From b9e2609169709f8aad257fa5e3a92cb780dfad3f Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Sun, 19 May 2024 01:45:37 +0200 Subject: weiter --- .cspell.yaml | 39 ++-- analytic2.tex | 135 ++++++++++++ bib/articles.bib | 2 +- bib/misc.bib | 28 +++ bib/much.bib | 29 +++ bib/test.bib | 32 +++ bib/test/distributions.bib | 170 +++++++++++++++ main.tex | 4 +- much.tex | 502 +++++++++++++++++++++++++++++++++++++++++++++ preamble.tex | 37 +++- stresstensor.tex | 302 ++++++++++++++++++++++----- 11 files changed, 1205 insertions(+), 75 deletions(-) create mode 100644 analytic2.tex create mode 100644 bib/misc.bib create mode 100644 bib/much.bib create mode 100644 bib/test.bib create mode 100644 bib/test/distributions.bib create mode 100644 much.tex diff --git a/.cspell.yaml b/.cspell.yaml index 8497152..16a83c3 100644 --- a/.cspell.yaml +++ b/.cspell.yaml @@ -1,38 +1,47 @@ ---- -version: "0.2" -language: "en-GB" +version: 0.2 +language: en_US +minWordLength: 3 dictionaryDefinitions: - name: project-code - path: "./.cspell/project-code.txt" addWords: true + path: ./.cspell/project-code.txt - name: project-words - path: "./.cspell/project-words.txt" addWords: true + path: ./.cspell/project-words.txt - name: project-citekeys - path: "./.cspell/project-citekeys.txt" addWords: true + path: ./.cspell/project-citekeys.txt - name: latex-pkgs - path: "./.cspell/my-cspell-dicts/latex-pkgs.txt" addWords: true + path: ./.cspell/my-cspell-dicts/latex-pkgs.txt - name: biblatex - path: "./.cspell/my-cspell-dicts/biblatex.txt" addWords: true + path: ./.cspell/my-cspell-dicts/biblatex.txt - name: names - path: "./.cspell/my-cspell-dicts/names.txt" 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 dictionaries: - latex - project-code - project-words - latex-pkgs + - names + - names2 + - math +ignorePaths: + - ./.cspell/ + - ./aux/ + - ./cd-label/ + - ./my/ + - ./my2/ overrides: - filename: "bib/**/*.bib" dictionaries: - biblatex - names -ignorePaths: - - "./.cspell/" - - "./aux/" - - "./my/" - - "./my2/" -minWordLength: 3 diff --git a/analytic2.tex b/analytic2.tex new file mode 100644 index 0000000..e0dc68f --- /dev/null +++ b/analytic2.tex @@ -0,0 +1,135 @@ +\chapter{Analytic Vectors} + +\info{Dies ist nur ein Relikt meines Studiums analytischer Vektoren. Wird wieder entfernt, falls nicht benötigt.} + +\begin{definition}{Analytic Vector for an Operator}{analytic-vector-operator} + Let $A : D(A) \to \hilb{H}$ be an unbounded linear operator in a complex Hilbert space $\hilb{H}$. + A vector $x \in \hilb{H}$ is said to be an \emph{analytic vector} for $A$, + if $x$ lies in the domain of the power $A^n$ for all $n \in \NN$, and the power series + \begin{equation*}\tag{power-series-analytic-vector} + \sum_{n=0}^{\infty} \frac{A^n x}{n!} \, z^n + \end{equation*} + has a nonzero radius of convergece. + If the power series converges for all $z \in \CC$, + we say that $x$ is an \emph{entire analytic vector} for $A$. +\end{definition} +Note that, if $x$ is analytic for $A$, then the power series +\begin{equation*} + \sum_{n=0}^{\infty} \frac{\norm{A^n x}}{n!} \, z^n +\end{equation*} +converges for all complex $z$ with $\abs{z} < t$, +that is, in the open disc with radius $t$ centered in the origin of the complex plane. +This is a well-known consequence of the convergence behavior of power series. + +\begin{definition}{Analyticity of Vector-Valued Functions}{} + Let $G \subset \CC$ be open and let $\hilb{H}$ be a Hilbert space. + A function $f : G \to \hilb{H}$ is called + \begin{itemize} + \item \emph{strongly analytic} at $a \in G$, if the limit + \begin{equation*} + \lim_{z \to a} \frac{f(z) - f(a)}{z-a} + \end{equation*} + exists in norm. + \item \emph{weakly analytic} in $a \in G$, if for each $w \in \CC$ the scalar-valued function + \begin{equation*} + G \longrightarrow \CC, \quad z \longmapsto \innerp{w}{f(z)} + \end{equation*} + is analytic in $a$. + \end{itemize} +\end{definition} + +\begin{lemma}{Equivalence of Weak and Strong Analyticity}{} + Let $G \subset \CC$ be open. + Then a Banach space-valued function is strongly analytic on $G$ if and only if it is weakly analytic on $G$. +\end{lemma} +\begin{myproof} + Let $X$ be a Banach space and suppose that the function $f : G \to X$ is weakly analytic. + By definition, for each $g \in X'$ the scalar valued function $g \circ f : G \to \CC$ is analytic on $G$. + Consider a point $a \in G$. + Since $G$ is open, there exists a circular contour $\gamma$ around $a$ such that $\gamma$ and its interior lie wholly inside of $G$. + By Cauchy’s Integral Formula we have + \begin{equation*} + g(f(z)) = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(f(w))}{w-z} \, dw + \end{equation*} + for any $z$ in the interior of $\gamma$. + Writing + \begin{equation*} + Q(z) = \frac{f(z) - f(a)}{z - a} + \end{equation*} + for the difference quotient, we get + \begin{equation*} + g(Q(z)) = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(f(w))}{(w-z)(w-a)} \, dw + \end{equation*} + and + \begin{equation*} + g \parens*{\frac{Q(z) - Q(z')}{z - z'}} = \frac{1}{2 \pi i} \int_{\gamma} \frac{g(f(w))}{(w-z)(w-z')(w-a)} \, dw + \end{equation*} + for all $z,z'$ in the interior of $\gamma$. + The family of vectors $f(w) \in X$, indexed by complex numbers $w$ on the contour $\gamma$, can be viewed as a family of bounded linear functionals $C(f(w)) : X' \to \CC$ + via the canonical embedding $C : X \to X''$ of $X$ into its bidual. For every fixed $g \in X'$ the set of values $C(f(w))(g) = g(f(w))$ is bounded, because the function $g \circ f$ is continous and the contour is compact. + In other words, the family of functionals $C(f(w))$, $w \in \gamma$, is pointwise bounded. + The Uniform Boundedness Theorem implies that there exists a constant $M > 0$ such that $\abs{g(f(w))} \le M \norm{g}$ for all $w$ on $\gamma$ and all $g \in X'$. + \begin{equation*} + \abs*{g \parens*{\frac{Q(z) - Q(z')}{z - z'}}} \le \frac{M}{2 \pi} \norm{g} \int_{\gamma} \frac{dw}{\abs{w-z}\abs{w-z'}\abs{w-a}} + \end{equation*} + If we restict $z,z'$ to a neighbourhood $N$ of $a$ that stays away from $\gamma$, then the integral on the right hand side is + bounded by a constant independent of $z$ and $z'$. + Absorbing all constants into $M' > 0$ we obtain + \begin{equation*} + \abs{g(Q(z) - Q(z'))} \le M' \norm{g} \abs{z-z'} \quad \forall z,z' \in N. + \end{equation*} + \begin{equation*} + \norm{Q(z) - Q(z')} = \sup_{\substack{g \in X'\\ \norm{g} \le 1}} \abs{g(Q(z) - Q(z'))} \le M' \norm{z - z'}. + \end{equation*} + Hence, the limit of $Q(z)$ for $z \to a$ exists by completeness of $X$. +\end{myproof} + +\begin{definition}{Analytic Vector for an Unitary Group}{analytic-vector-unitary-group} + Let $\sigma : \RR \to U(\hilb{H})$ be a strongly continuous one-parameter unitary group on a complex Hilbert space $\hilb{H}$. + A vector $x \in \hilb{H}$ is said to be an \emph{analytic vector} for $\sigma$, if there exist + \begin{itemize} + \item a number $\lambda > 0$, defining a strip $I_{\lambda} = \braces{z : \abs{\Im z} < 1}$, and + \item a vector-valued function $f : I_{\lambda} \to \hilb{H}$, + \end{itemize} + with the properties that + \begin{itemize} + \item $f(t) = \sigma_t(x)$ for all $t \in \RR$, + \item $f$ is weakly analytic on $I_{\lambda}$. + \end{itemize} + In this case we write $f(z) = \sigma_z(x)$ for $z \in I_{\lambda}$. +\end{definition} + +\begin{proposition}{}{} + Let $\sigma : \RR \to U(\hilb{H})$ be a strongly continuous one-parameter unitary group on a complex Hilbert space $\hilb{H}$ + and let $A$ be its infinitesimal generator. + Then a vector $x \in \hilb{H}$ is analytic for $\sigma$ if and only if it is analytic for $A$. +\end{proposition} + +\begin{myproof} + First, suppose that $x$ is an analytic vector for $\sigma$. + Then, there exist a number $\lambda > 0$ and a function $f : I_{\lambda} \to X$ as in \cref{definition:analytic-vector-unitary-group}. + In particular, $f$ is (strongly) analytic on the strip $I_{\lambda}$, which contains the disk $\braces{z : \abs{z} \le r}$ when $r \le \lambda$. + Hence we have Cauchy estimates + \begin{equation*} + \norm{f^{(n)}(0)} \le \frac{n!}{r^n} M \quad \forall n \in \NN, + \quad \text{where} \ M = \sup_{\abs{z} = r} \norm{f(z)}. + \end{equation*} + For real $t$ we have $f(t) = \sigma_t(x) = \exp(itA) x$ and + the mapping $t \mapsto f(t)$ is strongly differentiable with derivatives + $f^{(n)}(0) = (iA)^n x$. + This implies that the power series + \begin{equation*} + \sum_{n=0}^{\infty} \frac{\norm{A^n x}}{n!} t^n \le M \sum_{n=0}^{\infty} \frac{t^n}{r^n} + \end{equation*} + is convergent for $t \le \lambda$ by majorization. Hernce $x$ is an analytic vector for the operator $A$. + + Coversely, suppose that $x$ is analytic for the generator $A$ of $\sigma$. + Then, by \cref{definition:analytic-vector-operator}, $x$ lies in the domains of all powers $A^n$, $n \in \NN$, and the power series + \begin{equation*} + \sum_{n=0}^{\infty} \frac{\norm{A^n x}}{n!} z^n + \end{equation*} + has a positive radius of convergence $t>0$. +\end{myproof} + +\chapterbib +\cleardoublepage diff --git a/bib/articles.bib b/bib/articles.bib index 2368b40..60d4d4f 100644 --- a/bib/articles.bib +++ b/bib/articles.bib @@ -1,8 +1,8 @@ @article{Epstein1965, author = {Epstein, H. and Glaser, V. and Jaffe, A.}, title = {Nonpositivity of the Energy Density in Quantized Field Theories}, - publisher = {Società Italiana di Fisica}, journaltitle = {Il Nuovo Cimento}, + publisher = {Società Italiana di Fisica}, volume = {36}, issue = {3}, date = {1965}, diff --git a/bib/misc.bib b/bib/misc.bib new file mode 100644 index 0000000..b32be08 --- /dev/null +++ b/bib/misc.bib @@ -0,0 +1,28 @@ +@book{ReedSimon1, + maintitle = {Methods of Modern Mathematical Physics}, + title = {Functional Analysis}, + author = {Michael Reed and Barry Simon}, + publisher = {Academic Press}, + date = {1980}, + edition = {Revised and Enlarged Edition}, + volume = {1}, +} +@book{ReedSimon2, + maintitle = {Methods of Modern Mathematical Physics}, + title = {Fourier Analysis, Self-Adjointness}, + author = {Michael Reed and Barry Simon}, + publisher = {Academic Press}, + date = {1975}, + volume = {2}, +} +@article{Nelson1972, + title = {Time-Ordered Operator Products of Sharp-Time Quadratic Forms}, + author = {Edward Nelson}, + publisher = {Elsevier Science}, + journal = {Journal of Functional Analysis}, + issn = {0022-1236,1096-0783}, + date = {1972}, + volume = {11}, + number = {2}, + pages = {211--219}, +} diff --git a/bib/much.bib b/bib/much.bib new file mode 100644 index 0000000..c2f76f4 --- /dev/null +++ b/bib/much.bib @@ -0,0 +1,29 @@ +@misc{Much2022, + title={An approximate local modular quantum energy inequality in general quantum field theory}, + author={Albert Much and Albert Georg Passegger and Rainer Verch}, + year={2022}, + eprint={2210.01145}, + archivePrefix={arXiv}, + primaryClass={math-ph} +} +@article{Bisognano1975, + title = {On the duality condition for a Hermitian scalar field}, + author = {Joseph J. Bisognano and Eyvind H. Wichmann}, + publisher = {American Institute of Physics}, + journal = {Journal of Mathematical Physics}, + issn = {0022-2488,1089-7658}, + year = {1975}, + volume = {16}, + number = {4}, + pages = {985-1007}, +} +@book{Schmüdgen2012, + title = {Unbounded Self-adjoint Operators on Hilbert Space}, + author = {Konrad Schmüdgen}, + publisher = {Springer}, + isbn = {9789400747524}, + year = {2012}, + series = {Graduate Texts in Mathematics}, + edition = {1}, + volume = {265}, +} diff --git a/bib/test.bib b/bib/test.bib new file mode 100644 index 0000000..c6d07d6 --- /dev/null +++ b/bib/test.bib @@ -0,0 +1,32 @@ +@book{book:4004309, + title = {Introduction to Algebraic and Constructive Quantum Field Theory}, + author = {John C. Baez and Irving Ezra Segal and Zhengfang Zhou}, + publisher = {Princeton University Press}, + isbn = {9781400862504}, + year = {1992}, + series = {Princeton Series in Physics}, + volume = {}, +} +@article{Wick1950, + title = {The Evaluation of the Collision Matrix}, + author = {G. C. Wick}, + publisher = {American Physical Society}, + journal = {Physical Review}, + issn = {0031-899X,1536-6065}, + year = {1950}, + volume = {80}, + number = {2}, + pages = {268--272}, +} +@article{Fewster1998, + title = {Bounds on negative energy densities in flat spacetime}, + author = {C. J .Fewster and S. P. Eveson}, + publisher = {American Physical Society}, + journal = {Physical Review D}, + ISSN = {1089-4918}, + year = {1998}, + volume = {58}, + number = {8}, + month = sep, + pages = {084010}, +} diff --git a/bib/test/distributions.bib b/bib/test/distributions.bib new file mode 100644 index 0000000..81746dc --- /dev/null +++ b/bib/test/distributions.bib @@ -0,0 +1,170 @@ +@book{Halperin1952, + series = {Canadian Mathematical Congress, Lecture Series}, + volume = {1}, + title = {Introduction to the Theory of Distributions}, + author = {Israel Halperin}, + note = {Based on the lectures given by \textsc{L. Schwartz}.}, + publisher = {University of Toronto Press}, + year = {1952}, +} +@book{Lighthill1958, + title = {Introduction to Fourier Analysis and Generalised Functions}, + author = {M. J. Lighthill}, + publisher = {Cambridge University Press}, + year = {1958}, +} +@article{Gårding1959, + doi = {10.1007/bf02724838}, + title = {Functional Analysis}, + author = {L. Gårding and J. L. Lions}, + publisher = {Società Italiana di Fisica}, + journaltitle = {Il Nuovo Cimento}, + issn = {0029-6341,1827-6121}, + year = {1959}, + volume = {14}, + issue = {1 Supplement}, + pages = {9--66}, +} +@book{Erdélyi1962, + title = {Operational Calculus and Generalized Functions}, + author = {Arthur Erdélyi}, + publisher = {Holt, Rinehart and Winston}, + year = {1962}, +} +@book{Schwartz1966, + title = {Théorie des distributions}, + author = {Laurent Schwartz}, + publisher = {Hermann}, + year = {1966}, + language = {french}, +} +@book{Trèves1967, + title = {Topological Vector Spaces, Distributions and Kernels}, + author = {François Trèves}, + publisher = {Academic Press}, + year = {1967}, +} +@book{Jones1982, + title = {The Theory of Generalised Functions}, + author = {D. S. Jones}, + publisher = {Cambridge University Press}, + year = {1982}, + edition = {2}, +} +@book{GenFunc1, + maintitle = {Generalized Functions}, + volume = {1}, + title = {Properties and Operations}, + author = {I. M. Gel'fand and G. E. Shilov}, + publisher = {Academic Press}, + year = {1964}, + language = {english}, + origlanguage = {russian}, + translator = {Eugene Saletan}, +} +@book{GenFunc2, + maintitle = {Generalized Functions}, + volume = {2}, + title = {Spaces of Fundamental and Generalized Functions}, + author = {I. M. Gel'fand and G. E. Shilov}, + publisher = {Academic Press}, + year = {1968}, + language = {english}, + origlanguage = {russian}, + translator = {Morris D. Friedman and Amiel Feinstein and Christian P. Peltzer}, +} +@book{GenFunc3, + maintitle = {Generalized Functions}, + volume = {3}, + title = {Theory of Differential Equations }, + author = {I. M. Gel'fand and G. E. Shilov}, + publisher = {Academic Press}, + year = {1967}, + language = {english}, + origlanguage = {russian}, + translator = {Meinhard E. Mayer}, +} +@book{GenFunc4, + maintitle = {Generalized Functions}, + volume = {4}, + title = {Applications of Harmonic Analysis}, + author = {I. M. Gel'fand and N. Ya. Vilenkin}, + publisher = {Academic Press}, + year = {1964}, + language = {english}, + origlanguage = {russian}, + translator = {Amiel Feinstein}, +} +@book{GenFunc5, + maintitle = {Generalized Functions}, + volume = {5}, + title = {Integral Geometry and Representation Theory}, + author = {I. M. Gel'fand and M. I. Graev and N. Ya. Vilenkin}, + publisher = {Academic Press}, + year = {1966}, + language = {english}, + origlanguage = {russian}, + translator = {Eugene Saletan}, +} +@book{GenFunc6, + maintitle = {Generalized Functions}, + volume = {6}, + title = {Representation Theory and Automorphic Functions}, + author = {I. M. Gel'fand and M. I. Graev and I. I. Pyatetskii-Shapiro}, + publisher = {W. B. Saunders}, + year = {1969}, + language = {english}, + origlanguage = {russian}, + translator = {K. A. Hirsch}, +} +@book{Richards1990, + title = {The Theory of Distributions}, + subtitle = {A Nontechnical Introduction}, + author = {Ian Richards and Heekyung Youn}, + publisher = {Cambridge University Press}, + year = {1990}, +} +@book{Strichartz1994, + title = {A Guide to Distribution Theory and Fourier Transforms}, + author = {Robert Strichartz}, + publisher = {CRC Press}, + year = {1994}, + series = {Studies in Advanced Mathematics}, + edition = {}, + volume = {}, +} +@article{Wightman1996, + doi = {10.1002/prop.2190440204}, + title = {How It Was Learned that Quantized Fields Are Operator-Valued Distributions}, + author = {A. S. Wightman}, + publisher = {John Wiley and Sons}, + journaltitle = {Fortschritte der Physik}, + issn = {0015-8208,1521-3978}, + year = {1996}, + volume = {44}, + issue = {2}, + pages = {143--178}, +} +@book{Friedlander1999, + title = {Introduction to the Theory of Distributions}, + author = {F. G. Friedlander and M. Joshi}, + publisher = {Cambridge University Press}, + year = {1999}, + edition = {2}, +} +@book{Hoskins2005, + title = {Theories of Generalised Functions}, + title = {Distributions, Ultradistributions and Other Generalised Functions}, + author = {R. F. Hoskins and J. S. Pinto}, + publisher = {Woodhead Publishing}, + year = {2005}, + edition = {2}, +} +@book{Hoskins2009, + title = {Delta Functions}, + subtitle = {Introduction to Generalised Functions}, + author = {R. F. Hoskins}, + publisher = {Woodhead Publishing}, + year = {2009}, + edition = {2}, +} diff --git a/main.tex b/main.tex index fc44017..91aaaab 100644 --- a/main.tex +++ b/main.tex @@ -1,5 +1,5 @@ \input{preamble} -\includeonly{stresstensor,fewstereveson,commutatortheorem,index} +\includeonly{stresstensor,fewstereveson,much,commutatortheorem,analytic2,symbols,index} \begin{document} \frontmatter \include{titlepage} @@ -12,10 +12,12 @@ \include{standard} \include{stresstensor} \include{fewstereveson} +\include{much} \include{samplesection} \appendix \include{sampleappendix} \include{commutatortheorem} +\include{analytic2} \backmatter \include{bibliography} \include{symbols} diff --git a/much.tex b/much.tex new file mode 100644 index 0000000..58aeab3 --- /dev/null +++ b/much.tex @@ -0,0 +1,502 @@ +\chapter{A quantum energy inequality involving local modular data} + + +\cite{Much2022} + +\begin{equation*} + \innerp{\psi}{\energydensity(f)\psi} \ge + - \epsilon - \norm{\smash[b]{\Delta}_{\smash[t]{\sharp}}^{-1/2} \ft{g}_{\lambda}(K_{\raisebox{5pt}{\footnotesize$\sharp$}}) \energydensity(f) \fockvaccum} +\end{equation*} + + +\section{Misc} + +\todo{Put this somwhere else.} + +A \emph{Lorentz transform} is a linear automorphism of Minkowski spacetime +which preserves the Lorentz bilinear form. +Lorentz transforms are usually represented by (real) $4 \times 4$ matrices, +with respect to the standard basis. +the \emph{Lorentz group} $\FullLorentzGroup$. +\begin{equation*} + \FullPoincareGroup = \RR^4 \ltimes \FullLorentzGroup +\end{equation*} + +The relativistic transformation law for one-particle states is given by +\begin{equation*} + \parens[\big]{U(a,\Lambda) \psi}(p) = e^{ia \cdot p} \psi(\Lambda^{-1} p), + \quad \psi \in \hilb{H}, (a,\Lambda) \in \ProperOrthochronousPoincareGroup. +\end{equation*} +The mapping $(a,\Lambda) \mapsto U(a,\Lambda)$ is a (irreducible) unitary representation +of the proper orthochronous Poincaré group $\ProperOrthochronousPoincareGroup$ +on the one-particle Hilbert space. +By applying any of the second quantization functors we obtain a representation on the multi-particle state space. +\begin{equation*} + \parens[\big]{U(a,\Lambda) \psi}{}_n(p_1,\ldots,p_n) = e^{ia \cdot (p_1 + \cdots + p_n)} \psi_n(\Lambda^{-1} p_1, \ldots, \Lambda^{-1} p_n), +\end{equation*} + +Poincaré covariance +\begin{equation} + \label{equation:poincare-covariance-local-algebras} + U(g) \localalg{\spacetimeregion{O}} U(g)^* = \localalg{g\spacetimeregion{O}} + \qquad g \in \ProperOrthochronousPoincareGroup +\end{equation} + +\begin{definition}{Von Neumann Algebra of Local Observables}{} + \begin{equation*} + \localalg{\spacetimeregion{O}} = \braces{b(\varphi(f)) \mid b, f \in \realschwartz{M}, \supp f \subset \spacetimeregion{O}}'' + \end{equation*} +\end{definition} + +\section{Basic Concepts of Modular Theory} +\index{modular!theory} + +If $\hilb{H}$ is a Hilbert space +we shall denote the $C^*$-algebra of all bounded linear operators on $\hilb{H}$ by $B(\hilb{H})$. + +\begin{definition}{Cyclic and Separating Vectors}{} + Suppose $\hilb{H}$ is a Hilbert space and $\mathcal{A}$ is a $C^*$-subalgebra of $B(\hilb{H})$. + A vector $\Omega \in \hilb{H}$ is called + \begin{itemize} + \item \emph{cyclic}\index{cyclic vector} for $\mathcal{A}$ if the vector set $\mathcal{A} \Omega$ is dense in $\hilb{H}$. + \item \emph{separating}\index{separating vector} for $\mathcal{A}$ if the map $A \mapsto A \Omega$ from $\mathcal{A}$ into $\hilb{H}$ is injective. + \end{itemize} +\end{definition} +Occasionally, a vector that is both cyclic and separating is called \emph{standard}\index{standard vector}. + +Recall that the commutant of a set $\mathcal{S} \subset B(\hilb{H})$ of operators +is defined as the set of all operators $T \in B(\hilb{H})$ which commute with all operators $S$ in $\mathcal{S}$. +We shall denote the commutant of $\mathcal{S}$ by $\mathcal{S}'$.\nomenclature{$\mathcal{A}'$}{commutant of $\mathcal{A}$} + +\begin{proposition}{}{cyclic-separating} + \begin{enumerate}[label=(\roman*),nosep,leftmargin=*,widest=ii] + \item A vector is cyclic for $\mathcal{A}$ if and only if it is separating for $\mathcal{A}'$. + \item If $\vNa{M}$ is a von Neumann algebra, then a vector is cyclic and separating for $\vNa{M}$ + if and only if it is cyclic and separating for $\vNa{M}'$. + \end{enumerate} +\end{proposition} + +\begin{proof} + \todo{xxx} + The second assertion directly follows from the first and the fact that $\vNa{M}'' = \vNa{M}$. +\end{proof} + +If $\Omega$ is separating for $\mathcal{A}$, +then every element of $\mathcal{A}\Omega$ is of the form $A\Omega$ +with a unique $A \in \mathcal{A}$. +This allows us to define an (anti-linear) operator $S_0$ in $\hilb{H}$ with domain $\mathcal{A}\Omega$ by +\begin{equation} + \label{equation:definition-s0} + \quad S_0 A\Omega \defequal S_0 A^*\Omega \qquad A \in \mathcal{A}. +\end{equation} +The operator $S_0$ is densely defined if and only if $\Omega$ is cyclic for $\mathcal{A}$. +Since the $*$-operation on $\mathcal{A}$ is involutive, +the range of $S_0$ coincides with its domain. + +\begin{lemma}{}{} + If $\Omega$ is a cyclic and separating vector for a von Neumann algebra $\mathcal{A}$, + then the operator $S_0$ defined by~\eqref{equation:definition-s0} is closable. +\end{lemma} +\begin{proof} + By \cref{proposition:cyclic-separating}, + $\Omega$ is also cyclic and separating for the commutant $\vNa{A}'$. + Hence we may, analogously to $S_0$, + define another anti-linear operator $F_0$ in $\hilb{H}$ with dense domain $\mathcal{A}' \Omega$ by + \begin{equation*} + \quad F_0 B\Omega \defequal F_0 B^*\Omega \qquad B \in \mathcal{A'}. + \end{equation*} + By definition of $S_0$ and $F_0$ we have for every $A \in \mathcal{A}$ and $B \in \mathcal{A}'$ + \begin{equation*} + \innerp{S_0 A \Omega}{B \Omega} = + \innerp{\Omega}{AB \Omega} = + \innerp{\Omega}{BA \Omega} = + \innerp{F_0 B\Omega}{A \Omega}. + \end{equation*} + This adjoint identity establishes that $S_0 \subset F_0^*$. + (The \enquote{twisted} appearance of the identity is correct, + since it involves anti-linear operators on both sides.) + The Hilbert adjoint $F_0^*$ of $F_0$ is closed. + Hence, we have shown that $S_0$ has a closed extension, and + this implies that $S_0$ is closable. +\end{proof} + +\begin{definition}{Tomita operator}{} + Suppose $\Omega$ is a cyclic and separating vector for a von Neumann algebra $\mathcal{A}$. + The closure $S = \operatorclosure{S_0}$ + of the operator $S_0$ defined on $\mathcal{A}\Omega$ by + $S_0 A\Omega = S_0 A^*\Omega$ + for $A \in \mathcal{A}$ + is called the + \emph{Tomita operator}\index{Tomita operator}\index{operator!Tomita}\nomenclature{$S$}{Tomita operator} + for the pair $(\mathcal{A},\Omega)$. +\end{definition} + +It is a well-known fact that closed operators can be decomposed +in a similar fashion to the polar coordinate representation $z = e^{i\arg z} \abs{z}$ +of a complex number. +We state the theorem in its somewhat uncommon variant for anti-linear operators, +as this is our only use case. + +\begin{theorem}{Polar Decomposition for Anti-Linear Closed Operators}{polar-decomposition} + \index{polar decomposition} + Let $T$ be an arbitrary closed anti-linear operator in a Hilbert space $\hilb{H}$. + Then there exist + a positive selfadjoint linear operator $\abs{T}$ and + a partial anti-linear isometry $U$ + such that + \begin{equation*} + T = U \abs{T} \qquad \bracks[\big]{\text{in particular, $\Domain{T} = \Domain{\abs{T}}$}}. + \end{equation*} + The operators $U$ and $\abs{T}$ are uniquely determined given the additional conditions + \begin{equation*} + \ker\abs{T} = \ker T \qquad + (\ker U)^\perp = (\ker T)^\perp \qquad + \ran U = \overline{\ran T}. + \end{equation*} +\end{theorem} + +Proofs of this statement are contained in~\cite{ReedSimon1} and~\cite{Schmüdgen2012}. +When we speak of \emph{the} polar composition we tacitly assume that the additional conditions +ensuring uniqueness are satisfied. + +Now we are able to introduce the fundamental objects of modular theory. + +\begin{definition}{Modular Conjugation, Modular Operator}{} + Suppose $\vNa{M}$ is a von Neumann algebra acting on a Hilbert space $\hilb{H}$, + and suppose $\Omega \in \hilb{H}$ is a cyclic and separating vector for $\vNa{M}$. + Let $S$ be the Tomita operator for $(\vNa{M},\Omega)$ and let + \begin{equation*} + S = J \Delta^{1/2} + \end{equation*} + be its polar decomposition. + The anti-unitary operator $J$ is called + \emph{modular conjugation}\index{modular!conjugation}\nomenclature{$J$}{modular conjugation}. + The positive selfadjoint operator $\Delta$ is called + \emph{modular operator}\index{modular!operator}\index{operator!modular}\nomenclature{$\Delta$}{modular operator}. + The pair $(J,\Delta)$ is said to be the \emph{modular data}\index{modular!data}\index{modular!objects} associated to + the pair $(\vNa{M},\Omega)$. +\end{definition} + +\todo{clarify why $J$ is anti-unitary} + +\begin{definition}{Modular Group}{} + Adopt the notation of the foregoing definition. + The mapping $\RR \ni t \mapsto \Delta^{it}$ is called the \emph{modular group}\index{modular!group} associated to + $(\vNa{M},\Omega)$. +\end{definition} + +The modular group is a strongly continuous one-parameter unitary group on $\hilb{H}$. + +\newpage + +\begin{proposition}{}{modular-data-unitary} + Suppose $\vNa{M}$ is a von Neumann algebra acting on a Hilbert space $\hilb{H}$. + Let $U$ be a unitary operator on $\hilb{H}$. + Then $U\vNa{M}U^*$ is a von Neumann algebra on $\hilb{H}$. + Suppose further that $\Omega \in \hilb{H}$ is a cyclic and separating vector for $\vNa{M}$. + Then $U \Omega$ is cyclic and separating for $U\vNa{M}U^*$. + Let $(J,\Delta)$ be the modular data associated to $(\vNa{M},\Omega)$. + Then $(UJU^*,U{\Delta}U^*)$ is the modular data associated to $(U\vNa{M}U^*,U\Omega)$. +\end{proposition} + +\begin{proof} + To prove the first assertion, + consider any $A \in (U\vNa{M}U^*)''$. + By the double commutant theorem, + it suffices to show that $A \in U\vNa{M}U^*$. + As $\vNa{M}$ is a von Neumann algebra, + this is equivalent to $U^*\! AU \in \vNa{M}''$, + again by the double commutant theorem. + Let $B \in \vNa{M}'$. + It is easy to check that $UBU^* \in (U\vNa{M}U^*)'$. + By assumption, $A$ lies in the commutant of $(U\vNa{M}U^*)'$. + Thus we find that $[U^*\! AU,B] = U^* [A,UBU^*] U = 0$, as desired. + + The set of vectors $U\vNa{M}U^* U\Omega = U\vNa{M}\Omega$ is dense in $\hilb{H}$, + since it is the image of $\vNa{M} \Omega$ under the homeomorphism $U$. + Thus, the vector $U\Omega$ is cyclic for $U\vNa{M}U^*$. + Let us show that it is also separating. + Suppose $A$ is in $\vNa{M}$ and $UAU^*U\Omega = UA\Omega = 0$ + Since unitaries are injective, $A\Omega = 0$. + Now $A=0$ follows from the assumption that $\Omega$ is separating for $\vNa{M}$. + We have shown that the mapping $UAU^*U\Omega = UA\Omega$ from $U\vNa{M}U^* \to \hilb{H}$ is injective. + + Let $S = \overline{S_0}$ be the Tomita operator associated to $(\vNa{M},\Omega)$, + and let $S' = \overline{S'_0}$ be the Tomita operator associated to $(U\vNa{M}U^*,U\Omega)$. + Then we have + \begin{equation*} + (S'_0 U) A \Omega = + S'_0 (U A U^*) U \Omega = + (U A^* U^*) U \Omega = + U A^* \Omega = + U S_0 A \Omega + \end{equation*} + for all $A \in \vNa{M}$. Consequently, $S'_0 = U S_0 U^*$ as operators with domain $U\vNa{M}\Omega$. + Taking the closure, we obtain $S' = U S U^*$. + We can write this as $S' = UJU^* U\Delta^{1/2} U^*$, + where $S = J \Delta^{1/2}$ is the polar decomposition of the Tomita operator. + It is straightforward to check that + $UJU^*$ is anti-unitary and $U\Delta^{1/2} U^*$ is positive selfadjoint, + and satisfy the additional condition of \cref{theorem:polar-decomposition}. + The uniqueness of the polar decomposition implies that + $UJU^*$ is the modular conjugation and $U\Delta U^*$ is the modular conjugation + associated to the pair $(U\vNa{M}U^*,U\Omega)$. +\end{proof} + +\newpage + +Finally, let us outline how modular theory enters into algebraic quantum field theory. + +\begin{theorem}{Reeh-Schlieder Theorem}{reeh-schlieder} + \todo{spell it out} +\end{theorem} + +By Reeh-Schlieder (\cref{theorem:reeh-schlieder}), the vacuum $\Omega$ is cyclic and separating for $\localalg{\spacetimeregion{O}}$. +Thus, modular theory + + +\section{The Geometric Action of the Modular Operator Associated With a Wedge Domain} + +\begin{definition}{Right and Left Wedge, General Wedges}{} + The \emph{right wedge}\index{wedge!right}\nomenclature[WR]{$\rightwedge$}{right wedge} + and \emph{left wedge}\index{wedge!left}\nomenclature[WL]{$\leftwedge$}{left wedge} + in Minkowski space $M$ are the open subsets + \begin{equation*} + \rightwedge \defequal \braces[\big]{x \in M \vcentcolon x^1 > \abs{x^0}} + \quad \text{and} \quad + \leftwedge \defequal \braces[\big]{x \in M \vcentcolon x^1 < -\abs{x^0}}. + \end{equation*} +We say that a spacetime region $W \subset M$ is a \emph{wedge}\index{wedge} + if there exists an element $g$ of the Poincaré group + such that $W = g \rightwedge$. +\end{definition} + +Instead of the right wedge, +we could just as well have used the left wedge to define the notion of a general wedge, +since they are transformed into each other by space inversion. + +\begin{lemma}{}{general-wedge-from-right-wedge} + If a spacetime region $W$ is a wedge, + then there exists an element $g$ of the proper orthochronous Poincaré group + such that $W = g \rightwedge$. +\end{lemma} + +\begin{proof} + \todo{xxx} +\end{proof} + +In the standard representation of the Lorentz group, the boost (or velocity transformation) along the $x^1$-axis +with rapidity $2 \pi t$ is given by the matrix\footnote{ + This matrix depends on the choice of metric signature. + Ours is $(+,-,-,-)$. + For $(-,+,+,+)$, use + \begin{equation*} + \Lambda(t) = \begin{pmatrix} + \phantom{-}\cosh(2 \pi @ t) & -\sinh(2 \pi @ t) & \; 0 \; & \; 0 \; \\ + -\sinh(2 \pi @ t) & \phantom{-}\cosh(2 \pi @ t) & 0 & 0 \\ + 0 & 0 & 1 & 0 \\ + 0 & 0 & 0 & 1 \\ + \end{pmatrix}. + \end{equation*} + } + +\begin{equation*} + \Lambda(t) = \begin{pmatrix} + \cosh(2 \pi @ t) & \sinh(2 \pi @ t) & \; 0 \; & \; 0 \; \\ + \sinh(2 \pi @ t) & \cosh(2 \pi @ t) & 0 & 0 \\ + 0 & 0 & 1 & 0 \\ + 0 & 0 & 0 & 1 \\ + \end{pmatrix} +\end{equation*} + +The following proposition shows that $t \mapsto \Lambda(t)$ is +a one-parameter subgroup of the stabilizer group of the right wedge +with respect to the action of the Lorentz group on subsets of Minkowski space. + +\begin{proposition}{}{} + \begin{enumerate}[label=(\roman*),nosep,leftmargin=*,widest=ii] + \item $\Lambda(s + t) = \Lambda(s) \Lambda(t)$ for all $s,t \in \RR$. + \item $\Lambda(t) \rightwedge = \rightwedge$ for all $t \in \RR$. + \end{enumerate} +\end{proposition} + +\begin{proof} + The first property can be verified by direct computation. + Let us prove the second. + By definition, the image $\Lambda(t) x$ of a vector $x \in M$ lies in $\rightwedge$ if and only if + \begin{equation*} + x^0 \sinh(2 \pi t) + x^1 \cosh(2 \pi t) + > \abs{x^0 \cosh(2 \pi t) + x^1 \sinh(2 \pi t)}, + \end{equation*} + or equivalently + \begin{equation*} + x^0 \parens[\big]{\sinh(2 \pi t) \mp \cosh(2 \pi t)} + + x^1 \parens[\big]{\cosh(2 \pi t) \mp \sinh(2 \pi t)} > 0 + \end{equation*} + for both sign choices. + Using the definitions of the hyperbolic sine and cosine, this may be further simplified to $(x^1 \mp x^0) e^{2 \pi t} > 0$, + which holds if and only if $x^1 > \abs{x^0}$, + since the exponential is always positive. + So we have shown that + \begin{equation} + \label{equation:image-right-wedge} + \Lambda(t) x \in \rightwedge \iff x \in \rightwedge. + \end{equation} + This implies that $\Lambda(t)\rightwedge \subset \rightwedge$ for all $t \in \RR$. + Conversely, given an arbitrary vector $y \in \rightwedge$, + we have to find $x \in \rightwedge$ such that $\Lambda(t) x = y$. + Consider $x = \Lambda(-t) y$. Clearly, $x \in \rightwedge$, because of $y \in \rightwedge$ + and~\eqref{equation:image-right-wedge}. Now it follows from $\Lambda(-t) = \Lambda(t)^{-1}$ that in fact $\Lambda(t) x = y$. +\end{proof} + +\begin{theorem}{Bisognano-Wichmann Theorem \textmd{\cite{Bisognano1975}}}{} + For the theory of a free scalar field in Minkowski spacetime, + let $\spacetimeregion{O} \mapsto \localalg{\spacetimeregion{O}}$ be the net of von Neumann algebras of local observables. + If $(J,\Delta)$ is the modular data associated to the algebra $\localalg{\rightwedge}$ of the right wedge and the vacuum $\Omega$, then + \begin{equation*} + J = \Theta \cdot U\parens[\big]{0, R_{23}(\pi)} \qquad + \Delta^{it} = U\parens[\big]{0,\Lambda(t)}, + \end{equation*} + where $U$ is the theory's unitary representation of the proper orthochronous Poincaré group. +\end{theorem} + +\todo{give definition of $\Theta$ and $R_{23}$} + +Note that above statement is for the right wedge only. +Let us investigate how the modular group changes, if we consider another wedge region. +By \cref{lemma:general-wedge-from-right-wedge} any wedge $W$ can be obtained as $W = g\rightwedge$, where $g$ is a proper orthochronous Poincaré transformation. +The covariance property~\eqref{equation:poincare-covariance-local-algebras} of $\vNa{R}$ implies +\begin{equation*} + \localalg{W} = + U(g) \localalg{\rightwedge} U(g)^*. +\end{equation*} +The vacuum $\Omega$ is Poincaré invariant: +\begin{equation*} + U(g) \Omega = \Omega. +\end{equation*} +We write $(J_W,\Delta_W)$ for the modular data associated to $(\localalg{W},\Omega)$. +By \cref{proposition:modular-data-unitary} +\begin{equation*} + J_W = U(g) J U(g)^* \qquad + \Delta_W = U(g) \Delta U(g)^* +\end{equation*} +Recall that the modular group $\Delta_W^{it}$ is defined by means of functional calculus. +This raises the following problem: given a selfadjoint operator $A$, a unitary operator $U$ +and a suitable function $f$ we want to express $f(UAU^*)$ in terms of $f(A)$, if possible. +Note that two different functional calculi are at play here, the former is for $UAU^*$ and the latter for $A$. +Simple functions such as polynomials suggest $f(UAU^*) = Uf(A)U^*$. +That this is generally true is the statement of the following Lemma. + + +\begin{lemma}{}{functional-calclus-unitary-trafo} + Suppose that $A$ is a selfadjoint operator on a Hilbert space $\hilb{H}$, + with spectral measure $E_A$. + Suppose $U$ is an unitary operator on $\hilb{H}$, and + let $E_{U\! @AU^*}$ denote the spectral measure of the (selfadjoint) operator $UAU^*$. + Then we have $U E_A U^* = E_{U\! @AU^*}$, and + \begin{equation*} + U f(A) U^* = f(U\! @@AU^*) + \end{equation*} + for all Borel functions $f : \RR \to \CC$. +\end{lemma} + +\question{Ist diese Aussage korrekt? Ist mein Beweis richtig? Geht der auch einfacher?} + +\begin{proof} + For each regular value $\lambda \in \rho(A)$ let + \begin{equation*} + R_A(\lambda) = (A-\lambda)^{-1} + \end{equation*} + denote the resolvent operator of $A$. + This proof is based on Stone's Formula \todo{reference}, + which relates the resolvent to the spectral projections of $A$: + If $E_A$ is the spectral measure of $A$ and $\alpha < \beta$ are real numbers, then + \begin{equation*} + \stronglim_{\varepsilon \downarrow 0} + \frac{1}{\pi i} \int_{\alpha}^{\beta} \bracks{R_A(\lambda + i \varepsilon) - R_A(\lambda - i \varepsilon)} d\lambda + = E_A \parens[\big]{\bracks{\alpha,\beta}} + E_A \parens[\big]{\parens{\alpha,\beta}} + \end{equation*} + Recall that a spectral measure is countably additive. + As a consequence, + \begin{equation*} + \stronglim_{\alpha \uparrow a} + \stronglim_{\beta \downarrow b} + \stronglim_{\varepsilon \downarrow 0} + \frac{1}{2\pi i} \int_{\alpha}^{\beta} \bracks{R_A(\lambda + i \varepsilon) - R_A(\lambda - i \varepsilon)} d\lambda + = E_A \parens[\big]{\bracks{a,b}} + \end{equation*} + for all $a \in \RR \cup \braces{-\infty}$, $b \in \RR \cup \braces{\infty}$. + Observe that $\rho(A) = \rho(U\! @AU^*)$ and that for each (common) regular value $\lambda$ we have + \begin{equation*} + R_{U\! @AU^*}(\lambda) = U R_A(\lambda) @ U^*\!. + \end{equation*} + Since conjugation with an unitary commutes with the strong operator limit, we obtain + \begin{equation*} + E_{U\! @AU^*} \parens[\big]{\bracks{a,b}} + = U E_A \parens[\big]{\bracks{a,b}} U^* + \end{equation*} + for all $a,b \in \RR$. + The collection $\mathcal{A}$ of all subsets $S$ of $\RR$ such that + $E_{U\! @AU^*} \parens[\big]{S} = U E_A \parens[\big]{S} U^*$ + is a $\sigma$-algebra on $\RR$. + We have shown that all closed intervals belong to $\mathcal{A}$. + It is well known that the Borel-$\sigma$-algebra $\mathcal{B}$ of $\RR$ + is generated by the closed intervals. Hence, $\mathcal{B} \subset \mathcal{A}$. + This shows that the spectral measures $U E_A U^*$ and $E_{U\! @AU^*}$ coincide. +\end{proof} + +\begin{equation*} + \Delta_W^{it} + = U(g) \Delta^{it} U(g)^* + = U(g) U\parens[\big]{0,\Lambda(t)} U(g)^* + = U\parens[\big]{g(0,\Lambda(t))g^{-1}} +\end{equation*} + + +Recall that Stones Theorem \todo{add reference} states that +every strongly continuous one-parameter unitary group +is of the form $t \mapsto e^{itK}$ with a uniquely determined +selfadjoint operator $K$, which is called \emph{infinitesimal generator} of the group. + +\begin{definition}{Modular Hamiltonian}{} + The infinitesimal generator of the modular group associated to a spacetime region $\spacetimeregion{O}$ is called the + \emph{modular Hamiltonian}\index{modular!Hamiltonian}\nomenclature{$K_{\spacetimeregion{O}}$}{modular Hamiltonian for $\spacetimeregion{O}$} + for said region, and denoted $K_{\spacetimeregion{O}}$. +\end{definition} + +In other words, $K_{\spacetimeregion{O}}$ is the unique selfadjoint operator such that $\Delta_{\spacetimeregion{O}}^{it} = e^{itK_{\spacetimeregion{O}}}$ for all $t \in \RR$. + +\begin{proposition}{}{} + The modular Hamiltonian for the right wedge is given by $d \Gamma(A)$, where + \begin{equation*} + A\psi(p) = - \frac{2\pi}{i} \parens[\big]{\partial_0 \psi(p) \, p^1 + \partial_1 \psi(p) \, p^0} + \end{equation*} +\end{proposition} + +\section{Complex Lorentz Transformations} + +\subsection{Analytic Continuation of the Space-Time Translation Group} + +\subsection{Complex Lorentz Boosts} + +\begin{lemma}{}{} + Suppose $A$ is a selfadjoint operator on some Hilbert space $\hilb{H}$. + For all complex numbers $z$ define a closed normal operator $V(z) = e^{izA}$ by means of functional calculus. + Let $g$ be a xxx function. Then the range of the bounded operator $g(A)$ is contained in the domain of $V(z)$ for all $z$, and + \begin{equation*} + V(z) g(A) = \int e^{iz \lambda} g(\lambda) dE_A(\lambda). + \end{equation*} +\end{lemma} + +\subsection{A Convolution Theorem for Vector-Valued Tempered Distributions} + +\blockcquote{Bisognano1975}{% + The extension to vector-valued tempered distributions is trivial. +} + + + +\chapterbib +\cleardoublepage + +% vim: syntax=mytex diff --git a/preamble.tex b/preamble.tex index 4a88ef7..35d94ad 100644 --- a/preamble.tex +++ b/preamble.tex @@ -15,7 +15,7 @@ %\usepackage{graphicx} \usepackage{tcolorbox} \usepackage[style=ext-alphabetic]{biblatex} -\usepackage[intoc]{nomencl} +\usepackage[intoc,refpage]{nomencl} \usepackage{makeidx} \usepackage{idxlayout} \usepackage{hyperref} @@ -82,6 +82,7 @@ \numberwithin{equation}{chapter} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\dom}{dom} +\DeclareMathOperator{\ran}{ran} % extend amsmath's proof environment \NewDocumentEnvironment{myproof}{Ob}{\IfNoValueTF{#1}{\begin{proof}}{\begin{proof}[\proofname\ of \Cref{#1}]}}{\end{proof}} @@ -177,7 +178,8 @@ % ---------- nomencl \makenomenclature \renewcommand*{\nomname}{List of Symbols} -\def\pagedeclaration#1{, \hyperlink{page.#1}{page\nobreakspace#1}} +%\def\pagedeclaration#1{, \hyperlink{page.#1}{page\nobreakspace#1}} +\def\pagedeclaration#1{, \hyperlink{page.#1}{#1}} % ---------- makeidx \makeindex @@ -250,6 +252,8 @@ % Fourier transformation % ---------------------- \newcommand*{\ft}[1]{\hat{#1}} +\newcommand*{\wideft}[1]{\widehat{#1}} +\newcommand*{\ift}[1]{\check{#1}} \newcommand*{\FT}[1]{\mathcal{F}\parens*{#1}} \newcommand*{\iFT}[1]{\mathcal{F}^{-1}\parens*{#1}} @@ -269,6 +273,7 @@ \newcommand*{\schwartz}[1]{\mathcal{S}(#1)} \newcommand*{\realschwartz}[1]{\mathcal{S}_{\RR}(#1)} \newcommand*{\tempdistrib}[1]{\mathcal{S}'(#1)} +\newcommand*{\tempdistribnoarg}{\mathcal{S}'} % Fock spaces % ----------- @@ -289,6 +294,8 @@ % --------- \newcommand*{\QF}[1]{QF(#1)} \newcommand{\QFequal}{\overset{\text{\scriptsize QF}}{=}} +% operator associated to a quadratic form +\newcommand*{\QFop}[1]{{#1}_{\mathrm{op}}} % Standard Subspaces % ------------------ @@ -311,3 +318,29 @@ }} \newcommand{\defequal}{\overset{\text{\scriptsize def}}{=}} + +\newcommand*{\energydensity}{\varrho} +\newcommand*{\fockvaccum}{\Omega} + +% Observable Algebras +\newcommand*{\vNa}[1]{\mathcal{#1}} +\newcommand*{\localalg}[1]{\vNa{R}(#1)} + + +\newcommand*{\FullLorentzGroup}{\mathcal{L}} +\newcommand*{\ProperOrthochronousLorentzGroup}{\FullLorentzGroup_{+}^{\uparrow}} +\newcommand*{\FullPoincareGroup}{\mathcal{P}} +\newcommand*{\ProperOrthochronousPoincareGroup}{\FullPoincareGroup_{+}^{\uparrow}} + +% spacetime domains +\newcommand*{\spacetimeregion}[1]{\mathcal{#1}} +\newcommand*{\rightwedge}{W_{\! R}} +\newcommand*{\leftwedge}{W_{\! L}} + +\newcommand*{\todo}[1]{{\color{blue}TODO: #1}} +\newcommand*{\question}[1]{{\color{blue}Question: #1}} +\newcommand*{\info}[1]{{\color{blue}Info: #1}} + +\newcommand*{\operatorclosure}[1]{\overline{#1}} + +\DeclareMathOperator*{\stronglim}{s-lim} diff --git a/stresstensor.tex b/stresstensor.tex index a4bb6fb..4c128c2 100644 --- a/stresstensor.tex +++ b/stresstensor.tex @@ -35,9 +35,13 @@ as a service to the reader. \item Given a complex-valued function $f$ on $M$, we define its \emph{Fourier transform} $\ft{f}\,$ by \begin{equation} \label{fourier-transform} - \hat{f}(p) = \frac{1}{(2 \pi)^2} \int_{M} e^{i p \cdot x} f(x) \, dx + \ft{f}(p) \defequal \int_{M} e^{i p \cdot x} f(x) \, dx \end{equation} - whenever the integral converges. The \emph{inverse Fourier transform} is TODO + whenever the integral converges. The \emph{inverse Fourier transform} is defined by + \begin{equation*} + \label{inverse-fourier-transform} + \ift{f}(p) \defequal \frac{1}{(2 \pi)^2} \int_{M} e^{-i p \cdot x} f(x) \, dx. + \end{equation*} \item To a mathematician $\overline{\phantom{z}}$ usually means complex conjugation and ${}^*$ indicates the Hilbert adjoint of an operator, while a physicist may read ${}^*$ as complex conjugation and denotes the Hilbert adjoint with ${}^{\dagger}$. @@ -85,9 +89,15 @@ as a service to the reader. \begin{equation*} E : \schwartz{M} \to \hilb{H}, \quad f \mapsto Ef = \left.\ft{f}\,\right\vert {X_m^+} \end{equation*} + We define a $\RR$-linear mapping $\phi$ by + \begin{equation*} + \realschwartz{M} \ni f \mapsto \varphi(f) = \Phi_{\mathrm{S}}(Ef) = \frac{1}{\sqrt{2}} \parens*{a(Ef) + a(Ef)^\dagger} + \end{equation*} + This extedns to complex valued test functions $f \in \schwartz{M}$ \begin{equation*} - \realschwartz{M} \ni f \mapsto \Phi(f) = \Phi_{\mathrm{S}}(Ef) = \frac{1}{\sqrt{2}} \parens*{a(Ef) + a(Ef)^\dagger} + \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(E\bar{f}) + a(Ef)^\dagger} \end{equation*} + called the \emph{massive free scalar quantum field} \item annihilation and creation operators, $f \in \schwartz{M}$, $\psi \in \BosonFock{\hilb{H}}$ \begin{align*} @@ -176,7 +186,7 @@ leads to \parens[\big]{a(p)^\dagger \psi} {}_n (k_1, \ldots, k_n) = \frac{1}{\sqrt{n}} \sum_{i=1}^n \delta(p - k_i) \, \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n), \end{equation} -where the symmetization is necessary to obtain an expression that +where the symmetrization is necessary to obtain an expression that at least has a chance of being a $n$ Boson state. However, it clearly is not a $L^2$ function. Given any state $\psi'$, we can @@ -207,7 +217,7 @@ For completeness, we give a precise definition of quadratic form. q : D(q) \times D(q) \to \CC, \end{equation*} where $D(q)$ is a linear subspace of $\hilb{H}$, called the \emph{form domain}\index{form domain}\index{quadratic form!domain of a}, - such that $q$ is conjugate linear in its first agrument + such that $q$ is conjugate linear in its first argument and linear in its second argument (i.e.\ sesquilinear). We say that $q$ is \emph{densely defined} if $D(q)$ is dense in $\hilb{H}$. @@ -221,13 +231,13 @@ but one may obtain an operator with trivial domain. \begin{definition}{Operator Associated to a Quadratic Form}{} Suppose $q$ is a densely defined quadratic form on a complex Hilbert space $\hilb{H}$. - The linear \emph{operator associated to}\index{quadratic form!operator associated to a} $q$, denoted $q_{\mathrm{op}}$, + The linear \emph{operator associated to}\index{quadratic form!operator associated to a} $q$, denoted $\QFop{q}$, is defined on the domain \begin{equation*} - D(q_{\mathrm{op}}) = \braces{\psi \in D(q) \mid \text{the map $q(\cdot,\psi) : D(q) \to \CC$ is bounded}}, + D(\QFop{q}) = \braces{\psi \in D(q) \mid \text{the map $q(\cdot,\psi) \vcentcolon D(q) \to \CC$ is bounded}}, \end{equation*} - and maps $\psi \in D(q_{\mathrm{op}})$ to the vector $q_{\mathrm{op}}\psi$ in $\hilb{H}$ satisfying - $q(\psi',\psi) = \innerp{\psi'}{q_{\mathrm{op}}\psi}$, + and maps $\psi \in D(\QFop{q})$ to the vector $\QFop{q}\psi$ in $\hilb{H}$ satisfying + $q(\psi'\!,\psi) = \innerp{\psi'\!}{\QFop{q}\psi}$, which exists and is unique by Riesz’s Representation Theorem. \end{definition} @@ -299,7 +309,7 @@ the $\alpha^{(0)},\alpha^{(1)}_i,\alpha^{(2)}_{j,k},\ldots$ are complex numbers, of which only finitely many are nonzero, and $e$ is a special object representing an empty product of $z$'s. To make this mathematically precise: -we are speaking of the noncommutative associative algebra over $\CC$ +we are speaking of the non-commutative associative algebra over $\CC$ freely generated by the elements of $\hilb{H}$. The unit of the algebra is $e$. @@ -314,7 +324,7 @@ where $z,z' \in \hilb{H}$. \begin{definition}{Infinitesimal Weyl Algebra}{} Let $\hilb{H}$ be a complex Hilbert space. The \emph{infinitesimal Weyl algebra}\index{infinitesimal Weyl algebra} $\WeylAlg(\hilb{H})$ over $\hilb{H}$ - is the noncommutative associative algebra over $\CC$ + is the non-commutative associative algebra over $\CC$ generated by the elements of $\hilb{H}$, with the relations \begin{equation*} zz' - z'z = i \Imag \innerp{z}{z'} \, e \qquad z,z' \in \hilb{H}, @@ -371,10 +381,10 @@ the formula makes sense even for $r=0$ and asserts that $\normord{e} = e$. The cases $r=1$ and $r=2$ read \begin{align*} \normord{z} &= - \frac{1}{\sqrt{2}} \parens[\big]{\weylannihilator(z) + \weylcreator(z)} = z \\ + \frac{1}{\sqrt{2}} \parens[\big]{\weylannihilator(z) + \weylcreator(z)} = z, \\ \normord{z_1 z_2} &= \frac{1}{2} - \parens[\big]{\weylannihilator(z_1) \weylannihilator(z_2) + \weylannihilator(z_1) \weylcreator(z_2) - + \weylcreator(z_1) \weylannihilator(z_2) + \weylcreator(z_1) \weylcreator(z_2) } + \parens[\big]{ \weylannihilator(z_1) \weylannihilator(z_2) + \weylcreator(z_1) \weylannihilator(z_2) + + \weylcreator(z_2) \weylannihilator(z_1) + \weylcreator(z_1) \weylcreator(z_2) }. \end{align*} This suggests that the normally ordered product $\normord{z_1 \!\cdots z_r}$ is symmetric in $z_1,\ldots,z_n$. This is in fact true, and becomes evident @@ -470,20 +480,19 @@ and may be obtained via integration by parts. Naturally, we now define the \emph{distributional derivative} of the field by \begin{equation*} - D \varphi(f) = \varphi(D^{\dagger} f) \qquad \forall f \in \schwartz{\RR^d} + D \varphi(f) = \varphi(D^{\dagger} f) \qquad \forall f \in \schwartz{\RR^4} \end{equation*} -As one expects, $D\varphi$ is an operator-valued tempered distribution on $M=\RR^d$. -TODO +As one expects, $D\varphi$ is an operator-valued tempered distribution on $M=\RR^4$. +In terms of creation and annihilation operators we have \begin{equation} \label{derivative-free-field} - D \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(ED^{\dagger}f)^{\dagger} + a(ED^{\dagger}f)} + D \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(ED^{\dagger}f)^{\dagger} + a(E\overline{D^{\dagger}f})}. \end{equation} - - -The operator corresponding to $D$ in Fourier space is the multiplication operator +In Fourier space the operator $D^\dagger$ corresponds to muliplication with the polynomial \begin{equation*} - -i \sum_{\alpha} a_{\alpha} p_0^{\alpha_0} (-p_1)^{\alpha_1} (-p_2)^{\alpha_2} (-p_3)^{\alpha_3} + \ft{D}(p) \defequal \sum_{\alpha} i^{\abs{\alpha}} a_{\alpha} (+p^0)^{\alpha_0} (-p^1)^{\alpha_1} (-p^2)^{\alpha_2} (-p^3)^{\alpha_3} \end{equation*} +If $D=\partial^{\mu}$, then $\ft{D}(p) = i @ p_{\!\mu}$, were the potential sign is concealed by lowering the index. Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients. @@ -494,8 +503,8 @@ Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) \sum_{\sigma \in S_r} \sum_{s=0\vphantom{S}}^{r} \frac{1}{s!(r-s)!} - \prod_{i=1\vphantom{S}}^{s} a^\dagger(D^\dagger_{\sigma(i)}f) - \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} a(D^\dagger_{\sigma(j)}f) + \prod_{i=1\vphantom{S}}^{s} a^\dagger(ED^\dagger_{\sigma(i)}f) + \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} a(E\overline{D^\dagger_{\sigma(j)}f}) \end{gather} \section{Renormalized Products of the Free Field and~its~Derivatives} @@ -516,7 +525,7 @@ this approach incurs significant technical difficulties. \begin{lemma}{Integral Representation of the Renormalized Product}{renormalized-product-integral-representation} Let $\varphi$ be the free scalar quantum field with mass parameter $m > 0$. Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients. - Then, for arbitrary Schwartz functions $f \in \schwartz{M}$ and Fock states $\psi,\psi' \in \BosonFock{L^2(X_m^+,d\Omega_m)}$, + Then, for arbitrary Schwartz functions $f \in \realschwartz{M}$ and Fock states $\psi,\psi' \in \BosonFock{L^2(X_m^+,d\Omega_m)}$, we have \begin{equation*} \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} = @@ -552,7 +561,32 @@ this approach incurs significant technical difficulties. \end{multline*} \end{lemma} -TODO(Note about the remaining dependence of $K$ on $f$.) +Note that $K$ has a remaining dependence on $f$ via $\chi$ +even thogh the notation does not indicate this. +This is made explicit in the alternative integral representation + \begin{equation} + \label{equation:alternative-integral-representation} + \begin{multlined} + \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} =\\ + \hspace{1cm} \int dp_1 \!\cdots dp_r + \sum_{s=0}^{r} + \, \ft{f}(p_1) \cdots\! \ft{f}(p_s) + \, \overline{\ft{f}(p_{s+1}) \cdots\! \ft{f}(p_r)} + \, \tilde{K}^s_{\psi'\!,\psi}(p_1,\ldots,p_r) + \end{multlined} + \end{equation} + where + \begin{multline*} + \tilde{K}^s_{\psi'\!,\psi}(p_1,\ldots,p_r) = + P_s(p_1,\ldots,p_r) + \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \delta_{m-s}^{n-(r-s)} \\ + \cdot \sqrt{m(m-1) \cdots (m-s+1)} \sqrt{n(n-1) \cdots (n-(r-s)+1)} \\ + \cdot \int dk_1 \cdots dk_{m-s} + \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)} + \ \psi_n(k_1,\ldots,k_{n-(r-s)},p_{s+1},\ldots,p_r). + \end{multline*} + This will be more convenient for xxx \begin{myproof}[lemma:renormalized-product-integral-representation] From equation~\eqref{equation:renormalized-product}, @@ -617,24 +651,30 @@ In particular, for squares ($r=2$) we have The following assertion is key to realizing the idea of taking the limit $f \to \delta_x$. -\begin{lemma}{}{integral-kernel-h-bound} +\begin{lemma}{H-bounds for the Integral Kernel}{integral-kernel-h-bound} In the setting of \cref{lemma:renormalized-product-integral-representation}, there exist a constant $C$, and a positive integer $l$, - such that for arbitrary states $\psi,\psi' \in \BosonFock{\hilb{H}}$, - and test functions $f \in \schwartz{M}$, + such that for arbitrary test functions $f \in \schwartz{M}$ + and states $\psi,\psi' \in \Domain{H^l}$, the function $K_{\psi'\!,\psi}$ is integrable (that is, $L^1$) and satisfies the $H$-bound \begin{equation*} \norm{K_{\psi'\!,\psi}}_1 \le C \norm{(1+H)^l \psi'} \norm{(1+H)^l \psi}. \end{equation*} + More specifically, it is sufficient to choose $l > rd + r/2$, + where $d$ is the highest order of differentiation occuring in $D_1, \ldots, D_r$. \end{lemma} -The Hamilton operator $H$ acts on $n$-particle states $\psi_n$ -by multiplication with $\omega(p_1)$ -In the following proof it will we convenient to use the abbreviation +The Hamilton operator $H$ acts on $n$-particle states $\psi_n$ as follows: +\begin{gather*} + H \psi_n(p_1,\ldots,p_n) = \parens[\big]{\omega(p_1) + \cdots + \omega(p_n)} \psi_n(p_1,\ldots,p_n) \\ + \shortintertext{where} + \omega(p) = \omega(\symbfit{p}) = \sqrt{m^2 + \abs{\symbfit{p}}^2} = \sqrt{m^2 + (p^1)^2 + (p^2)^2 + (p^3)^2}. +\end{gather*} +In the following proof it will be convenient to use the abbreviation \begin{equation*} - \omega(p_1,\ldots,p_s) = \omega(p_1) + \cdots + \omega(p_s) + \omega(p_1,\ldots,p_s) \defequal \omega(p_1) + \cdots + \omega(p_n). \end{equation*} \begin{myproof}[lemma:integral-kernel-h-bound] @@ -778,28 +818,71 @@ In the following proof it will we convenient to use the abbreviation \le \frac{\omega(p_1) + \cdots + \omega(p_s)}{s} \le \omega(p') \le 1 + \omega(k,p'), \nonumber\\ \shortintertext{hence} - \label{equation:one-plus-omega-estimate} + \label{equation:one-plus-omega-estimate1} \parens[\big]{1+\omega(k,p')} {}^{-a} - \le \parens[\big]{\omega(p_1) \cdots \omega(p_s)} {}^{-a/s}. + \le \parens[\big]{\omega(p_1) \cdots \omega(p_s)} {}^{-a/s}, \\ + \shortintertext{and similarly} + \label{equation:one-plus-omega-estimate2} + \parens[\big]{1+\omega(k,p)} {}^{-a} + \le \parens[\big]{\omega(p_1) \cdots \omega(p_{r-s})} {}^{-a/(r-s)}. \end{gather} - The estimates~\eqref{equation:polynomial-estimate} and~\eqref{equation:one-plus-omega-estimate} entail + The estimates~\eqref{equation:polynomial-estimate},~\eqref{equation:one-plus-omega-estimate1} and~\eqref{equation:one-plus-omega-estimate2} entail \begin{equation*} \abs{F(k,p',p)} \le C_s \prod_{i=1}^{s} \omega(p_i)^{d_i-a/s} - \prod_{j=s+1}^{r} \omega(p_j)^{d_j-a/(r-s)} + \prod_{j=s+1}^{r} \omega(p_j)^{d_j-a/(r-s)}. \end{equation*} + Since the right hand side does not depend on $k$, and the $p$-variables are separated, + the problem reduces to proving that + \begin{equation} + \label{equation:integral-finite} + \int \omega(q)^{-2b} \,d\Omega(q) < \infty + \end{equation} + for $b$ large enough. + Recall that $d \Omega(q) = \omega(q)^{-1} d^3 \symbfit{q}$. + By transformation to spherical coordinates, we find that~\eqref{equation:integral-finite} + is equivalent to + \begin{equation*} + \int \frac{r^2}{(m^2 + r^2)^{b+1/2}} \,dr < \infty + \end{equation*} + It it well known that this holds for $b > 1$. + $a > r d$ + + \begin{equation} + \label{equation:intermediate-result} + \norm{K_{\psi'\!,\psi}}_1 \le + \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \sum_{s=0}^{r} \delta_{m-s}^{n-(r-s)} C_s + \underbracket{\norm{(1+H)^l \psi'_m}_2}_{a'_m} + \underbracket{\norm{(1+H)^l \psi_n}_2}_{a_n} + \end{equation} + We introduce auxiliary variables $a'_m, a_n$ as shown above, and for convenience set $a_n = 0$ whenever $n < 0$. + Using this, we rewrite the right hand side of~\eqref{equation:intermediate-result} + and apply the Cauchy-Schwarz Inequality for sequences as follows: + \begin{equation*} + \sum_{s=0}^{r} C_s \sum_{m = 0}^{\infty} a'_m \cdot a_{m+r-2s} \le + \sum_{s=0}^{r} C_s \sqrt{\sum_{m=0}^{\infty} a'^2_m \sum_{n=0}^{\infty} a^2_n} + \end{equation*} + To complete the proof, observe that + \begin{equation*} + \norm{(1+H)^l \psi'}_2 = + \sqrt{\sum_{m=0}^{\infty} \norm{(1+H)^l \psi'_m}_2^2} = + \sqrt{\sum_{m=0}^{\infty} a'^2_m}, + \end{equation*} + and similar for $\psi$, by definition of the inner prouct + and because $((1+H)^l \psi')_m = (1+H)^l \psi'_m$ for all $m$. \end{myproof} \begin{lemma}{Renormalized Product at a Point}{} In the setting of \cref{lemma:renormalized-product-integral-representation}, - assume that $\psi,\psi'$ are in $D^l(H)$. + assume that $\psi,\psi'$ are in $\Domain{H^l}$. Let $x$ be any point in $M$ and let $\delta_x \in \tempdistrib{M}$ be the Dirac distribution supported in $x$. Then the limit \begin{equation*} \lim_{f \to \delta_x} \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} \end{equation*} - exists and depends continously on $x$. + exists and depends continuously on $x$. \end{lemma} \begin{proof} @@ -813,8 +896,22 @@ In the following proof it will we convenient to use the abbreviation The integrand is dominated by the function $\abs{K_{\psi'\!,\psi}(p_1,\ldots,p_r)}$, which has finite integral as it is $L^1$ by \cref{lemma:integral-kernel-h-bound}. + Moreover, the integrand converges pointwise to $K_{\psi'\!,\psi}(p_1,\ldots,p_r)$, since $\ft{f} \to 1$ when $f \to \delta_x$. TODO(With of choice of FT constants, $\ft{f} \to 1/(2\pi)^2$. Change here or change def?) + + Since the Fourier transformation of tempered distribution + is a continuous mapping $\tempdistribnoarg \to \tempdistribnoarg$, + we have $\ft{f} \to \FT{\delta_x}$ whenever $f \to \delta_x$ in the topology of $\tempdistribnoarg$. + Recall that $\ft{\delta} = 1$, and thus $\FT{\delta_x}(p) = e^{ix \cdot p}$ for all $p \in M$. + This shows that the integrand converges pointwise to + \begin{equation*} + \sum_{s=0}^r + e^{ix \cdot (p_1 + \cdots + p_s)} + e^{-ix \cdot (p_{s+1} + \cdots + p_r)} + \tilde{K}_{\psi'\!,\psi}(p_1,\ldots,p_r) + \end{equation*} + The Dominated Convergence Theorem implies \end{proof} @@ -847,20 +944,21 @@ In the following proof it will we convenient to use the abbreviation \begin{lemma}{TODO}{} Let $\varphi$ be a free quantum field. Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients. - Then we have for all states $\psi,\psi' \in D^l(H)$ + Suppose that $l$ is a positive integer large enough to satisfy the + Then we have for all states $\psi,\psi' \in \Domain{H^l}$ \begin{multline*} \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f) \,\psi} = \\ = \int dp_1 \!\cdots dp_r \sum_{s=0}^{r} \, \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r) - \, L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) + \, \tilde{K}_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) \end{multline*} where \begin{multline*} - L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) = + \tilde{K}_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) = + P_s(p_1,\ldots,p_r) \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} - \delta_{m-s}^{n-(r-s)} - \ P_s(p_1,\ldots,p_r) \\ + \delta_{m-s}^{n-(r-s)} \\ \cdot \sqrt{m(m-1) \cdots (m-s+1)} \sqrt{n(n-1) \cdots (n-(r-s)+1)} \\ \cdot \int dk_1 \cdots dk_{m-s} \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)} @@ -869,10 +967,12 @@ In the following proof it will we convenient to use the abbreviation and $P_s(p_1,\ldots,p_r)$ is defined as before. \end{lemma} -%\[ - %f(T), f\left( T \right), - %\int_{a}^{b} f\left( x \right) d x, \frac{1}{T}, -%\] +\begin{proof} + a +\end{proof} + + +\section{Definition of the Stress Tensor} In the theory of a real scalar field $\phi$ of mass $m$, the Lagrangian density of the Klein-Gordon action is given by @@ -890,7 +990,7 @@ Raising the index $\nu$ and inserting \cref{lagrangian-density} yields \end{equation*} The \emph{energy density}: \begin{equation*} - \rho = T^{00} = \frac{1}{2} \parens*{\sum_{\mu=0}^{3} (\partial^{\mu}\phi)^2 + m^2 \phi^2} + \energydensity = T^{00} = \frac{1}{2} \parens*{\sum_{\mu=0}^{3} (\partial^{\mu}\phi)^2 + m^2 \phi^2} \end{equation*} The discussion in the previous section enables us to define the \emph{renormalized stress-energy tensor} of a free scalar field $\varphi$ by @@ -900,11 +1000,11 @@ the \emph{renormalized stress-energy tensor} of a free scalar field $\varphi$ by and this is a quadratic form. In particular, the energy density is \begin{equation*} - \rho = \frac{1}{2} \sum_{\mu=0}^{3} \normord{(\partial^{\mu}\phi)^2} + \frac{1}{2} m^2 \normord{\phi^2} + \energydensity = \frac{1}{2} \sum_{\mu=0}^{3} \normord{(\partial^{\mu}\varphi)^2} + \frac{1}{2} m^2 \normord{\varphi^2} \end{equation*} \begin{multline*} - \innerp{\psi'\!}{\rho(f) \,\psi} = \\ + \innerp{\psi'\!}{\energydensity(f) \,\psi} = \\ = \int dp_1 dp_2 \parens{p_1^{\mu} p_2^{\mu} + m^2} \sum_{s=0}^{r} (-1)^{s+1} @@ -925,13 +1025,13 @@ where \begin{theorem}{TODO}{} Let $\varphi$ be a free quantum field. Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients. - Then we have for all states $\psi,\psi' \in D^l(H)$ + Then we have for all test functions $f \in \schwartz{M}$ \begin{multline*} \normord{D_1 \varphi \cdots D_r \varphi}(f) \QFequal \int dp_1 \!\cdots dp_r \sum_{s=0}^{r} P_s(p_1,\ldots,p_r) \, \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r) \\ \cdot a^\dagger(p_1) \cdots a^\dagger(p_s) a(p_{s+1}) \cdots a(p_r) \end{multline*} - as quadratic forms on $D^l(H)$, where + as quadratic forms on $\Domain{H^l}$, where \begin{multline*} \quad P_s(p_1,\ldots,p_r) = \frac{1}{\sqrt{2^r}} @@ -944,17 +1044,107 @@ where \begin{definition}{}{} \begin{multline*} - \rho(f) \QFequal \frac{1}{4} \int dp dp' (p \cdot p' + m^2) + \energydensity(f) \QFequal \frac{1}{4} \int dp dp' (p \cdot p' + m^2) \Big\lbrack \ft{f}(p+p') a(p) a(p') + {}\\ + 2\ft{f}(p-p') a^\dagger(p) a(p') + \ft{f}(-p-p') a^\dagger(p) a^\dagger(p') \Big\rbrack \end{multline*} \end{definition} +\begin{equation*} + \bar{p} := \eta p = (p^0,-\symbfit{p}) +\end{equation*} + +\begin{proposition}{}{} + \begin{multline*} + \innerp{\psi'}{\energydensity(f) \psi} = + \frac{1}{4} \int dp dp' + (\bar{p} \cdot p' + m^2) + \bracks[\big]{2 \ft{f}(p - p') L^1_{\psi'\!,\psi}(p,p')} \\ + + (-\bar{p} \cdot p' + m^2) + \bracks[\big]{\ft{f}(- p - p') L^0_{\psi'\!,\psi}(p,p') + \ft{f}(p + p') L^2_{\psi'\!,\psi}(p,p')} + \end{multline*} +\end{proposition} + +\begin{proposition}{}{} + \begin{multline*} + \energydensity(f) \QFequal \frac{1}{4} \int dp dp' + (m^2 + \bar{p} \cdot p') + \bracks[\Big]{2\ft{f}(p-p') a^\dagger(p) a(p')} \\ + + (m^2 - \bar{p} \cdot p') + \bracks[\Big]{\ft{f}(p+p') a(p) a(p') + \ft{f}(-p-p') a^\dagger(p) a^\dagger(p')} + \end{multline*} +\end{proposition} + +\begin{proposition}{}{} + The Fock vaccum $\fockvaccum$ lies in the domain of $\energydensity(f)\QFop{}$ + for all test functions $f \in \schwartz{M}$ + and $\energydensity(f)\QFop{}\fockvaccum$ is the vector $\psi$ defined by + \begin{equation*} + \psi_2(p,p') = \frac{\sqrt{2}}{4} (m^2 - \bar{p} \cdot p') \ft{f}(-p-p') + \end{equation*} + and $\psi_n \equiv 0$ for $n \ne 2$. +\end{proposition} + +\begin{equation*} + \energydensity(f) \Omega = ? +\end{equation*} + \section{Essential Selfadjointness of Renormalized Products} -a +\begin{lemma}{H-Bounds for the Renormalized Product}{} + \begin{equation*} + \abs{\innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f)\psi}} \le + C \norm{(I+H)^l \psi} \norm{(I+H)^l \psi'} + \end{equation*} +\end{lemma} -%\nocite{*} +\begin{proof} + This proof is nearly identical to that of \cref{lemma:integral-kernel-h-bound} + and we will only cover the differences. + \begin{equation*} + \begin{multlined}[c] + \abs{\innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f)\psi}} \le + \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{s=0}^{r} + \delta_{m-s}^{n-(r-s)} \\ + \hspace{2.5cm} \cdot + \int \!dk \int \!dp'\! \int \!dp \, \abs*{F(k,p',p) \, G'(k,p') \, G(k,p)}, + \end{multlined} + \end{equation*} + where + \begin{multline} + F(k,p',p) = \parens[\big]{1+\omega(k,p')} {}^{-a} \parens[\big]{1+\omega(k,p)} {}^{-a} P_s(p',p) \\ + \cdot \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r) + \end{multline} + and $G$ and $G'$ are defined as before. + All we have to do, is verifying that + \begin{equation*} + \sup_k \norm{F(k,\cdot,\cdot)}_2 < \infty + \end{equation*} + for a sufficiently large integer $a$. + Then we obtain the desired $H$-bound with $l=a+r/2$. + + Recall that the Schwartz class is preserved by Fourier transform, translation and multiplication with polynomials. + Moreover, it is well known that Schwartz functions are square-integrable with repect to the Lorentz invariant measure on the mass shell. + Hence, + \begin{equation*} + \int dp_1 \abs{\ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r)}^2 + \end{equation*} + is bounded by a constant independent of $p_2,\ldots,p_r$. +\end{proof} + +\section{Covariance} + +\begin{equation*} + f_g(x) \defequal f(g^{-1} x) \qquad + x \in M, \quad g \in \ProperOrthochronousPoincareGroup. +\end{equation*} + +\begin{theorem}{Covariance}{covariance-renormalized-product} + \begin{equation*} + U(g) \,\normord{D_1 \varphi \cdots D_r \varphi}(f)\, U(g)^{-1} + = \normord{D_1 \varphi \cdots D_r \varphi}(f_g) + \end{equation*} +\end{theorem} \chapterbib \cleardoublepage -- cgit v1.2.3-54-g00ecf