces
 % -----------
 \newcommand*{\FullFock}[1]{\mathcal{F}(#1)}
@@ -258,9 +271,14 @@
 
 % Operators
 % ---------
-\newcommand{\Domain}[1]{\mathcal{D}(#1)}
-\newcommand{\Range}[1]{\mathcal{R}(#1)}
-\newcommand{\Graph}[1]{\mathcal{G}(#1)}
+\newcommand*{\Domain}[1]{\mathcal{D}(#1)}
+\newcommand*{\Range}[1]{\mathcal{R}(#1)}
+\newcommand*{\Graph}[1]{\mathcal{G}(#1)}
+
+% Quadratic forms
+% ---------
+\newcommand*{\QF}[1]{QF(#1)}
+\newcommand{\QFequal}{\overset{\text{\scriptsize QF}}{=}}
 
 % Standard Subspaces
 % ------------------
@@ -273,4 +291,13 @@
 \newcommand*{\realorthcomp}{^\bot}
 
 % normal ordering (aka Wick ordering)
-\newcommand{\normord}[1]{{\vcentcolon\mathrel{#1}\vcentcolon}}
+\newcommand*{\normord}[1]{{\vcentcolon\mathrel{#1}\vcentcolon}}
+
+\newcommand\restr[2]{{% we make the whole thing an ordinary symbol
+  \left.\kern-\nulldelimiterspace % automatically resize the bar with \right
+  #1 % the function
+  \vphantom{\big|} % pretend it's a little taller at normal size
+  \right| {#2} % this is the delimiter
+  }}
+
+\newcommand{\defequal}{\overset{\text{\scriptsize def}}{=}}
diff --git a/stresstensor.tex b/stresstensor.tex
new file mode 100644
index 0000000..4cdc321
--- /dev/null
+++ b/stresstensor.tex
@@ -0,0 +1,600 @@
+\chapter{Construction of the Stress Tensor of a Free Scalar Quantum Field}
+\label{chapter:stress-tensor}
+
+\begin{equation*}
+  H = \tfrac{1}{2} \parens*{(\partial_t \phi)^2 + \abs{\nabla_{\!\!\symbfit{x}} \phi}^2 + m^2 \phi^2} 
+\end{equation*}
+
+At the end of this \namecref{chapter:stress-tensor} 
+we will have gained the ability to rigorously define
+arbitrary renormalized products of the free field and its derivatives
+as a densely defined quadratic-form valued tempered distribution,
+which on the dense subspace of the smooth vectors of the Hamiltonian
+is realized by essentially self-adjoint operators.
+
+\section{Choosing Conventions and Fixing Notation}
+\label{section:conventions}
+
+It is an unfortunate reality of quantum physics literature that
+there is a great deal of variation in notation and choice of signs and constants.
+While this does not affect the physical or mathematical content,
+it is a hindrance when working with formulas from multiple sources.
+In the present \namecref{section:conventions} we detail our choices
+as a service to the reader.
+
+\begin{itemize}
+  \item \emph{Minkowski space} $M=\RR^4$ equipped with the \emph{Lorentz bilinear form} (or metric)
+    \begin{equation*}
+      x \cdot y = g_{\mu \nu} x^{\mu} y^{\nu} = x^0y^0 - x^1 y^1 - x^2 y^2 - x^3 y^3
+    \end{equation*}
+    points $x = (x^0,x^1,x^2,x^3) \in M$ are sometimes written $x = (x^0,\symbfit{x})$ with separated time and space coordinates
+  \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
+    \end{equation}
+    whenever the integral converges. The \emph{inverse Fourier transform} is TODO
+  \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 ${}^{\dagger}$.
+    We avoid confusion by using $\overline{\phantom{z}}$ for complex conjugation
+    and ${}^{\dagger}$ for the Hilbert adjoint exclusively.
+  \item $\schwartz{M}$ is the space of complex-valued Schwartz functions on $M$ \\
+    $\realschwartz{M}$ is the space of real-valued Schwartz functions on $M$ 
+  \item The \emph{Bosonic Fock space} over a Hilbert space $\hilb{H}$ is denoted $\BosonFock{\hilb{H}}$. \\
+    Its \emph{finite particle subspace} is denoted $\BosonFockFinite{\hilb{H}}$.
+  \item Abstract free field: The \emph{Segal quantization} $\Phi$ assigns to every
+    $g \in \hilb{H}$, a selfadjoint (unbounded) operator $\Phi(g)$ in $\BosonFock{\hilb{H} }$,
+    which on the the finite particle subspace is given by
+    \begin{equation*}
+      \Phi_{\mathrm{S}} (g) = \frac{1}{\sqrt{2}} \parens*{a(g) + a(g)^\dagger} 
+    \end{equation*}
+    annihilation and creation operators, $g \in \hilb{H}$, $\psi \in \BosonFock{\hilb{H}}$ for $\hilb{H} = L^2(R^4,\Omega_m)$
+    \begin{align*}
+      \parens[\big]{a(g) \psi} {}_n (k_1, \ldots, k_n)
+      &= \sqrt{n+1} \int_M \! \bar{g}(p) \, \psi_{n+1} (p,k_1, \ldots, k_n) \, d\Omega_m(p) \\
+      \parens[\big]{a(g)^\dagger \psi} {}_n (k_1, \ldots, k_n)
+      &= \frac{1}{\sqrt{n}} \sum_{i=1}^n g(k_i) \, \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n)
+    \end{align*}
+    The symbol $\widehat{\hphantom{k_i}}$ over $k_i$ indicates omission.
+    \begin{align*}
+      \parens[\big]{a(g) \psi} {}_{n-1} (k_1, \ldots, k_{n-1}) 
+      &= \sqrt{n} \int_M \! \bar{g}(p) \, \psi_n (p,k_1, \ldots, k_{n-1}) \, d\Omega_m(p) \\
+      \parens[\big]{a(g)^\dagger \psi} {}_{n+1} (k_1, \ldots, k_{n+1}) 
+      &= \frac{1}{\sqrt{n+1}} \sum_{i=1}^{n+1} g(k_i) \, \psi_n (k_1, \ldots, \widehat{k_i}, \ldots, k_{n+1})
+    \end{align*}
+    \begin{multline*}
+      \parens[\big]{a(g) a(g) \psi} {}_{n-2} (k_1, \ldots, k_{n-2}) = \\
+      \sqrt{n} \sqrt{n-1} \int_M \int_M \! \bar{g}(p_1) \bar{g}(p_2) \, \psi_n (p_1,p_2,k_1, \ldots, k_{n-s}) \, d\Omega_m(p_1) d\Omega_m(p_2) \\
+    \end{multline*}
+    \begin{multline}
+      \parens[\big]{a(g_1) \cdots a(g_s) \psi} {}_{n-s} (k_1, \ldots, k_{n-s}) = 
+      \sqrt{n (n-1) \cdots (n-s+1)} \cdot {} \\
+      \cdot \int_M \!\! d\Omega_m(p_1) \cdots \!\! \int_M \!\! d\Omega_m(p_s) \ \bar{g_1}(p_1) \cdots \bar{g_s}(p_s) \ \psi_n (k_1, \ldots, k_{n-2},p_1,\ldots,p_s)
+    \end{multline}
+  \item We consider the free Hermitian scalar field of mass $m > 0$. \\
+    \emph{mass hyperboloid} $X_m^+ = \braces{p \in M \mid p^2 = m^2, p^0 > 0 }$ 
+    with normalized Lorentz invariant measure $\Omega_m$
+  \item single particle state space: $\hilb{H} = L^2(X_m^+, \Omega_m)$
+    \begin{equation*}
+      E : \schwartz{M} \to \hilb{H}, \quad f \mapsto Ef = \left.\ft{f}\,\right\vert {X_m^+}
+    \end{equation*}
+    \begin{equation*}
+      \realschwartz{M} \ni f \mapsto \Phi(f) = \Phi_{\mathrm{S}}(Ef) = \frac{1}{\sqrt{2}} \parens*{a(Ef) + a(Ef)^\dagger}
+    \end{equation*}
+  \item annihilation and creation operators, $f \in \schwartz{M}$, $\psi \in \BosonFock{\hilb{H}}$
+    \begin{align*}
+      \parens[\big]{a(f) \psi} {}_n (k_1, \ldots, k_n)
+      &= \sqrt{n+1} \int_M \! \overline{Ef(p)} \, \psi_{n+1} (p,k_1, \ldots, k_n) \, d\Omega_m(p) \\
+      \parens[\big]{a(f)^\dagger \psi} {}_n (k_1, \ldots, k_n)
+      &= \frac{1}{\sqrt{n}} \sum_{i=1}^n Ef(k_i) \, \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n)
+    \end{align*}
+    annihilation operator a point $p$ in momentum space.
+    \begin{equation*}
+      \parens[\big]{a(p) \psi} {}_n (k_1, \ldots, k_n)
+      = \sqrt{n+1} \, \psi_{n+1} (p,k_1, \ldots, k_n)
+    \end{equation*}
+    creation \enquote{operator} a point $p$ in momentum space.
+    \begin{equation*}
+      \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*}
+    \begin{gather*}
+      a(p)^\dagger : F \times F \longrightarrow \CC \\
+      \innerp[\big]{\psi'}{a(p)^\dagger \psi}
+      \defequal
+      \begin{multlined}[t]
+        \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \sum_{i=1}^n
+        \int_M \! d\Omega_m(k_1) \cdots \widehat{d\Omega_m(k_i)} \cdots d\Omega_m(k_n) \\
+        \cdot \overline{\psi'_{n} (k_1, \ldots, \underset{i}{p}, \ldots, k_n)}
+        \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n)
+      \end{multlined}
+    \end{gather*}
+    \begin{equation*}
+      \innerp[\big]{\psi'}{a(p)^\dagger \psi} =
+      \innerp[\big]{a(p) \psi'}{\psi}
+    \end{equation*}
+    Define ...
+    \begin{equation*}
+      a(p_1)^\dagger \cdots a(p_s)^\dagger a(p_{s+1}) \cdots a(p_r)
+    \end{equation*}
+    \begin{equation*}
+      \innerp[\big]{\psi'}{a(p_1)^\dagger \cdots a(p_s)^\dagger a(p_{s+1}) \cdots a(p_r) \psi}
+      = \innerp[\big]{a(p_1) \cdots a(p_s) \psi'}{a(p_{s+1}) \cdots a(p_r) \psi}
+    \end{equation*}
+    abc
+\end{itemize}
+
+\begin{equation*}
+  \normord{\varphi(f)^2} = \tfrac{1}{2} \parens[\big]{a^{\dagger}(Ef) a^{\dagger}(Ef) + a(Ef) a(Ef)} + a^{\dagger}(Ef) a(Ef) 
+\end{equation*}
+
+\begin{equation*}
+  \normord{\varphi(f)^2} = \tfrac{1}{2} \parens[\big]{a^{\dagger} a^{\dagger} + a a} + a^{\dagger} a \quad \text{where} \quad a = a(Ef)
+\end{equation*}
+
+\begin{equation*}
+  \innerp{\psi'}{\normord{\varphi(f)^2} \,\psi} = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \innerp{\psi'_m}{\normord{\varphi(f)^2} \,\psi_n}
+\end{equation*}
+
+There will be no contribution to the sum
+unless either $m=n+2$ or $m=n$ or $m=n-2$.
+Now let us calculate those contributions.
+To avoid the introduction of sums,
+we use the adjoint identity
+to transform creation operators on the right
+into annihilation operators on the left.
+
+\begin{align*}
+  \innerp{\psi'_{n+2}}{\normord{\varphi(f)^2} \,\psi_n} 
+  &= \tfrac{1}{2} \innerp{a(Ef) a(Ef) \psi'_{n+2}}{\psi_n} \\
+  &= \begin{multlined}[t][10cm]
+    \tfrac{1}{2} \sqrt{n+2} \sqrt{n+1} \int dp_1 dp_2 \, \ft{f}(p_1) \ft{f}(p_2) \int dk_1 \cdots dk_n \\
+    \overline{\psi'_{n+2}(p_1,p_2,k_1,\ldots,k_n)} \, \psi_n(k_1,\ldots,k_n)
+  \end{multlined} \\[1ex]
+  \innerp{\psi'_{n}}{\normord{\varphi(f)^2} \,\psi_n} 
+  &= \innerp{a(Ef) \psi'_{n}}{a(Ef) \psi_n} \\
+  &= \begin{multlined}[t][10cm]
+    \sqrt{n} \sqrt{n} \int dp_1 dp_2 \, \ft{f}(p_1) \overline{\ft{f}(p_2)} \int dk_1 \cdots dk_{n-1} \\
+    \overline{\psi'_{n}(p_1,k_1,\ldots,k_{n-1})} \, \psi_n(p_2,k_1,\ldots,k_{n-1})
+  \end{multlined} \\[1ex]
+  \innerp{\psi'_{n-2}}{\normord{\varphi(f)^2} \,\psi_n} 
+  &= \tfrac{1}{2} \innerp{\psi'_{n-2}}{a(Ef) a(Ef) \psi_n} \\
+  &= \begin{multlined}[t][10cm]
+    \tfrac{1}{2} \sqrt{n} \sqrt{n-1} \int dp_1 dp_2 \, \overline{\ft{f}(p_1)} \overline{\ft{f}(p_2)} \int dk_1 \cdots dk_{n-2} \\
+    \overline{\psi'_{n-2}(k_1,\ldots,k_{n-2})} \psi_n(p_1,p_2,k_1,\ldots,k_{n-2})
+  \end{multlined}
+\end{align*}
+
+\begin{align*}
+  \innerp{\psi'_{m}}{\normord{\varphi(f)^2} \,\psi_n} 
+  &= \begin{multlined}[t][10cm]
+    \tfrac{1}{2} \sqrt{n+2} \sqrt{n+1} \int dp_1 dp_2 \, \ft{f}(p_1) \ft{f}(p_2) \int dk_1 \cdots dk_n \\
+    \chi(p_{s+1}) \cdots \chi(p_2) \\
+    \overline{\psi'_{m}\parens{k_1,\ldots,k_{n-s},p_1,\ldots,p_s}} \, \psi_n\parens{k_1,\ldots,k_{n-(2-s)},p_{s+1},\ldots,p_2}
+  \end{multlined}
+\end{align*}
+
+\begin{proposition}{}{}
+  asdf
+\end{proposition}
+
+\subsubsection{Linear Differential Operators and their Formal Adjoint}
+
+Before we turn to the problem of defining renormalized products of a quantum field and its derivatives
+we must clarify what is meant mathematically by the derivative of a field.
+For this, we recall that in Wightmans approach to quantum field theory,
+a quantum field $\varphi$ on a spacetime manifold $M$ is modeled by an operator valued tempered distribution,
+that is a mapping that assigns to each (Schwatz class) test function $f$ on $M$ an unbounded operator $\varphi(f)$
+in the Fock space xxx over some Hilbert space $\hilb{H}$, such that for each fixed pair of states $\psi,\psi'$
+the mapping
+\begin{equation*}
+  \schwartz{M} \to \CC, \quad
+  f \mapsto \innerp{\psi'}{\varphi(f) \psi}
+\end{equation*}
+is a (scalar-valued) tempered distibution on $M$.
+It is well known that tempered distibutions have partial derivatives of any order.
+Suppose we work with $M = \RR^d$ for simplicity,
+and let $\partial_i$ denote the partial derivative with respect to the $i$-th coordinate.
+Then a general \emph{linear differential operator with constant coefficients} on $M$ looks like
+\begin{equation*}
+  D = \sum_{\alpha} a_{\alpha} \partial^{\alpha},
+\end{equation*}
+where the sum runs over all multi-indices $\alpha = (\alpha_1,\ldots,\alpha_d) \in \NN^d$,
+the coefficients $a_{\alpha}$ are complex numbers,
+and $\partial^{\alpha} = \partial_1^{\alpha_1} \!\cdots \partial_d^{\alpha_d}$.
+Then the \emph{distributional derivative} of a tempered distribution $\eta \in \tempdistrib{\RR^d}$
+is defined by
+\begin{equation*}
+  (D\eta)(f) = \eta(D^{\dagger}f) \quad \forall f \in \mathcal{S},
+\end{equation*}
+where the \emph{formal adjoint} of $D$ is the linear differential operator with constant coefficients given by
+\begin{equation*}
+  D^\dagger = \sum_{\alpha} (-1)^{\abs{\alpha}} a_{\alpha} \partial^{\alpha}.
+\end{equation*}
+Here we use the notation $\abs{\alpha} = \alpha_1 + \cdots + \alpha_d$.
+The functional $D \eta$ is well defined, because the Schwartz class is stable under the application of
+linear differential operators with constant coefficients.
+It can be shown that $D \eta$ is again a tempered distribution.
+The appearance of $-1$ in $D^{\dagger}$ is justified by the adjoint identity
+\begin{equation*}
+  \int (Df)(x) g(x) dx = \int f(x) (D^{\dagger}g)(x) dx,
+\end{equation*}
+which holds for all functions $f,g \in \schwartz{\RR^d}$
+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}
+\end{equation*}
+As one expects, $D\varphi$ is an operator-valued tempered distribution on $M=\RR^d$.
+TODO
+\begin{equation}
+  \label{derivative-free-field}
+  D \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(ED^{\dagger}f)^{\dagger} + a(ED^{\dagger}f)}
+\end{equation}
+
+
+The operator corresponding to $D$ in Fourier space is the multiplication operator
+\begin{equation*}
+  -i \sum_{\alpha} a_{\alpha} p_0^{\alpha_0} (-p_1)^{\alpha_1} (-p_2)^{\alpha_2} (-p_3)^{\alpha_3}
+\end{equation*}
+
+
+Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients.
+\begin{equation*}
+  \normord{D_1 \varphi(f) \cdots D_r \varphi(f)}
+\end{equation*}
+
+\section{Renormalized Products of the Free Field and its Derivatives}
+
+
+
+\begin{lemma}{Integral Representation of the Renormalized Product}{renormalized-product-integral-representation}  
+  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 $\psi,\psi'$
+  \begin{equation*}
+    \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} =
+    \int dp_1 \!\cdots dp_r
+    \, \ft{f}(p_1) \cdots\! \ft{f}(p_r)
+    \, K_{\psi'\!,\psi}(p_1,\ldots,p_r)
+  \end{equation*}
+  where the \enquote{integral kernel} is given by
+  \begin{multline*}
+    K_{\psi'\!,\psi}(p_1,\ldots,p_r) =
+    \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{s=0}^{r}
+    \delta_{m-s}^{n-(r-s)}
+    \ \chi(p_{s+1}) \cdots \chi(p_{r})
+    \ P_s(p_1,\ldots,p_r) \\
+    \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*}
+  where $\chi(p) = \overline{\ft{f}(p)} / \ft{f}(p)$ and
+  \begin{equation*}
+    P_s(p_1,\ldots,p_r) =
+    \frac{1}{\sqrt{2^r}} \sum_{\sigma \in S_r}
+    \ft{D}_{\sigma(1)}(p_1) \cdots \ft{D}_{\sigma(s)}(p_s)
+    \overline{\ft{D}_{\sigma(s+1)}(p_{s+1}) \cdots \ft{D}_{\sigma(r)}(p_r)}.
+  \end{equation*}
+\end{lemma}
+
+In the special case that $D_1 = \cdots = D_n = D$ we have
+\begin{equation*}
+  P_s(p_1,\ldots,p_r) =
+  \sqrt{2^r}
+  \ft{D}(p_1) \cdots \ft{D}(p_s)
+  \overline{\ft{D}(p_{s+1}) \cdots \ft{D}(p_r)}.
+\end{equation*}
+For squares, that is $r=2$
+\begin{equation*}
+  P_s(p_1,p_2) = \begin{cases}
+    2 \, \ft{D}(p_1)\ft{D}(p_2) & s=0 \\
+    2 \, \ft{D}(p_1)\overline{\ft{D}(p_2)} & s=1 \\
+    2 \, \overline{\ft{D}(p_1)\ft{D}(p_2)} & s=2
+  \end{cases}
+\end{equation*}
+
+\begin{myproof}[lemma:renormalized-product-integral-representation]
+\begin{multline*}
+  \innerp{\psi'}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} =
+  \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
+  \frac{1}{\sqrt{2^r}} \sum_{s=0}^{r} \sum_{\sigma \in S_r} \\
+  \big\langle
+  a(ED_{\sigma(1)}^{\dagger}f) \cdots a(ED_{\sigma(s)}^{\dagger}f) \psi_m,
+  a(ED_{\sigma(s+1)}^{\dagger}f) \cdots a(ED_{\sigma(r)}^{\dagger}f) \psi_n
+  \big\rangle
+\end{multline*}
+  \begin{gather*}
+    \sqrt{m(m-1) \cdots (m-s+1)}
+    \sqrt{n(n-1) \cdots (n-(r-s)+1)}
+    \int dk_1 \cdots dk_{m-s} \\
+    \int dp_1 \cdots dp_s
+    \ ED_{\sigma(1)}^{\dagger}f(p_1) \cdots ED_{\sigma(s)}^{\dagger}f(p_s)
+    \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)} \\
+    \int dp_{s+1} \cdots dp_r
+    \ \overline{ED_{\sigma(s+1)}^{\dagger}f(p_{s+1}) \cdots ED_{\sigma(r)}^{\dagger}f(p_r)}
+    \ \psi_n(k_1,\ldots,k_{n-(r-s)},p_{s+1},\ldots,p_r)
+  \end{gather*}
+\end{myproof}
+
+The following assertion is key
+
+\begin{lemma}{}{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 arbitary states $\psi,\psi' \in xxx$,
+  and test functions $f \in \schwartz{M}$,
+  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*}
+\end{lemma}
+
+\begin{myproof}
+  We have to find an estimate for
+  \begin{equation*}
+    \norm{K_{\psi'\!,\psi}}_1 =
+    \int dp_1 \!\cdots dp_r
+    \, \abs{K_{\psi'\!,\psi}(p_1,\ldots,p_r)}.
+  \end{equation*}
+   We apply the triangle inequalities for sums and integrals
+   to the expression for $K_{\psi'\!,\psi}$ given in \cref{lemma:renormalized-product-integral-representation},
+   use the fact that $\chi(p)$ has modulus one, make the estimates
+   \begin{equation*}
+    m(m-1) \cdots (m-s+1) \le m^r
+    \quad \text{and} \quad 
+    n(n-1) \cdots (n-(r-s)+1) \le n^r,
+   \end{equation*}
+   and finally reorder the integration with Fubini’s theorem
+   to obtain
+  \begin{equation}
+    \label{first-estimate}
+    \begin{multlined}[c]
+      \norm{K_{\psi'\!,\psi}}_1 \le
+      \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{s=0}^{r}
+      \delta_{m-s}^{n-(r-s)} \sqrt{m^r n^r} \\
+      \hspace{2.5cm} \cdot \abs*{\int \!dk \int \!dp'\! \int \!dp \,
+      P_s(p',p) \, \psi'_m(k,p') \, \psi_n(k,p)},
+    \end{multlined}
+  \end{equation}
+  where we have used the abbreviations
+  \begin{align*}
+    k &= (k_1,\ldots,k_{m-s})
+    \quad p' = (p_1,\ldots,p_s)
+    \quad p = (p_{s+1},\ldots,p_r) \\
+    dk &= dk_1 \cdots dk_{m-s}
+    \quad \text{and so on.}
+  \end{align*}
+  Observe that $P_s(p_1,\ldots,p_r)$ is a (complex) polynomial
+  in the $4r$ variables $p_i^\mu$, $i=1,\ldots,r$, $\mu=0,\ldots,3$.
+  Its degree is given by
+  \begin{equation*}
+    \deg P_s = \sum_{i=1}^r \deg \ft{D}_i
+  \end{equation*}
+  that is the sum of the highest orders of differentiation
+  occurring in each of the operators $D_1, \ldots, D_r$.
+  There is no reason to expect arbitary states $\psi,\psi'$ to temper fast enough
+  to counteract this polynomial growth.
+  Thus, the integral in \cref{first-estimate} will not converge, in general.
+  However, if $\psi$ lies in the domain of $H^l$ for some positive integer $l$,
+  then we can be sure that $(1+H)^l \psi$ is square integrable, and we have
+  \begin{equation*}
+    \psi_n(k,p) = \parens[\big]{1+\omega(k,p)} {}^{-l} (1+H)^l \psi_n(k,p)
+  \end{equation*}
+  \begin{equation*}
+    (1+H)^l psi
+  \end{equation*}
+  \begin{equation*}
+    \abs*{\int \!dk \int \!dp'\! \int \!dp \, F(k,p',p) \, G'(k,p') \, G(k,p)}
+  \end{equation*}
+  where
+  \begin{align}
+    F(k,p',p) &= \parens[\big]{1+\omega(k,p')} {}^{-l} \parens[\big]{1+\omega(k,p)} {}^{-l} P_s(p',p) \\
+    G'(k,p') &= \sqrt{m^r} (1+H)^l \psi_m(k,p') \\
+    G(k,p) &= \sqrt{n^r} (1+H)^l \psi_n(k,p)
+  \end{align}
+  \begin{equation*}
+    \abs*{\int dp G(k,p) F(k,p',p)}^2
+    \le \int dp \abs{F(k,p',p)}^2
+    \cdot \int dp \abs{G(k,p)}^2
+  \end{equation*}
+  \begin{equation*}
+    \int dp' \abs*{\int dp G(k,p) F(k,p',p)}^2
+    \le \int dp \abs{G(k,p)}^2
+    \sup_{k} \norm{F(k,\cdot,\cdot)}_2^2
+  \end{equation*}
+  \begin{align*}
+    &\quad \abs*{\int \!dk \int \!dp'\! \int \!dp \, F(k,p',p) \, G'(k,p') \, G(k,p)} \\
+    &\le \int \!dk \int \!dp' \abs{G'(k,p')} \abs*{\int \!dp \, F(k,p',p) \, G(k,p)} \\
+    &\le \norm{G'}_2 \parens*{\int dk \int dp' \abs*{\int dp G(k,p) F(k,p',p)}^2} \\
+    &\le \norm{G'}_2 \norm{G}_2 \sup_{k} \norm{F(k,\cdot,\cdot)}_2
+  \end{align*}
+  We claim that there exists a positive constant $C_1$ independent of $m$ and $n$ such that
+  \begin{equation*}
+    \norm{G}_2 \le C_1 \norm{(1+H)^{l+r/2} \psi_n}_2
+  \end{equation*}
+  and similary for $G'$.
+  This follows from $N \psi_n = n\psi_n$, where $N$ is the number operator, and the fact that $\omega(q)$ has a positive lower bound $M$
+
+  $H \psi_n(k,p) = (1+\omega(k,p) \psi_n(k,p)$ 
+
+  $1+\omega(k,p) \ge n \epsilon$ 
+
+  $\norm{(1+H)\psi_n}_2 \ge n \epsilon \norm{\psi_n} = \epsilon \norm{N \psi_n}$ 
+
+  In order to determine conditions for the finiteness of the remaining factor involving $F$,
+  it is desireable to have an estimate of the growth of $P_s$ in terms of $\omega(p_1),\ldots,\omega(p_r)$.
+  Notice that it is sufficient to make an estimate that is valid on the support of the measure $\Omega_m$, that is, the mass shell $X_m^+$,
+  since $F$ appears in an integral with respect to $p_1,\ldots,p_r$.
+  For an arbitrary point $q$ on the mass shell $X_m^+$ we have
+  \begin{align*}
+    q^{0} &= \omega(q) \\
+    \abs{q^{\mu}} &\le \omega(q) \quad \mu = 1,2,3.
+  \end{align*}
+  Moreover, $\omega(q)$ has a positive lower bound, namely $m$, so that
+  for all exponents $a,b \in \NN$ with $a < b$ there
+  exists a constant $c_{a,b}$ such that $\omega(q)^a \le c_{a,b}\, \omega(q)^b$.
+  This allows us to make the estimate
+  \begin{equation*}
+    \abs{P_s(p_1,\ldots,p_r)} \le C_s \prod_{i=1}^r \omega(p_i)^{d_i} \quad \text{where}\ d_i = \deg \ft{D}_i.
+  \end{equation*}
+
+  \begin{equation*}
+    \sqrt[s]{\omega(p_1) \cdots \omega(p_s)} 
+    \le \frac{\omega(p_1) + \cdots + \omega(p_s)}{s} 
+    \le \omega(p') \le 1 + \omega(k,p') 
+  \end{equation*}
+  \begin{equation*}
+    \parens[\big]{1+\omega(k,p')} {}^{-l}
+    \le \parens[\big]{\omega(p_1) \cdots \omega(p_s)} {}^{-l/s}
+  \end{equation*}
+  \begin{equation*}
+    \abs{F(k,p',p)} \le
+    \prod_{i=1}^{s} \omega(p_i)^{d_i-l/s} 
+    \prod_{j=s}^{r-s} \omega(p_j)^{d_j-l/(r-s)} 
+  \end{equation*}
+\end{myproof}
+
+\begin{lemma}{Renormalized Product at a Point}{}
+  In the setting of \cref{lemma:renormalized-product-integral-representation},
+  \begin{equation*}
+    \lim_{f \to \delta_x}
+    \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi}
+    = \int dp_1 \!\cdots dp_r
+    \, K_{\psi'\!,\psi}(p_1,\ldots,p_r)
+  \end{equation*}
+\end{lemma}
+
+\begin{definition}{Renormalized Product at a Point}{}
+  In the setting of \cref{lemma:renormalized-product-integral-representation},
+  \begin{equation*}
+    \normord{D_1 \varphi \cdots D_r \varphi} \ \vcentcolon \ 
+    M \to \QF{fock}
+  \end{equation*}
+  \begin{equation*}
+    \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(x) \,\psi}
+    = \int dp_1 \!\cdots dp_r
+    \, K_{\psi'\!,\psi}(p_1,\ldots,p_r)
+  \end{equation*}
+\end{definition}
+
+\begin{proof}
+  According to \cref{lemma:renormalized-product-integral-representation} we have
+  \begin{equation*}
+    \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi}
+    = \int dp_1 \!\cdots dp_r
+    \, \ft{f}(p_1) \cdots\! \ft{f}(p_r)
+    \, K_{\psi'\!,\psi}(p_1,\ldots,p_r)
+  \end{equation*}
+  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 $f \to \delta_x$.
+  The Dominated Convergence Theorem implies
+\end{proof}
+
+\begin{lemma}{Renormalized Product as a QF-valued distribution}{}
+  In the setting of \cref{lemma:renormalized-product-integral-representation},
+  \begin{equation*}
+    \normord{D_1 \varphi \cdots D_r \varphi} \ \vcentcolon \ 
+    \schwartz{M} \to \QF{fock}
+  \end{equation*}
+  \begin{equation*}
+    \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f) \,\psi} =
+    \int_M \!dx \ f(x) \ \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(x) \,\psi}
+  \end{equation*}
+\end{lemma}
+
+\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 $\psi,\psi'$
+  \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)
+  \end{multline*}
+  where
+  \begin{multline*}
+    L_{\psi'\!,\psi}^{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) \\
+    \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*}
+  and $P_s(p_1,\ldots,p_r)$ is defined as before.
+\end{lemma}
+
+\[
+    f(T), f\left( T \right),
+    \int_{a}^{b} f\left( x \right) d x, \frac{1}{T},
+\]
+
+In the theory of a real scalar field $\phi$ of mass $m$,
+the Lagrangian density of the Klein-Gordon action is given by
+\begin{equation}
+  \label{lagrangian-density}
+  \mathcal{L} = \frac{1}{2} \parens{\partial^{\mu} \phi \partial_{\mu} \phi - m^2 \phi^2} 
+\end{equation}
+and the \emph{canonical stress-energy tensor} is defined by
+\begin{equation*}
+  T^{\mu}_{\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \phi)} \partial_{\nu} \phi - \delta^{\mu}_{\nu} \mathcal{L}
+\end{equation*}
+Raising the index $\nu$ and inserting \cref{lagrangian-density} yields
+\begin{equation*}
+  T^{\mu\nu} = \partial^{\mu}\phi \partial^{\nu}\phi + \frac{1}{2} \eta^{\mu\nu} \parens*{m^2 \phi^2 - \partial_{\lambda}\phi \partial^{\lambda}\phi} 
+\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} 
+\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
+\begin{equation*}
+  T^{\mu\nu} = \normord{\partial^{\mu}\varphi \partial^{\nu}\varphi + \frac{1}{2} \eta^{\mu\nu} \parens*{m^2 \varphi^2 - \partial_{\lambda}\varphi \partial^{\lambda}\varphi}}
+\end{equation*}
+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}
+\end{equation*}
+
+\begin{multline*}
+  \innerp{\psi'\!}{\rho(f) \,\psi} = \\
+  = \int dp_1 dp_2
+  \parens{p_1^{\mu} p_2^{\mu} + m^2} 
+  \sum_{s=0}^{r} (-1)^{s+1}
+  \, \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r)
+  \, L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r)
+\end{multline*}
+where
+\begin{multline*}
+  L_{\psi'\!,\psi}^{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*}
+
+\begin{equation*}
+  A \QFequal B
+\end{equation*}
+
+\section{Essential Self-Adjointness of Renormalized Products}
+
+\nocite{*}
+
+\chapterbib
+\cleardoublepage
+
+% vim: syntax=mytex