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-rw-r--r--.cspell.yaml4
-rw-r--r--.cspell/project-words.txt7
-rw-r--r--analytic2.tex9
-rw-r--r--bib/stresstensor.bib21
-rw-r--r--conclusion.tex4
-rw-r--r--convolution.tex28
-rw-r--r--declaration.tex22
-rw-r--r--images/FAU_NatFak_EN_Q_RGB_black.pdf1270
-rw-r--r--index.tex2
-rw-r--r--intro.tex21
-rw-r--r--main.tex5
-rw-r--r--much.tex117
-rw-r--r--preamble.tex17
-rw-r--r--sampleappendix.tex17
-rw-r--r--samplesection.tex8
-rw-r--r--second.tex6
-rw-r--r--standard.tex21
-rw-r--r--stresstensor.tex464
-rw-r--r--titlepage.tex78
19 files changed, 524 insertions, 1597 deletions
diff --git a/.cspell.yaml b/.cspell.yaml
index 16a83c3..add1b47 100644
--- a/.cspell.yaml
+++ b/.cspell.yaml
@@ -20,9 +20,6 @@ dictionaryDefinitions:
- name: names
addWords: true
path: ./.cspell/my-cspell-dicts/names.txt
- - name: names2
- addWords: true
- path: ./.cspell/my-cspell-dicts/names2.txt
- name: math
addWords: true
path: ./.cspell/my-cspell-dicts/math.txt
@@ -32,7 +29,6 @@ dictionaries:
- project-words
- latex-pkgs
- names
- - names2
- math
ignorePaths:
- ./.cspell/
diff --git a/.cspell/project-words.txt b/.cspell/project-words.txt
index 3986d22..7656e56 100644
--- a/.cspell/project-words.txt
+++ b/.cspell/project-words.txt
@@ -11,3 +11,10 @@ sahlmann
epsteinglaserjaffe
distri
fau
+Erlangen
+chatbots
+lechner
+universität
+functorial
+preimage
+mytex
diff --git a/analytic2.tex b/analytic2.tex
index 13b6700..0648130 100644
--- a/analytic2.tex
+++ b/analytic2.tex
@@ -22,7 +22,7 @@ that is, in the open disc with radius $t$ centered in the origin of the complex
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.
+ Let $G \subset \CC$ be open and let $\hilb{X}$ be a Banach space.
A function $f : G \to \hilb{H}$ is called
\begin{itemize}
\item \emph{strongly analytic} at $a \in G$, if the limit
@@ -38,6 +38,10 @@ This is a well-known consequence of the convergence behavior of power series.
\end{itemize}
\end{definition}
+\begin{lemma}{Uniform Boundedness Theorem}{uniform-boundedness-theorem}
+ If a collection of bounded linear operators from a Banach space into a normed space is pointwise bounded, then it is uniformly bounded.
+\end{lemma}
+
\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 $G$.
@@ -130,6 +134,3 @@ This is a well-known consequence of the convergence behavior of power series.
\end{equation*}
has a positive radius of convergence $t>0$.
\end{myproof}
-
-\chapterbib
-\cleardoublepage
diff --git a/bib/stresstensor.bib b/bib/stresstensor.bib
index 38066ee..a87cb20 100644
--- a/bib/stresstensor.bib
+++ b/bib/stresstensor.bib
@@ -53,3 +53,24 @@
number = {8},
pages = {084010},
}
+@book{Baez1992,
+ title = {Introduction to Algebraic and Constructive Quantum Field Theory},
+ author = {John Baez and Irving Segal and Zhengfang Zhou},
+ publisher = {Princeton University Press},
+ isbn = {},
+ year = {1992},
+ series = {Princeton Series in Physics},
+ edition = {},
+ number = {56},
+}
+@article{Fewster2012,
+ title = {Dynamical Locality of the Free Scalar Field},
+ author = {Christopher J. Fewster and Rainer Verch},
+ publisher = {Springer},
+ journal = {Annales Henri Poincaré},
+ issn = {1424-0637},
+ year = {2012},
+ volume = {13},
+ issue = {7},
+ pages = {1675--1709},
+}
diff --git a/conclusion.tex b/conclusion.tex
new file mode 100644
index 0000000..7fe04a1
--- /dev/null
+++ b/conclusion.tex
@@ -0,0 +1,4 @@
+\addchap{Conclusion}
+\todo{xxx}
+
+% vim: syntax=mytex
diff --git a/convolution.tex b/convolution.tex
index eee6e16..66120f2 100644
--- a/convolution.tex
+++ b/convolution.tex
@@ -42,14 +42,18 @@ as this will facilitate our proof of the convolution formula.
We consider a $\sigma$-finite measure space $(X,\SigmaAlgebra{A},\mu)$,
a separable Fréchet space $Y$ (over $\CC$) and the task is
to define the integral of functions $f \vcentcolon X \to Y$.
-Recall that a measure space is said to be \emph{$\sigma$-finite}
+Recall that a measure space is said to be
+\emph{$\sigma$-finite}
+%\index{sigma-finite@$sigma$-finite} TODO: fix @
+%\nomenclature[A]{$\mathcal{A}'$}{commutant of $\mathcal{A}$}
if it can be exhausted by a countable number of measurable subsets of finite measure.
-By \emph{Fréchet space} we mean a complete Hausdorff locally convex (topological vector) space
+By \emph{Fréchet space}\index{Fréchet space}
+we mean a complete Hausdorff locally convex (topological vector) space
which possesses countable neighborhood bases.
We will make use of a countable family $P@@$ of seminorms that generates the topology of $@@Y$.
A topological space is called \emph{separable} if it contains a countable dense subset.
-A function $f \vcentcolon X \to Y$ will be called \emph{simple}
+A function $f \vcentcolon X \to Y$ will be called \emph{simple}\index{simple function}
if it is of the form $\sum_{i=1}^n \chi_{A_i} y_i$
where $n \in \NN$, $A_i \in \SigmaAlgebra{A}$ with $\mu(A_i) < \infty$, and $y_i \in Y$.
Naturally, the \emph{integral} of $f$ is defined to be the vector $\int f = \sum_{i=1}^n \mu(A_i) y_i \in Y$.
@@ -60,14 +64,14 @@ if it is the $\mu$-almost everywhere pointwise limit of simple functions.
Suppose $(X,\SigmaAlgebra{A},\mu)$ is a $\sigma$-finite measure space,
and $Y@@$ is a separable Fréchet space
whose topology is generated by a family $P@@$ of seminorms.
-A strongly measurable function $f \vcentcolon X \to Y$ is called \emph{(generalized Bochner) integrable}
+ A strongly measurable function $f \vcentcolon X \to Y$ is called \emph{(generalized Bochner) integrable}
if there exists a sequence $(f_n)$ of simple functions such that
\begin{equation}
\label{equation:bochner-integrable}
\lim_{n \to \infty} \int_X p \circ (f_n - f) \, d\mu = 0
\qquad \forall p \in P.
\end{equation}
-In this case, the \emph{(generalized Bochner) integral} of $f$ is defined by
+ In this case, the \emph{(generalized Bochner) integral}\index{Bochner integral!generalized} of $f$ is defined by
\begin{equation}
\label{equation:bochner-integral}
\int_X f \ d\mu \defequal
@@ -226,7 +230,7 @@ Denote by $\TestFunctions{\RR^n}$ the vector space of all functions $f \vcentco
such that the derivatives $\partial^{\alpha} f$ exist and are continuous for all multi-indices $\alpha \in \NN^n$.
Recall that the space $\SchwartzFunctions{\RR^n}$ of \emph{Schwartz functions} is defined to be the vector space
\begin{equation*}
- \SchwartzFunctions{\RR^n,X} \defequal \braces{f \in \TestFunctions{\RR^n} \vcentcolon \norm{f}_{\alpha,\beta} < \infty \ \forall \alpha,\beta \in \NN^n}
+ \SchwartzFunctions{\RR^n} \defequal \braces{f \in \TestFunctions{\RR^n} \vcentcolon \norm{f}_{\alpha,\beta} < \infty \ \forall \alpha,\beta \in \NN^n}
\end{equation*}
equipped with the locally convex topology induced by the family of seminorms
\begin{equation*}
@@ -247,7 +251,7 @@ is defined in the same way as in \cref{definition:convolution-distribution-test-
\end{equation*}
\begin{proposition}{Vector-Valued Convolution Formula}{vector-valued-convolution-formula}
- Let $v \in \TemperedDistributions{\RR^n\!,X}$ be a tempered distribution with values in a separable Fréchet space $X$, and
+ Let $v \in \TemperedDistributions{\RR^n\!,Y}$ be a tempered distribution with values in a separable Fréchet space $Y$, and
let $f \in \SchwartzFunctions{\RR^n}$ be a Schwartz test function. Then one has
\begin{equation*}
(v * f)(g) = \int v(\tau_x \tilde{f}@@) g(x) \, dx \qquad g \in \SchwartzFunctions{\RR^n}.
@@ -256,7 +260,7 @@ is defined in the same way as in \cref{definition:convolution-distribution-test-
\begin{proof}
We fix a Schwartz function $g$, and consider the finite measure $\mu = \abs{g} \lambda$ on $\RR^n$,
- where $\lambda(x) = dx$ is the Lebesgue measure.
+ where $\lambda = dx$ is the Lebesgue measure.
We show that the mapping $x \mapsto \tau_x \tilde{f}$ is a generalized Bochner $\mu$-integrable function $\RR^n \to \SchwartzFunctions{\RR^n}$
using \cref{theorem:generalized-bochner}.
For all $\alpha,\beta \in \NN^n$ we see by substituting $x+y$ for $y$ that
@@ -275,9 +279,9 @@ is defined in the same way as in \cref{definition:convolution-distribution-test-
because $g$ is Schwartz class.
Hence, $x \mapsto \tau_x \tilde{f}$ defines an integrable function.
- The mapping $v \vcentcolon \SchwartzFunctions{\RR^n} \to X$ is linear and continuous by definition.
+ The mapping $v \vcentcolon \SchwartzFunctions{\RR^n} \to Y$ is linear and continuous by definition.
By \cref{theorem:integral-commutes-with-operator},
- the composite mapping $x \mapsto v(\tau_x \tilde{f})$ is a $\mu$-integrable function $\RR^n \to X$, and
+ the composite mapping $x \mapsto v(\tau_x \tilde{f})$ is a $\mu$-integrable function $\RR^n \to Y$, and
\begin{equation}
\label{equation:general-bochner-appears}
\int v(\tau_x \tilde{f}) \, d\mu(x) = v \parens[\bigg]{\int \tau_x \tilde{f} \, d\mu(x)}
@@ -293,11 +297,11 @@ is defined in the same way as in \cref{definition:convolution-distribution-test-
and the proof is complete.
\end{proof}
-Let us point out that even in the special case that $X$ is a Banach space
+Let us point out that even in the special case that $Y$ is a Banach space
the integral on the right hand side of~\eqref{equation:general-bochner-appears}
only has meaning as a generalized Bochner integral,
since the integrand takes values in $\SchwartzFunctions{\RR^n}$,
which is not a Banach space.
We could not have performed this step with the ordinary Bochner integral.
-%\nomenclature[B]{$\BoundedLinearOperators{X,Y}$}{bounded linear operators from $X$ to $Y$\nomnorefpage}
+% vim: syntax=mytex
diff --git a/declaration.tex b/declaration.tex
new file mode 100644
index 0000000..970fc8d
--- /dev/null
+++ b/declaration.tex
@@ -0,0 +1,22 @@
+\chapter*{Declaration of Originality}
+\thispagestyle{empty}
+
+I hereby confirm that I completed the submitted work independently and without
+the unauthorized assistance of third parties and without the use of undisclosed
+and, in particular, unauthorized aids. This work has not been previously
+submitted in its current form or in a similar form to any other examination
+authorities and has not been accepted as part of an examination by any other
+examination authority.
+
+Where the wording has been taken from other people’s work or ideas, this has
+been properly acknowledged and referenced. This also applies to drawings,
+sketches, diagrams and sources from the Internet.
+
+In particular, I am aware that the use of artificial intelligence is forbidden
+unless its use an aid has been expressly permitted by the examiner. This
+applies in particular to chatbots and such programs in general that can
+complete the tasks of the examination or parts thereof on my behalf.
+
+\vspace{2cm}
+\noindent Erlangen, 30.09.2024 \hfill \underline{\hspace{5cm}}\\
+\phantom{x} \hfill {\footnotesize Justin Gassner}
diff --git a/images/FAU_NatFak_EN_Q_RGB_black.pdf b/images/FAU_NatFak_EN_Q_RGB_black.pdf
deleted file mode 100644
index b3e5919..0000000
--- a/images/FAU_NatFak_EN_Q_RGB_black.pdf
+++ /dev/null
@@ -1,1270 +0,0 @@
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diff --git a/index.tex b/index.tex
index c51baa7..2132e66 100644
--- a/index.tex
+++ b/index.tex
@@ -1,4 +1,4 @@
\setindexpreamble{%
- This is the index preamble.
+ This is the index preamble. % TODO remove?
}
\printindex
diff --git a/intro.tex b/intro.tex
index 7a77ecb..e2a8f69 100644
--- a/intro.tex
+++ b/intro.tex
@@ -1,21 +1,4 @@
\chapter{Introduction}
-asdfjd asdjflds asldfkjadsl
+\todo{xxx}
-$\FullFock{\hilb{H}}$
-$\FullFockFinite{\hilb{H}}$
-$\BosonFock{\hilb{H}}$
-$\BosonFockFinite{\hilb{H}}$
-
-$\normord{\phi(x)}$
-
-$V\sympcomp V\realorthcomp$
-
-$\realscalarp{v}{w}$
-$\symplecticp{v}{w}$
-
-$\Domain{T}$
-
-$\AlgebraicDirectSum{_{a}^{b} c}$
-
-$\Re x$
-\cleardoublepage
+% vim: syntax=mytex
diff --git a/main.tex b/main.tex
index 9158399..f2a22e6 100644
--- a/main.tex
+++ b/main.tex
@@ -1,6 +1,6 @@
\input{preamble}
\input{ushyphex}
-\includeonly{stresstensor,fewstereveson,much,commutatortheorem,convolution,analytic2,symbols,index}
+\includeonly{titlepage,contents,intro,stresstensor,fewstereveson,much,conclusion,convolution,analytic2,symbols,index,declaration}
\begin{document}
\frontmatter
\include{titlepage}
@@ -15,6 +15,7 @@
\include{fewstereveson}
\include{much}
\include{samplesection}
+\include{conclusion}
\appendix
\include{sampleappendix}
\include{commutatortheorem}
@@ -24,4 +25,6 @@
\include{bibliography}
\include{symbols}
\include{index}
+% maybe acknowledgements
+\include{declaration}
\end{document}
diff --git a/much.tex b/much.tex
index 249dc73..d11b04a 100644
--- a/much.tex
+++ b/much.tex
@@ -50,6 +50,10 @@ Poincaré covariance
\section{Basic Concepts of Modular Theory}
\index{modular!theory}
+It is a distinctive feature of the quantum energy inequality \todo{ref}, which is at the center of our investigation,
+that the modular operator $\Delta$ associated to a local algebra of observables and the vacuum vector appears in its lower bound.
+In this short section we will review the the definition of $\Delta$ and other basic concepts of the Tomita--Takesaki modular theory of von Neumann algebras.
+
If $\hilb{H}$ is a Hilbert space
we shall denote the $C^*$-algebra of all bounded linear operators on $\hilb{H}$ by $\BoundedLinearOperators{\hilb{H}}$.
@@ -159,7 +163,7 @@ as this will be our only use case.
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 $U$
+a anti-linear partial isometry $U$
such that
\begin{equation*}
T = U \abs{T} \qquad \bracks[\big]{\text{in particular, $\Domain{T} = \Domain{\abs{T}}$}}.
@@ -194,7 +198,12 @@ Now we are able to introduce the fundamental objects of modular theory.
the pair $(\vNa{M},\Omega)$.
\end{definition}
-\todo{clarify why $J$ is anti-unitary}
+The anti-linear partial isometry $J$ satisfies
+$(\ker J)^\perp = (\ker S)^\perp$ and $\ran J = \overline{\ran S}$
+by \cref{theorem:polar-decomposition}.
+Since $S@@$ is injective and has dense range,
+it follows that $J@@$ has $\hilb{H}$ as both initial and final space,
+and thus is in fact anti-unitary.
\begin{definition}{Modular Group}{}
Adopt the notation of the foregoing definition.
@@ -522,7 +531,7 @@ there exists a unique projection-valued measure $E$ on $\RR^4$ such that
\label{equation:spectral-resolution-translation}
U(a) = \int_{\RR^4} \exp(ia \cdot k) \, dE(k) \qquad a \in \RR^4.
\end{equation}
-Then one can define a vector $P$ of unbounded selfadjoint operators
+Then one can define a vector $P = (P_0,P_1,P_2,P_3)$ of unbounded selfadjoint operators
\begin{equation*}
P_i = \int_{\RR^4} k_i \, dE(k) \qquad i=0,\ldots,3
\end{equation*}
@@ -639,7 +648,7 @@ Next we consider an operator-valued tempered distribution $u$ that is \emph{cova
in the sense that it obeys the relativistic transformation law
\begin{equation}
\label{equation:covariance-distribution}
- U(g) u(f) U(g)^* = u(f_g) \qquad g \in \RestrictedPoincareGroup, f \in \schwartz{\RR^4},
+ U(g) u(f) U(g)^* = u(f_g) \qquad g \in \RestrictedPoincareGroup, f \in \SchwartzFunctions{\RR^4},
\end{equation}
where $f_g(x) = f(g^{-1} x)$ for all $x \in M$.
In particular, if $g=(a,I)$ is the translation by a vector $a \in \RR^4$,
@@ -658,12 +667,12 @@ However, thanks to the spectrum condition we may modify this function outside of
\begin{lemma}{}{depends-only-on-restriction}
Let $u$ be a covariant operator-valued tempered distribution.
- Then the vector $u(f) \FockVacuum$, where $f \in \schwartz{\RR^4}$,
+ Then the vector $u(f) \FockVacuum$, where $f \in \SchwartzFunctions{\RR^4}$,
depends only on the restriction of $\ft{f}$ to $\ClosedForwardCone$.
\end{lemma}
\begin{proof}
- We consider a Schwartz function $g \in \schwartz{\RR^4}$ and
+ We consider a Schwartz function $g \in \SchwartzFunctions{\RR^4}$ and
the operator $G = \int g(k) dE(k)$,
where $E$ is the unique projection-valued measure on $\RR^4$ such that
$U(a) = \int \exp(ik \cdot a) dE(k)$ for all $a \in \RR^4$.
@@ -675,7 +684,7 @@ However, thanks to the spectrum condition we may modify this function outside of
\end{multline*}
\question{Darf ich hier wirklich die Integrationsreihenfolge vertauschen?}
- Recall that the Fourier transform of $u$ is defined by $\ft{u}(f) = u(\ft{f}@@)$ for $f \in \schwartz{\RR^4}$.
+ Recall that the Fourier transform of $u$ is defined by $\ft{u}(f) = u(\ft{f}@@)$ for $f \in \SchwartzFunctions{\RR^4}$.
We obtain the action of the translation group on $\ft{u}(\ft{f}@@)\FockVacuum$ by definition chasing and~\eqref{equation:real-translation-law}:
\begin{equation*}
U(a) \ft{u}(\ft{f}@@)\FockVacuum
@@ -706,13 +715,13 @@ Such a function does exist \todo{elaborate, smooth cutoff}. Then $f_z$ will be S
Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$, by~\cref{lemma:depends-only-on-restriction}.
%\begin{lemma}{}{}
- %For every $z \in \ClosedForwardTube$ there exists a Schwartz function $d_z \in \schwartz{\RR^4}$
- %such that $\ft{e_z} \in \schwartz{\RR^4}$ and $\ft{e_z}(p) = \exp(iz \cdot p)$ for $p \in \ClosedForwardCone$.
+ %For every $z \in \ClosedForwardTube$ there exists a Schwartz function $d_z \in \SchwartzFunctions{\RR^4}$
+ %such that $\ft{e_z} \in \SchwartzFunctions{\RR^4}$ and $\ft{e_z}(p) = \exp(iz \cdot p)$ for $p \in \ClosedForwardCone$.
%\end{lemma}
\begin{proposition}{}{prp}
Let $u$ be a covariant operator-valued tempered distribution,
- and let $f \in \schwartz{\RR^4}$ be a test function. Then we have,
+ and let $f \in \SchwartzFunctions{\RR^4}$ be a test function. Then we have,
in generalization of~\eqref{equation:real-translation-law},
\begin{equation*}
U(z) u(f) \FockVacuum = u(f_z) \FockVacuum \qquad \forall z \in T_+.
@@ -726,14 +735,17 @@ Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$,
Dann folgt die Behauptung wohl mit Edge of the Wedge~\cite[Theorem 2-17]{Streater1964}}
\end{proof}
-\begin{corollary}{}{}
+\begin{corollary}{}{convolution2}
Let $u$ be a covariant operator-valued tempered distribution,
- and let $f \in \schwartz{\RR^4}$ be a test function. Then we have,
+ and let $f \in \SchwartzFunctions{\RR^4}$ be a test function. Then we have,
\begin{equation*}
U(z) u(f) \FockVacuum = \int dx \, f(x) \, u(d_{z+x}) \FockVacuum \qquad \forall z \in T_+.
\end{equation*}
\end{corollary}
+As discussed in \cref{chapter:convolution},
+the vector-valued integral on the right-hand side exists in the strong sense of Bochner.
+
\begin{proof}
The convolution formula \cref{proposition:vector-valued-convolution-formula} applied to the vector-valued distribution defined by $f \mapsto \alpha(f) = u(f) \FockVacuum$ yields
\begin{equation*}
@@ -786,9 +798,7 @@ For later use, we give the action of $\Lambda(is)$ on a complex four-vector $x+i
y^3
\end{pmatrix}
\end{equation}
-
-We
-
+The purely imaginary Lorentz boost matrix
\begin{equation*}
\mathcal{J} \defequal \Lambda(i/2) = \begin{pmatrix}
-1 & 0 & \; 0 \; & \; 0 \; \\
@@ -797,22 +807,23 @@ We
0 & 0 & 0 & 1 \\
\end{pmatrix}
\end{equation*}
+will play a special role because it maps the right and left wedges onto each other.
-\begin{equation*}
- \mathcal{J}_{\pm} \defequal \Lambda(\pm i/4) = \begin{pmatrix}
- 0 & \pm i & \; 0 \; & \; 0 \; \\
- \pm i & 0 & 0 & 0 \\
- 0 & 0 & 1 & 0 \\
- 0 & 0 & 0 & 1 \\
- \end{pmatrix}
-\end{equation*}
+%\begin{equation*}
+ %\mathcal{J}_{\pm} \defequal \Lambda(\pm i/4) = \begin{pmatrix}
+ %0 & \pm i & \; 0 \; & \; 0 \; \\
+ %\pm i & 0 & 0 & 0 \\
+ %0 & 0 & 1 & 0 \\
+ %0 & 0 & 0 & 1 \\
+ %\end{pmatrix}
+%\end{equation*}
We now turn to the unitary representation of (real) Lorentz boosts
\begin{equation*}
V(t) \defequal U \parens[\big]{0,\Lambda(t)} \qquad t \in \RR
\end{equation*}
on Fock space and aim for an analytic extension similar to the previous section.
-By Stone's theorem theorem there exists a unique selfadjoint operator $K$ such that
+By Stone's Theorem there exists a unique selfadjoint operator $K$ such that
\begin{equation*}
V(t) = \exp(itK) = \int_{\RR} \exp(it \lambda) \,dE_K(\lambda),
\end{equation*}
@@ -825,10 +836,10 @@ Now we define \emph{complex Lorentz boosts} to be the operators
In contrast to the previous section, we
-\begin{lemma}{}{}
+\begin{lemma}{}{complex-lorentz-boosts}
Suppose $A$ is a selfadjoint unbounded operator on some Hilbert space $\hilb{H}$.
For each complex number $z$ define the closed normal operator $V(z) = e^{izA}$ by means of functional calculus.
- Let $g \in \schwartz{\RR}$ be a Schwartz function.
+ Let $g \in \SchwartzFunctions{\RR}$ be a Schwartz function.
\begin{enumerate}
\item $V(z) V(w) = V(z + w)$ for all $z,w \in \CC$.
\item The operator $g(A)$ is bounded, and its range is contained in the domain of $V(z)$ for all $z \in \CC$.
@@ -850,7 +861,7 @@ but a dense subspace of $\Domain{T}$ need not be a core for $T$.
\begin{lemma}{A Common Core for All Complex Lorentz Boosts}{common-core-for-complex-lorentz-boots}
Adopt the notation of the foregoing lemma. The linear subspace
\begin{equation*}
- \mathcal{D}_0 = \Span \Set{\ran g(K) \given g \in \schwartz{\RR}}
+ \mathcal{D}_0 = \Span \Set{\ran g(K) \given g \in \SchwartzFunctions{\RR}}
\end{equation*}
is a core for $V(z)$ for every $z \in \CC$.
\end{lemma}
@@ -971,18 +982,56 @@ Remember that $\mathcal{J} = \Lambda(i/2) = \diag(-1,-1,1,1)$.
\begin{lemma}{}{biso3}
Suppose that $u$ is a covariant operator-valued tempered distribution.
- Let $f \in \schwartz{M}$ with $\supp f \subset \rightwedge$, and
- let $g \in \schwartz{M}$ be arbitrary. Then
- \begin{equation*}
+ Let $f \in \SchwartzFunctions{M}$ with $\supp f \subset \rightwedge$, and
+ let $g \in \SchwartzFunctions{M}$ be arbitrary. Then
+ \begin{equation}
+ \label{equation:biso3-claim}
V(i/2) g(K) u(f) \FockVacuum = g(K) u(f_{\mathcal{J}}) \FockVacuum
- \end{equation*}
+ \end{equation}
\end{lemma}
Here, $K$ is the infinitesimal generator of the group $t \mapsto V(t)$ of real Lorentz boosts,
$\FockVacuum$ is the Fock vacuum, and $\mathcal{J}$ is the Lorentz transformation given by the diagonal matrix $\diag(-1,-1,1,1)$.
\begin{proof}
- a
+ Instead of~\eqref{equation:biso3-claim} we prove
+ \begin{equation}
+ \label{equation:modified-claim}
+ V(i/4) g(K) u(f) \FockVacuum = V(-i/4) g(K) u(f_{\mathcal{J}}) \FockVacuum.
+ \end{equation}
+ This is equivalent due to \cref{lemma:complex-lorentz-boosts}(i).
+ As before, we write $e_0 = (1,0,0,0)$ for the positive time-like unit vector,
+ and introduce a complex translation as follows:
+ \begin{equation}
+ \label{equation:step1}
+ u(f) \FockVacuum = \stronglim_{\epsilon \downarrow 0} U(i \epsilon e_0) u(f) \FockVacuum.
+ \end{equation}
+ \cref{corollary:convolution2}
+ \begin{equation}
+ \label{equation:step2}
+ U(i \epsilon e_0) u(f) \FockVacuum =
+ \int dx f(x) u(d_{x + i \epsilon e_0}) \FockVacuum
+ \end{equation}
+ The operator $V(i/4) g(K)$ is bounded by Lemma xxx,
+ and therefore,
+ when applied to ~\eqref{equation:step1}
+ can be moved inside the strong limit,
+ and when applied to ~\eqref{equation:step2}
+ can be moved inside the integrand using \cref{theorem:integral-commutes-with-operator}.
+ Taken together, we obtain
+ \begin{equation}
+ \label{equation:step3}
+ V(i/4) g(K) u(f) \FockVacuum =
+ \stronglim_{\epsilon \downarrow 0} \int dx f(x) V(i/4) g(K) U(i \epsilon e_0) u(d_x) \FockVacuum
+ \end{equation}
+ Next we aim to bring the right hand side of~\eqref{equation:step3} into a form where \cref{lemma:biso2} is applicable.
+ The strong limit commutes with the strong integral (\todo{Why?}).
+ Moreoverk
+ \begin{equation*}
+ V(i/4) g(K) u(f) \FockVacuum =
+ \int dx f_{\delta}(x) V(i/4) g(K) V(-i/4) \stronglim_{\epsilon \downarrow 0} V(i/4) U(x + i \epsilon e_0) u(d_{\delta}) \FockVacuum
+ \end{equation*}
+ Now, performing all transformations in reverse yields~\eqref{equation:modified-claim}, as desired.
\end{proof}
\begin{equation*}
@@ -993,8 +1042,8 @@ Die Anwendung auf die Energiedichte $\energydensity$:
\begin{proposition}{}{main-result}
Suppose $W \subset M$ is any wedge domain, with associated modular operator $\Delta_W$ and modular Hamiltonian $K_W$.
- Let $f \in \schwartz{M}$ with $\supp f \subset W$, and
- let $h \in \schwartz{M}$ be arbitrary. Then
+ Let $f \in \SchwartzFunctions{M}$ with $\supp f \subset W$, and
+ let $h \in \SchwartzFunctions{M}$ be arbitrary. Then
\begin{equation*}
\norm{\Delta_W^{-1/2} h(K_W) \energydensity(f) \FockVacuum}
= \norm{h(K) \energydensity(f_{\mathcal{J}g}) \FockVacuum},
diff --git a/preamble.tex b/preamble.tex
index fb67d0a..2d358ae 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -11,6 +11,7 @@
\usepackage{amsmath,amsthm}
\usepackage{mathtools}
\usepackage[colon=literal]{unicode-math} % TODO get rid of this since it messes up math italic correction
+\usepackage{uni-titlepage}
\usepackage[inline]{enumitem}
\usepackage{multicol}
%\usepackage{graphicx}
@@ -89,8 +90,8 @@
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\ev}{ev}
% extend amsmath's proof environment
-\newenvironment{myproof}[1]{\proof[\proofname\ of \Cref{#1}]}{\endproof}
-%\NewDocumentEnvironment{myproof}{Ob}{\IfNoValueTF{#1}{\begin{proof}}{\begin{proof}[\proofname\ of \Cref{#1}]}}{\end{proof}}
+%\newenvironment{myproof}[1]{\proof[\proofname\ of \Cref{#1}]}{\endproof}
+\NewDocumentEnvironment{myproof}{m}{\begin{proof}[\proofname\ of \Cref{#1}]}{\end{proof}}
% ---------- mathtools
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
@@ -189,7 +190,6 @@
\renewcommand{\nomname}{List of Symbols}
\renewcommand{\nompreamble}{\begin{multicols}{2}}
\renewcommand{\nompostamble}{\end{multicols}}
-%\def\pagedeclaration#1{, \hyperlink{page.#1}{page\nobreakspace#1}}
\def\pagedeclaration#1{, \hyperlink{page.#1}{#1}}
% ---------- makeidx
@@ -203,13 +203,13 @@
bookmarksnumbered,
bookmarksopen,
bookmarksopenlevel=2,
- hypertexnames=false,
pdfpagelabels,
pdftitle={Title},
pdfauthor={Justin Gassner},
pdfsubject={Subject},
pdfkeywords={Keyword1, Keyword2},
}
+% TODO: insert title, subject and keywords
% ---------- more pdf stuff
\pdfvariable minorversion 6
@@ -257,7 +257,7 @@
\newcommand{\Mon}{\operatorname{Mon}}
% Infinitesimal Weyl algebra
-\newcommand{\WeylAlg}{\mathcal{W}}
+\newcommand{\InfinitesimalWeylAlg}{\mathcal{W}}
\newcommand{\weylannihilator}{A}
\newcommand{\weylcreator}{\weylannihilator^\dagger}
@@ -379,11 +379,18 @@
\newcommand*{\ProperComplexPoincareGroup}{\ProperPoincareGroup(\CC)}
\newcommand*{\ImproperComplexPoincareTransformations}{\PoincareGroup_{\!-}(\CC)}
+% Set Theory
+\newcommand*{\PowerSet}[2][]{\mathcal{P}\parens[#1]{#2}}
+
% Functional Analysis
\newcommand*{\BoundedLinearOperators}[2][]{B\parens[#1]{#2}}
\newcommand*{\ContinousLinearOperators}[2][]{L\parens[#1]{#2}}
\DeclareMathOperator*{\stronglim}{s-lim}
\DeclareMathOperator*{\weaklim}{w-lim}
+\DeclareMathOperator*{\End}{End}
+
+\newcommand*{\SymmetricGroup}[1]{S_{#1}}
+\newcommand*{\Vector}[1]{\symbfit{#1}}
% spacetime regions
\newcommand*{\spacetimeregion}[1]{\mathcal{#1}}
diff --git a/sampleappendix.tex b/sampleappendix.tex
deleted file mode 100644
index 6cdf6cb..0000000
--- a/sampleappendix.tex
+++ /dev/null
@@ -1,17 +0,0 @@
-\chapter{Sample Appendix}
-
-Just more \index{sample text}sample text.
-
-$x \equiv y$
-
-\begin{equation}
- \label{eq:test}
- a + b
-\end{equation}
-\ref{eq:test}
-\eqref{eq:test}
-\cref{eq:test}
-
-\nocite{*}
-
-\cref{lemma:xxx}
diff --git a/samplesection.tex b/samplesection.tex
deleted file mode 100644
index 205fa6f..0000000
--- a/samplesection.tex
+++ /dev/null
@@ -1,8 +0,0 @@
-\chapter{Sample Section}
-
-Just some \index{sample text}sample text.
-\nomenclature{$\mathcal{S}$}{Schwartz test function space\nomrefpage}
-
-\section{Subsection}
-
-\section{Another Subsection}
diff --git a/second.tex b/second.tex
deleted file mode 100644
index d7e1a38..0000000
--- a/second.tex
+++ /dev/null
@@ -1,6 +0,0 @@
-\chapter{Second Quantisation}
-
-\blockquote[Edward Nelson]{%
- First quantization is a mystery, but second quantization is a functor.
-}
-Just more text.
diff --git a/standard.tex b/standard.tex
deleted file mode 100644
index 86bd1ab..0000000
--- a/standard.tex
+++ /dev/null
@@ -1,21 +0,0 @@
-\chapter{Standard Subspaces}
-
-\cite{Longo2008}
-\cite{Neeb2017}
-
-\section{Test}
-\begin{definition}{Symplectic Complement}{abc}
- test
- \nomenclature{$X$}{a Banach space, see \cref{definition:abc} asdf asdfasdf askdfj askdfj akdjf askdjf aksdfj aksdjfk asdjfksajf kasjdfkasjkfjsa kjsd ksjfaksjfksdjf}
-\end{definition}
-\begin{lemma}{Important}{xxx}
- test
-\end{lemma}
-
-\cref{lemma:xxx}
-test
-
-\section{Test}
-\begin{definition}{cyclic, separating}{}
- test
-\end{definition}
diff --git a/stresstensor.tex b/stresstensor.tex
index 70a79ea..a03b10b 100644
--- a/stresstensor.tex
+++ b/stresstensor.tex
@@ -161,6 +161,7 @@ into annihilation operators on the left.
\end{align*}
\section{Quadratic Forms}
+\label{section:quadratic-forms}
In a typical physics literature treatment of second quantization,
the annihilation and creation operators and the quantized field are treated as
@@ -169,13 +170,20 @@ disregarding the fact that these may not be operators, in a strict sense, and wi
Nonetheless, this notational fiction is useful, and we can uphold it with little effort by giving
the pointwise \enquote{operators} rigorous meaning as quadratic forms.
+Let us consider a real scalar quantum field with mass parameter $m>0$.
+Thus, the single-particle state space is the Hilbert space $\hilb{H} = L^2(X_m^+,\Omega_m)$,
+where $X_m^+$ is the upper half of the hyperboloid defined by the condition $p \cdot p = m^2$ in momentum space,
+and $\Omega_m$ is the unique normalized Lorentz-invariant measure on it.
Given a point $p$ in momentum space,
-we define the annihilation operator $a(p)$ with domain TODO by
+we define the annihilation operator $a(p)$ to be the operator
+whose domain is the finite particle subspace $\BosonFockFinite{\hilb{H}}$ of Bosonic Fock space
+and whose action on a vector $\psi = (\psi_n)_{n \ge 0}$ in $\BosonFockFinite{\hilb{H}}$ is given by
\begin{equation*}
\parens[\big]{a(p) \psi} {}_n (k_1, \ldots, k_n)
- = \sqrt{n+1} \, \psi_{n+1} (k_1, \ldots, k_n,p)
+ = \sqrt{n+1} \, \psi_{n+1} (k_1, \ldots, k_n,p).
\end{equation*}
-The issue arises when one looks for an adjoint to this operator.
+While this is a perfectly well-defined operator,
+an issue arises when one looks for an adjoint to this operator.
A formal calculation based on the adjoint identity
\begin{equation}
\label{equation:adjoint-identity}
@@ -193,12 +201,12 @@ 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
formally calculate the inner product of $\psi'$ with~\eqref{equation:creation-operator-at-point}
-and we use the result to define the $a^\dagger(p)$
+and use the result to define $a^\dagger(p)$
as a mapping that assigns a number to each \emph{pair} of states.
That is, we define the creation \enquote{operator} $a^\dagger(p)$
to be the quadratic form
\begin{gather*}
- a(p)^\dagger : F \times F \longrightarrow \CC \\
+ a(p)^\dagger \vcentcolon \BosonFockFinite{\hilb{H}} \times \BosonFockFinite{\hilb{H}} \longrightarrow \CC \\
\innerp[\big]{\psi'}{a(p)^\dagger \psi}
\defequal
\begin{multlined}[t]
@@ -210,13 +218,13 @@ to be the quadratic form
\end{gather*}
One can verify directly that
with this definition the adjoint identity~\eqref{equation:adjoint-identity}
-holds for all $\psi,\psi' \in F$.
+holds for all $\psi,\psi' \in \BosonFockFinite{\hilb{H}}$.
For completeness, we give a precise definition of quadratic form.
-\begin{definition}{Quadratic Form}{}
+\begin{definition}{Quadratic Form}{quadratic-form}
A \emph{quadratic form}\index{quadratic form} $q$ on a complex Hilbert space $\hilb{H}$ is a mapping
\begin{equation*}
- q : D(q) \times D(q) \to \CC,
+ q \vcentcolon 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 argument
@@ -225,6 +233,11 @@ For completeness, we give a precise definition of quadratic form.
if $D(q)$ is dense in $\hilb{H}$.
\end{definition}
+The creation \enquote{operator} $a^\dagger(p)$ we considered above
+is a densely defined quadratic form on Bosonic Fock space
+in the sense of \cref{definition:quadratic-form}
+and its form domain is the finite particle subspace.
+
Any linear operator on a complex Hilbert space $\hilb{H}$ has an
obvious interpretation as a quadratic form on $\hilb{H}$,
and the form domain agrees with the domain of the operator.
@@ -247,8 +260,8 @@ We will use the symbol $\QFequal$ between quadratic forms or operators
to indicate their equality as quadratic forms.
\todo{statement about domains?}
-
A natural question is how the smeared operators relate to the pointwise ones.
+The answer is simple:
\begin{equation*}
a(g) \QFequal \int \overline{g(p)} a(p) \, d\Omega_m(p)
@@ -258,40 +271,53 @@ A natural question is how the smeared operators relate to the pointwise ones.
a^\dagger(g) \QFequal \int g(p) a^\dagger(p) \, d\Omega_m(p)
\end{equation*}
We have to explain what is meant by the integral on the right hand side.
-Suppose $q(p)$ is a quadratic form on $\BosonFock{\hilb{H}}$ for each $q \in \RR^4$,
+Suppose $q(p)$ is a quadratic form on $\BosonFock{\hilb{H}}$ for each $p \in \RR^4$,
that share a common domain $D \subset D(q(p))$,
and $g$ is in $\hilb{H} = L^2(\RR^4,\Omega_m)$. Then we define
a quadratic form by
\begin{equation*}
- \parens{\int g(p) q(p) \, d\Omega_m(p)}(\psi',\psi)
- = \int g(p) \parens{q(p)}(\psi',\psi) \, d \Omega_m(p)
+ \parens[\bigg]{\int g(p) q(p) \, d\Omega_m(p)}(\psi',\psi)
+ \defequal \int g(p) \parens[\big]{q(p)}(\psi',\psi) \, d \Omega_m(p)
\end{equation*}
for all $\psi,\psi' \in D$.
+A product of the form
\begin{equation*}
a(p_1)^\dagger \cdots a(p_s)^\dagger a(p_{s+1}) \cdots a(p_r)
\end{equation*}
+can be rigorously defined as quadratic form by setting
\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}
+ \defequal \innerp[\big]{a(p_1) \cdots a(p_s) \psi'}{a(p_{s+1}) \cdots a(p_r) \psi}.
\end{equation*}
\section{Normal Ordering}
% The Renormalization Map?
-%\blockcquote{Wick1950}{%
- %\textelp{} we then proceed to rearrange such a product so as to carry all
- %creation operators to the left of all destruction operators \textelp{}. The
- %main problem to be solved in carrying out this idea is one of algebraic
- %technique \textelp{}
-%}
-
+\blockcquote{Wick1950}{%
+ \textelp{} we then proceed to rearrange such a product so as to carry all
+ creation operators to the left of all destruction operators \textelp{}. The
+ main problem to be solved in carrying out this idea is one of algebraic
+ technique \textelp{}
+}
The process of renormalizing a product of field operators
has the purpose of discarding infinite constants
that occur when calculating the vacuum expectation value.
-\todo{present physicists way of introducing normal ordering}
-Now let us extract the algebraic essence of the situation.
+In physics texts the standard prescription for doing this goes as follows:
+(1) Express all field operators by creation and annihilation operators
+and expand the expression into a sum of products.
+(2) Within each product, move all creation operators to the left of all annihilation operators
+while ignoring their commutation relations.
+
+This ad-hoc way of dealing with infinities is not entirely satisfactory.
+For once, it leaves the uniqueness question (whether there is another way to
+achieve the desired effect) open.
+Moreover, it obscures the facts that we are renormalizing in relation to the vacuum state
+(in principle, one could use another state for reference), and that the process
+is representation-independent and purely algebraic.
+
+With this in mind, let us extract the algebraic essence of the situation.
The objects of our calculations are the field operators $\Phi(f)$,
but it does not matter that these are realized as linear maps on Fock space.
Forming the product $\Phi(f)\Phi(g)$ might as well be done purely symbolically,
@@ -300,70 +326,128 @@ having the meaning of operator composition;
and similar for the other two arithmetic operations,
addition and multiplication with a complex scalar.
Thus we should calculate with abstract objects $\Phi(f)$ labeled by Hilbert space vectors $f \in \hilb{H}$.
-Considering that here $\Phi$ carries no meaning, we can use the label $f$ itself to represent the object.
-
+Considering that here the symbol $\Phi$ carries no meaning, we can drop it and use the label $f$ itself to represent the object.
This leads us to consider formal expressions
\begin{equation*}
- \alpha^{(0)} e + \sum_{i} \alpha^{(1)}_i z^{(1)}_i + \sum_{j,k} \alpha^{(2)}_{j,k} z^{(2)}_j z^{(2)}_k + \cdots
+ \alpha^{(0)} e + \sum_{i} \alpha^{(1)}_i z^{(1)}_i + \sum_{j,k} \alpha^{(2)}_{j,k} \, z^{(2)}_j z^{(2)}_k + \cdots
\end{equation*}
where the $z^{(1)}_i,z^{(2)}_j,z^{(2)}_k,\ldots$ are in $\hilb{H}$,
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 non-commutative associative algebra over $\CC$
+we are speaking of the unital non-commutative associative algebra over $\CC$
freely generated by the elements of $\hilb{H}$.
The unit of the algebra is $e$.
-This in not quite what we want
-\todo{explain need for commutation relations}
+This is not quite what we want,
+as we yet need to account for the commutation relations
+$\bracks{\Phi(f),\Phi(g)} = i \Imag \innerp{f}{g}$.
By abstract algebra, this is viable
by forming the quotient of the free algebra
with respect to the two-sided ideal
-generated by all elements $zz' - z'z = i \Imag \innerp{z}{z'} \, e$,
+generated by all elements $zz' - z'z = i \Imag \innerp{z}{z'} \cdot e$,
where $z,z' \in \hilb{H}$.
+In addition, we must implement the $\RR$-linear dependence of $\Phi(f)$ on $f$.
\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 non-commutative associative algebra over $\CC$
+ The \emph{infinitesimal Weyl algebra}\index{infinitesimal Weyl algebra} $\InfinitesimalWeylAlg(\hilb{H})$ over $\hilb{H}$
+ is the unital 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},
- \end{equation*}
+ \begin{align*}
+ \alpha \cdot z &= 1 \cdot (\alpha z) \qquad \alpha \in \RR, z \in \hilb{H}, \\
+ zz' - z'z &= i \Imag \innerp{z}{z'} \, e \qquad z,z' \in \hilb{H},
+ \end{align*}
where $e$ is the unit of the algebra.
+ Moreover, the mapping $z \mapsto z^* = z$
+ extends to a $*$-operation that turns $\InfinitesimalWeylAlg(\hilb{H})$ into a $*$-algebra.
\end{definition}
-
-\todo{introduce $\Phi$ as representation of $\WeylAlg$}
-
+This construction is an instance of what is known as a quantization functor.
+For a longer discussion of its functorial properties we refer the reader to~\cite{Fewster2012}.
+
+To avoid any confusion, we emphasize that
+$1 \cdot iz$ (coefficient $1$, vector $iz$) and $i \cdot z$ (coefficient $i$, vector $z$)
+are always different algebra elements (unless $z=0$), and there no rule that allows one to move imaginary numbers from the coefficient part to the vector part.
+
+Now let $\Phi = \Phi_{\mathrm{S}}$ be
+the Segal quantization for the free Boson field over $\hilb{H}$,
+which was discussed in the preceding section.
+For each $z \in \hilb{H}$,
+we consider $\Phi(z)$ as an everywhere-defined linear operator
+on the finite particle space $\BosonFockFinite{\hilb{H}}$ by restriction.
+The mapping $z \mapsto \Phi(z)$ thus induces a natural $*$-representation
+of the infinitesimal Weyl algebra,
+\begin{align*}
+ \Phi \vcentcolon \InfinitesimalWeylAlg(\hilb{H}) &\longrightarrow \End \parens[\big]{\BosonFockFinite{\hilb{H}}} \\
+ w = z_1 \!\cdots z_n &\longmapsto \Phi(w) = \Phi(z_1) \circ \cdots \circ \Phi(z_n),
+\end{align*}
+since $\InfinitesimalWeylAlg(\hilb{H})$ is generated by elements of $\hilb{H}$
+and $\bracks{\Phi(z),\Phi(z')} = i \Imag \innerp{z}{z'}$ on $\BosonFockFinite{\hilb{H}}$ for each $z,z' \in \hilb{H}$.
+
+In general, any renormalization scheme for the infinitesimal Weyl algebra $\InfinitesimalWeylAlg$
+will be in relation to a given linear functional, i.e.\ an element $E$ of the dual space $\InfinitesimalWeylAlg'$,
+which we interpret to be yielding expectation values.
+Each element of $\InfinitesimalWeylAlg$ should then be mapped to an element of $\InfinitesimalWeylAlg$
+that is renormalized in the sense that it has zero expectation value in relation to $E$.
+We will see that this can be accomplished by bringing each product into a special order called normal order.
+In the case of the free Boson field,
+the Fock vacuum state $\FockVacuum$ gives rise to the linear functional
+\begin{equation*}
+ \InfinitesimalWeylAlg(\hilb{H}) \ni w \mapsto E(w) \defequal \innerp[\big]{\FockVacuum}{\Phi(w) \FockVacuum},
+\end{equation*}
+which we call the \emph{normal vacuum}.
+The normal vacuum has two essential properties: Firstly,
+\begin{equation}
+ \label{equation:normal-vaccum-1}
+ E(e) = 1
+\end{equation}
+since $\Phi(e)$ is the identity operator and the vacuum $\Omega$ is a unit vector.
+For stating the second property we need to introduce some notation:
\begin{definition}{Annihilator and Creator}{}
- Suppose $\WeylAlg$ is the infinitesimal Weyl algebra
+ Suppose $\InfinitesimalWeylAlg(\hilb{H})$ is the infinitesimal Weyl algebra
over some complex Hilbert space $\hilb{H}$.
- For all $z \in \hilb{H}$,
- we define, as elements of $\WeylAlg$, the \emph{annihilator}
+ For all $z \in \hilb{H}$
+ we define, as elements of $\InfinitesimalWeylAlg(\hilb{H})$, the \emph{annihilator}
\begin{equation*}
- \weylannihilator(z) = \frac{1}{\sqrt{2}} \parens{z+iz},
+ \weylannihilator(z) = \frac{1}{\sqrt{2}} \parens{z+i \cdot iz},
\end{equation*}
and the \emph{creator}
\begin{equation*}
- \weylcreator(z) = \frac{1}{\sqrt{2}} \parens{z-iz}.
+ \weylcreator(z) = \frac{1}{\sqrt{2}} \parens{z-i \cdot iz}.
\end{equation*}
\end{definition}
+The second important property of the normal vacuum $E$ is
+\begin{equation}
+ \label{equation:normal-vaccum-2}
+ E\parens[\Big]{\prod_{i=1\vphantom{S}}^{s} \weylcreator(z_i)
+ \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} \weylannihilator(z_j)} = 0
+ \qquad \forall z_1,\ldots,z_r \in \hilb{H}, r \ge 1, 1 \le s \le r.
+\end{equation}
+This follows from $\Phi(\weylannihilator(z)) = a(z)$, $\Phi(\weylcreator(z)) = a^{\dagger}(z)$.
+%XXX more detail
-\begin{equation*}
- z = \frac{1}{\sqrt{2}} \parens[\big]{\weylannihilator(z) + \weylcreator(z)}
-\end{equation*}
+It is easy to verify that for all $z,z' \in \hilb{H}$
+\begin{gather*}
+ z = \frac{1}{\sqrt{2}} \parens[\big]{\weylannihilator(z) + \weylcreator(z)}, \\
+ \bracks[\big]{\weylannihilator(z),\weylcreator(z')} = i \Imag \innerp{z}{z'} e.
+\end{gather*}
+Taken together, these identities show that $\InfinitesimalWeylAlg$ is linearly generated by expressions of the form
+$\prod_{i=1}^{s} \weylcreator(z_i) \prod_{j=s+1}^{r} \weylannihilator(z_j)$.
+We conclude that the normal vacuum is the \emph{unique} functional on $\InfinitesimalWeylAlg$
+satisfying the conditions~\eqref{equation:normal-vaccum-1} and~\eqref{equation:normal-vaccum-2}, which are physically justified.
-A \emph{monomial} in the Weyl algebra $\WeylAlg$ over a complex Hilbert space $\hilb{H}$ is an element of the form
-$z_1 \cdots z_r \in \WeylAlg$, where $r \ge 0$ and $z_1,\ldots,z_r$ are in $\hilb{H}$.
+A \emph{monomial} in the Weyl algebra $\InfinitesimalWeylAlg$ over a complex Hilbert space $\hilb{H}$ is an element of the form
+$z_1 \!\cdots z_r \in \InfinitesimalWeylAlg$, where $r \ge 0$ and $z_1,\ldots,z_r$ are in $\hilb{H}$.
We allow $r=0$, meaning that the unit $e$ is a monomial.
-The set of all monomials in $\WeylAlg$ is denoted $\Mon(\WeylAlg)$.
+The set of all monomials in $\InfinitesimalWeylAlg$ is denoted $\Mon(\InfinitesimalWeylAlg)$.
\begin{definition}{Normal Ordering}{}
- Let $\hilb{H}$ be a complex Hilbert space and $\WeylAlg$ its associated infinitesimal Weyl algebra.
- The mapping $\normord{\,\,}$, defined by
+ Let $\hilb{H}$ be a complex Hilbert space and $\InfinitesimalWeylAlg$ its associated infinitesimal Weyl algebra.
+ The mapping $\normord{\,\,}$\nomenclature[:]{$\normord{\,\,}$}{normal ordering},
+ defined by linear extension of the mapping
\begin{gather}
- \Mon(\WeylAlg) \longrightarrow \WeylAlg \nonumber\\
+ \Mon(\InfinitesimalWeylAlg) \longrightarrow \InfinitesimalWeylAlg \nonumber\\
\label{equation:normal-ordering}
\normord{z_1 \!\cdots z_r} =
\frac{1}{\sqrt{2^r}}
@@ -372,8 +456,8 @@ The set of all monomials in $\WeylAlg$ is denoted $\Mon(\WeylAlg)$.
\prod_{\mathclap{j \in \braces{1,\ldots,r} \setminus I}} \weylannihilator(z_j),
\end{gather}
is called the \emph{normal} (or \emph{Wick}) \emph{ordering}\index{normal ordering}\index{Wick ordering} on $\hilb{H}$.
- A monomial $z_1 \cdots z_r \in \Mon(\WeylAlg)$ is said to be in \emph{normal} (or \emph{Wick}) \emph{order},
- if $\normord{z_1 \cdots z_r} = z_1 \cdots z_r$.
+ An element $w \in \InfinitesimalWeylAlg$ is said to be in \emph{normal} (or \emph{Wick}) \emph{order},
+ if $\normord{w} = w$.
\end{definition}
The products in~\eqref{equation:normal-ordering} are well defined
@@ -395,35 +479,50 @@ if one brings~\eqref{equation:normal-ordering} into the equivalent form
\label{equation:normal-ordering-symmetric}
\normord{z_1 \!\cdots z_r} =
\frac{1}{\sqrt{2^r}}
- \sum_{\sigma \in S_r}
+ \sum_{\sigma \in \SymmetricGroup{r}}
\sum_{s=0\vphantom{S}}^{r}
\frac{1}{s!(r-s)!}
\prod_{i=1\vphantom{S}}^{s} \weylcreator(z_{\sigma(i)})
\prod_{\mathclap{j=s+1\vphantom{S}}}^{r} \weylannihilator(z_{\sigma(j)})
\end{gather}
-by basic combinatorial arguments \todo{further explanation?}.
-In~\cite{Klein1973}, the factor $\frac{1}{s!(r-s)!}$ is erroneously missing.
-
+by basic combinatorial arguments:
+\nomenclature[S]{$\SymmetricGroup{r}$}{symmetric group on the set $\Set{1,\ldots,r}$}
+It is easiest to start from~\eqref{equation:normal-ordering-symmetric}
+and observe that the product remains invariant
+when we permute creators with creators and annihilators with annihilators.
+There are precisely $s!(r-s)!$ such permutations.
+Let $\sigma$ be any of these.
+The associated product corresponds to that summand of~\eqref{equation:normal-ordering}
+for which $I = \Set{\sigma(1),\ldots,\sigma(s)}$.
+The term $1 / s!(r-s)!$ cancels out.
+In more technical language, we are using the fact that the mapping
+\begin{align*}
+ \SymmetricGroup{r} \times \Set{1,\ldots,r} &\longrightarrow \PowerSet[\big]{\Set{1,\ldots,r}} \\
+ (\sigma,s) &\longmapsto \Set{\sigma(1),\ldots,\sigma(s)}
+\end{align*}
+is surjective and that the preimage of each $s$-element set has cardinality $s!(r-s)!$.
+\nomenclature[PX]{$\PowerSet{X}$}{power set of a set $X$}
+In~\cite{Klein1973}, the factor $1 / s!(r-s)!$ is erroneously missing
+(which makes no difference for $r < 2$).
+As a direct consequence of~\eqref{equation:normal-vaccum-2} we obtain
\begin{equation*}
E(\normord{z_1 \!\cdots z_r}) = 0 \qquad \forall z_1,\ldots,z_r \in \hilb{H}, r \ge 1
\end{equation*}
-\begin{equation*}
- E\parens[\Big]{\prod_{i=1\vphantom{S}}^{s} \weylcreator(z_i)
- \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} \weylannihilator(z_j)} = 0
- \qquad \forall z_1,\ldots,z_r \in \hilb{H}, r \ge 1, 1 \le s \le r
-\end{equation*}
-
+as desired.
The normal ordered product is supposed to represent the identical quantity as before ordering,
-except that we have adjusted our point of reference, such that measurements yield finite results.
-It is therefore \emph{physically reasonable} that the commutation relations
-of the normal ordered product with the field are analogous.
+except that we have adjusted our point of reference, so that the vacuum expectation value is zero.
+It is therefore \emph{physically reasonable} to demand that the commutation relations
+of the normal ordered product with the field remain unchanged.
As it turns out, this additional property makes the construction of normal ordering
\emph{mathematically unique}.
+This is the content of the following theorem.
-\begin{theorem}{Uniqueness of the Normal Order}{}
- Normal ordering is the unique mapping $N : \Mon(\WeylAlg) \to \WeylAlg$ such that
+\begin{theorem}{Characterization of the Normal Order}{}
+ Let $\hilb{H}$ be a complex Hilbert space and
+ let $E$ be the normal vacuum on $\InfinitesimalWeylAlg(\hilb{H})$.
+ Then, normal ordering is the unique mapping $N : \Mon(\InfinitesimalWeylAlg) \to \InfinitesimalWeylAlg$ such that
\begin{gather*}
E\parens[\big]{N(z_1 \!\cdots z_r)} = 0 \\
\bracks{N(z_1 \!\cdots z_r), z'} =
@@ -431,6 +530,7 @@ As it turns out, this additional property makes the construction of normal order
\end{gather*}
for all $z_1,\ldots,z_r,z' \in \hilb{H}$ and all $r \ge 1$.
\end{theorem}
+A proof of this statement is contained in the proof of~\cite[Theorem 7.1]{Baez1992}.
%\begin{theorem}{}{}
%The normal ordering is the renormalization with respect to the normal vacuum.
@@ -440,10 +540,10 @@ As it turns out, this additional property makes the construction of normal order
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,
+For this, we recall that in Wightman’s 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 (Schwartz 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 $\psi,\psi'$
+that is, a mapping that assigns to each (Schwartz class) test function $f$ on $M$ an unbounded operator $\varphi(f)$
+in the Fock space $\BosonFock{\hilb{H}}$ 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
@@ -482,32 +582,35 @@ 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^4}
+ 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 $M=\RR^4$.
In terms of creation and annihilation operators we have
\begin{equation}
- \label{derivative-free-field}
+ \label{equation:derivative-free-field}
D \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(ED^{\dagger}f)^{\dagger} + a(E\overline{D^{\dagger}f})}.
\end{equation}
In Fourier space the operator $D^\dagger$ corresponds to multiplication with the polynomial
-\begin{equation*}
+\begin{equation}
+ \label{equation:d-hat}
\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*}
+\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.
+Plugging~\eqref{equation:derivative-free-field} into the renormalization formula~\eqref{equation:normal-ordering-symmetric} yields
\begin{gather}
\label{equation:renormalized-product}
\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} =
\frac{1}{\sqrt{2^r}}
- \sum_{\sigma \in S_r}
+ \sum_{\sigma \in \SymmetricGroup{r}}
\sum_{s=0\vphantom{S}}^{r}
\frac{1}{s!(r-s)!}
\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})
+ \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} a(E\overline{D^\dagger_{\sigma(j)}f}).
\end{gather}
+%As discussed in \cref{section:quadratic-forms},
+%this is a well-defined quadratic form for each fixed test function $f$.
\section{Renormalized Products of the Free Field and~its~Derivatives}
@@ -557,7 +660,7 @@ this approach incurs significant technical difficulties.
\text{and} \quad P_s(p_1,\ldots,p_r) =
\frac{1}{\sqrt{2^r}}
\frac{1}{s!(r-s)!}
- \sum_{\sigma \in S_r}
+ \sum_{\sigma \in \SymmetricGroup{r}}
\ft{D}_{\sigma(1)}(p_1) \cdots \ft{D}_{\sigma(s)}(p_s) \hspace{1.5cm} \\[-1.5ex]
\cdot \overline{\ft{D}_{\sigma(s+1)}(p_{s+1}) \cdots \ft{D}_{\sigma(r)}(p_r)}.
\end{multline*}
@@ -570,14 +673,15 @@ This is made explicit in the alternative integral representation
\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
+ \hspace{1cm}
\sum_{s=0}^{r}
+ \int dp_1 \!\cdots dp_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
+ where the integral kernels are given by
\begin{multline*}
\tilde{K}^s_{\psi'\!,\psi}(p_1,\ldots,p_r) =
P_s(p_1,\ldots,p_r)
@@ -588,9 +692,9 @@ This is made explicit in the alternative integral representation
\ \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
+ This representation will be more convenient for xxx
-\begin{myproof}[lemma:renormalized-product-integral-representation]
+\begin{myproof}{lemma:renormalized-product-integral-representation}
From equation~\eqref{equation:renormalized-product},
applying the definition of the Fock space inner product,
and moving all creation operators to the left hand side,
@@ -598,7 +702,7 @@ This is made explicit in the alternative 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} \frac{1}{s!(r-s)!} \\
+ \frac{1}{\sqrt{2^r}} \sum_{s=0}^{r} \sum_{\sigma \in \SymmetricGroup{r}} \frac{1}{s!(r-s)!} \\
\cdot \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
@@ -608,7 +712,7 @@ This is made explicit in the alternative integral representation
can only be nonzero if the particle numbers match up
after the application of the annihilation operators in each argument,
that is if $m-s=n-(r-s)$.
- With~\eqref{equation:multiple-annihilation-operators}
+ By applying~\eqref{equation:multiple-annihilation-operators} twice,
this expression may be further expanded into
\begin{gather*}
\sqrt{m(m-1) \cdots (m-s+1)}
@@ -623,16 +727,20 @@ This is made explicit in the alternative integral representation
\end{gather*}
Now recall that $E$ stands for Fourier transformation (followed by restriction to the mass shell)
and that in Fourier space the linear differential operator $D^\dagger$ corresponds to a
- multiplication with the function $\hat{D}$, so that
+ multiplication with the function $\hat{D}$ defined by~\eqref{equation:d-hat}, so that
\begin{equation*}
ED_{\sigma(i)}^{\dagger}f(p_i) = \hat{D}_{\sigma(i)}(p_i) \cdot \ft{f} (p_i)
\qquad \forall i
\end{equation*}
- By Fubini’s Theorem, the we may interchange the integrals with respect to the variables $p_i$
- with the $k$-integrals.
- This allows us to move all factors involving $\ft{f}$ in front of the $k$-integrals.
- Finally, we introduce the $\chi$s through the substitution $\overline{\ft{f}} = \chi \ft{f}$,
- and combine all terms depending on $\sigma$ into $P_s$.
+ According to Fubini’s Theorem, we may move the $p$-integrals past the $k$-integrals to the very left of the expression.
+ This also allows us to move all factors involving $\ft{f}$ or one of the $\ft{D}_{\sigma(i)}$ in front of the $k$-integrals.
+ Finally, we move the summation over $s$ to the front.
+ By packaging everything not depending on $\ft{f}$ in new functions $\tilde{K}^{s}_{\psi',\psi}$,
+ where $s$ runs from $0$ to $r$,
+ we arrive at the representation~\eqref{equation:alternative-integral-representation}.
+
+ A single-integral representation is achieved by
+ introducing the functions $\chi$s through the substitution $\overline{\ft{f}} = \chi \ft{f}$,
\end{myproof}
In the special case that $D_1 = \cdots = D_r = D$ we have
@@ -679,7 +787,7 @@ In the following proof it will be convenient to use the abbreviation
\omega(p_1,\ldots,p_s) \defequal \omega(p_1) + \cdots + \omega(p_n).
\end{equation*}
-\begin{myproof}[lemma:integral-kernel-h-bound]
+\begin{myproof}{lemma:integral-kernel-h-bound}
We have to find an estimate for
\begin{equation*}
\norm{K_{\psi'\!,\psi}}_1 =
@@ -797,7 +905,7 @@ In the following proof it will be convenient to use the abbreviation
%$\norm{(1+H)\psi_n}_2 \ge n \epsilon \norm{\psi_n} = \epsilon \norm{N \psi_n}$
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 $\omega(p_1),\ldots,\omega(p_r)$.
+ it is desirable 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
@@ -875,10 +983,13 @@ In the following proof it will be convenient to use the abbreviation
and because $((1+H)^l \psi')_m = (1+H)^l \psi'_m$ for all $m$.
\end{myproof}
+Now we are in a position to implement the idea of taking the limit $f \to \delta_x$.
+Recall that any tempered distribution, and in particular Dirac distributions, may be approximated by Schwarz test functions.
+
\begin{lemma}{Renormalized Product at a Point}{}
In the setting of \cref{lemma:renormalized-product-integral-representation},
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 $x$.
+ Let $x$ be any point in $M$ and let $\delta_x \in \tempdistrib{M}$ be the Dirac distribution supported at $x$.
Then the limit
\begin{equation*}
\lim_{f \to \delta_x}
@@ -888,6 +999,11 @@ In the following proof it will be convenient to use the abbreviation
\end{lemma}
\begin{proof}
+ 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 $\tempdistribnoarg$.
+ Consequently, $\abs{\ft{f}}$ remains bounded by some constant $C$ while taking the limit.
+
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}
@@ -895,58 +1011,56 @@ In the following proof it will be convenient to use the abbreviation
\, \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}.
+ The integrand is dominated by the function $C^r\abs{K_{\psi'\!,\psi}(p_1,\ldots,p_r)}$,
+ which has finite integral 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?}
+ %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
+ This shows that the integrand converges pointwise to the function
\begin{equation*}
+ F(p_1,\ldots,p_r) =
\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)
+ \tilde{K}^s_{\psi'\!,\psi}(p_1,\ldots,p_r)
\end{equation*}
-
- The Dominated Convergence Theorem implies
+ The Dominated Convergence Theorem implies that $F$ is integrable and that
+ \begin{equation}
+ \label{equation:intx}
+ \lim_{f \to \delta_x}
+ \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi}
+ = \int dp_1 \!\cdots dp_r \, F(p_1,\ldots,p_r).
+ \end{equation}
+ In particular, the limit exists.
\end{proof}
-\begin{definition}{Renormalized Product at a Point}{}
- In the setting of \cref{lemma:renormalized-product-integral-representation},
- the mapping defined by
+\begin{definition}{Renormalized Product as a QF-valued distribution}{}
+ Adopt the assumptions of the foregoing Lemma.
+ We define two mappings that intentionally share the same name by
\begin{gather*}
\normord{D_1 \varphi \cdots D_r \varphi} \ \vcentcolon \
- M \to \QF{fock} \\
+ M \to \QF{\mathcal{D}(H^l)} \\
\innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(x) \,\psi}
= \lim_{f \to \delta_x}
\innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi}
\end{gather*}
- is called the xxx
-\end{definition}
-
-
-\begin{lemma}{Renormalized Product as a QF-valued distribution}{}
- In the setting of \cref{lemma:renormalized-product-integral-representation},
+ and
\begin{equation*}
\normord{D_1 \varphi \cdots D_r \varphi} \ \vcentcolon \
- \schwartz{M} \to \QF{fock}
+ \schwartz{M} \to \QF{\mathcal{D}(H^l)}
\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}
+\end{definition}
-\begin{lemma}{TODO}{}
+\begin{proposition}{}{}
Let $\varphi$ be a free quantum field.
Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients.
- Suppose that $l$ is a positive integer large enough to satisfy the
+ Suppose that $l$ is a large enough positive integer.
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} = \\
@@ -967,10 +1081,10 @@ In the following proof it will be convenient to use the abbreviation
\ \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}
+\end{proposition}
\begin{proof}
- a
+ This follows directly from the definition and~\eqref{equation:intx}.
\end{proof}
@@ -1005,15 +1119,9 @@ In particular, the energy density is
\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'\!}{\energydensity(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
+Now let $f$ be a real-valued Schwarz function,
+and let $\psi,\psi' \in \Domain{H^l}$ for a large enough integer $l$.
+For convenience we introduce the functions
\begin{multline*}
L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) =
\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
@@ -1023,6 +1131,43 @@ where
\ \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*}
+so that
+\begin{equation*}
+ \tilde{K}_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) =
+ P_s(p_1,\ldots,p_r) \,
+ L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r).
+\end{equation*}
+Since the energy density only contains squares, it suffices to consider $r=2$.
+
+\begin{multline*}
+ \innerp{\psi'\!}{\normord{\varphi^2}(f) \,\psi} = \int dp_1 dp_2 \,
+ \ft{f}(-p_1 - p_2) \, \tfrac{1}{2} L_{\psi'\!,\psi}^{0}(p_1,p_2) \\
+ + \ft{f}(p_1 - p_2) \, L_{\psi'\!,\psi}^{1}(p_1,p_2)
+ + \ft{f}(p_1 + p_2) \, \tfrac{1}{2} L_{\psi'\!,\psi}^{2}(p_1,p_2) \\
+ \innerp{\psi'\!}{\normord{(\partial^{\mu}\varphi)^2}(f) \,\psi} = \int dp_1 dp_2 \,
+ \ft{f}(-p_1 - p_2) \, \parens[\big]{-\tfrac{1}{2} (p_1)_{\mu} (p_2)_{\mu}} \, L_{\psi'\!,\psi}^{0}(p_1,p_2) \\
+ + \ft{f}(p_1 - p_2) \, \parens[\big]{+(p_1)_{\mu} (p_2)_{\mu}} \, L_{\psi'\!,\psi}^{1}(p_1,p_2)
+ + \ft{f}(p_1 + p_2) \, \parens[\big]{-\tfrac{1}{2} (p_1)_{\mu} (p_2)_{\mu}} \, L_{\psi'\!,\psi}^{2}(p_1,p_2)
+\end{multline*}
+
+\begin{equation*}
+ \bar{p} \defequal \eta p = (p^0,-\Vector{p})
+\end{equation*}
+
+\begin{equation*}
+ \sum_{\mu = 0}^{3} p_{\mu} p'_{\mu} = \bar{p} \cdot p'
+\end{equation*}
+
+\begin{proposition}{}{}
+ \begin{multline*}
+ \innerp{\psi'\!}{\energydensity(f) \, \psi} =
+ \frac{1}{4} \int dp \, dp'
+ (m^2 + \bar{p} \cdot p')
+ \bracks[\big]{2 \ft{f}(p - p') L^1_{\psi'\!,\psi}(p,p')} \\
+ + (m^2 -\bar{p} \cdot p')
+ \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{theorem}{TODO}{}
Let $\varphi$ be a free quantum field.
@@ -1038,45 +1183,25 @@ where
\quad P_s(p_1,\ldots,p_r) =
\frac{1}{\sqrt{2^r}}
\frac{1}{s!(r-s)!}
- \sum_{\sigma \in S_r}
+ \sum_{\sigma \in \SymmetricGroup{r}}
\ft{D}_{\sigma(1)}(p_1) \cdots \ft{D}_{\sigma(s)}(p_s) \hspace{1.5cm} \\[-1.5ex]
\cdot \overline{\ft{D}_{\sigma(s+1)}(p_{s+1}) \cdots \ft{D}_{\sigma(r)}(p_r)}.
\end{multline*}
\end{theorem}
-\begin{definition}{}{}
+\begin{proposition}{}{energy-density}
\begin{multline*}
- \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'
+ \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')} \\
+ \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}
+Observe that $\bar{p} \cdot p' = p^0p'^0 + p^1p'^1 + p^2p'^2 + p^3p'^3$ is symmetric in $p$ and $p'$.
+Consequently, we could rewrite the first bracketed expression as $2\ft{f}(p-p') a^\dagger(p) a(p')$
+
\begin{proposition}{}{}
The Fock vacuum $\FockVacuum$ lies in the domain of $\energydensity(f)\QFop{}$
for all test functions $f \in \schwartz{M}$
@@ -1087,6 +1212,29 @@ where
and $\psi_n \equiv 0$ for $n \ne 2$.
\end{proposition}
+\begin{proof}
+ Since $\ft{f}$ is a Schwarz function, there exists for each $N \in \NN$ a positive constant $C_N$ such that
+ \begin{equation*}
+ \abs{\ft{f}(q)}^2 \le \frac{C_N}{1+(q^0)^{2N}}
+ \end{equation*}
+ For $p,p'$ on the mass shell we have
+ \begin{equation*}
+ m^2 - \bar{p} \cdot p' =
+ m^2 - \sqrt{m^2 + \norm{\Vector{p}}^2} \sqrt{m^2 + \norm{\Vector{p'}}^2} + \Vector{p} \cdot \Vector{p}',
+ \end{equation*}
+ where on the right hand side \enquote*{$\cdot$} stands for the Euclidean scalar product for three-vectors.
+ The Cauchy-Schwarz Inequality (applied twice) shows
+ \begin{equation*}
+ m^2 + \Vector{p} \cdot \Vector{p}'
+ \le m^2 + \norm{\Vector{p}} \norm{\Vector{p}'}
+ \le \sqrt{m^2 + \norm{\Vector{p}}^2} \sqrt{m^2 + \norm{\Vector{p'}}^2}.
+ \end{equation*}
+ \begin{equation*}
+ \abs{m^2 - \bar{p} \cdot p'}
+ \le 2 \sqrt{m^2 + \norm{\Vector{p}}^2} \sqrt{m^2 + \norm{\Vector{p'}}^2}
+ \end{equation*}
+\end{proof}
+
\section{Essential Selfadjointness of Renormalized Products}
\begin{lemma}{H-Bounds for the Renormalized Product}{}
diff --git a/titlepage.tex b/titlepage.tex
index 219bd8a..8ce57de 100644
--- a/titlepage.tex
+++ b/titlepage.tex
@@ -1,40 +1,44 @@
-\begin{titlepage}
+\begin{fullsizetitle}
\bookmarksetupnext{level=section,italic}
\pdfbookmark{Title}{title}
- \begin{center}
- \fausansoffice
-
- \includegraphics[width=0.5\textwidth]{images/FAU_NatFak_EN_Q_RGB_black.pdf}
- \vspace{1.5cm}
-
- \textbf{\Huge Title Title Title\medskip\\ Title Title}
- \vspace{0.5cm}
-
- \directlua{% cspell:disable
- local handle = io.popen("git rev-parse HEAD")
- local commit = handle:read("*line")
- handle:close()
- output = 'commit \string\\href{https://git.jxir.de/master/commit/?id=' .. commit .. '}{' .. commit .. '}\\\\ compiled on ' .. os.date()
- tex.print(output)
- }% cspell:enable
- \vspace{1.5cm}
-
- A thesis presented for the degree of\\
- \textit{Master in Physics}
- \vspace{1cm}
-
- by\\
- \textbf{Justin Gassner}
- \vspace{\fill}
- %\vfill
-
- Supervised by\\
- \textbf{Prof.\ Dr.\ Hanno Sahlmann}\\
- \textbf{Prof.\ Dr.\ Gandalf Lechner}
- \vspace{1.5cm}
-
- Erlangen\\
- March 2024
- \end{center}
-\end{titlepage}
+ \centering
+ \fausansoffice
+
+ \phantom{.}
+ \vspace{4cm}
+
+ \textbf{\Huge A Local Modular\\ Quantum Energy Inequality\medskip\\ for the Scalar Free Field}
+ % TODO find a good title
+ \vspace{1cm}
+
+ %\directlua{% cspell:disable
+ %local handle = io.popen("git rev-parse HEAD")
+ %local commit = handle:read("*line")
+ %handle:close()
+ %output = 'commit \string\\href{https://git.jxir.de/master/commit/?id=' .. commit .. '}{' .. commit .. '}\\\\ compiled on ' .. os.date()
+ %tex.print(output)
+ %}% cspell:enable
+ %\vspace{1.5cm}
+
+ \textit{Master’s Thesis in Physics}
+ \vspace{1cm}
+
+ by\\
+ \textbf{Justin Gassner}
+ \vspace{1cm}
+
+ 30.09.2024
+ \vspace{1cm}
+
+ Friedrich-Alexander-Universität Erlangen-Nürnberg\\
+ \vspace{2cm}
+ \includegraphics[width=8.5cm]{images/fau-siegel.png}
+ \vspace{2cm}
+
+ Supervised by\\
+ \textbf{Prof.\ Dr.\ Hanno Sahlmann}\\
+ \textbf{Prof.\ Dr.\ Gandalf Lechner}
+ \vspace{1.5cm}
+
+\end{fullsizetitle}
\cleardoublepage