From e1a26e4528eb7b9c2f462562c8265cf963f34dfb Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Thu, 20 Jun 2024 03:56:45 +0200 Subject: weiter --- commutatortheorem.tex | 2 - convolution.tex | 286 ++++++++++++++++++++++++++++++++++++++++++++++---- distributions.tex | 4 - fewstereveson.tex | 3 - much.tex | 150 ++++++++++++++++++-------- preamble.tex | 36 ++++--- sampleappendix.tex | 2 - samplesection.tex | 2 - second.tex | 3 - standard.tex | 3 - stresstensor.tex | 7 -- 11 files changed, 392 insertions(+), 106 deletions(-) diff --git a/commutatortheorem.tex b/commutatortheorem.tex index 9f6384f..fa8cc59 100644 --- a/commutatortheorem.tex +++ b/commutatortheorem.tex @@ -16,6 +16,4 @@ \cite{ReedSimon2} \cite{Nelson1972} -\chapterbib - %vim: syntax=mytex diff --git a/convolution.tex b/convolution.tex index d3fb8bc..eee6e16 100644 --- a/convolution.tex +++ b/convolution.tex @@ -1,27 +1,32 @@ \chapter{A Convolution Formula for Vector-Valued Tempered Distributions} \label{chapter:convolution} - \blockcquote{Bisognano1975}{% The extension to vector-valued tempered distributions is trivial. } -Recall that the class $\schwartz{\RR^n}$ of complex-valued Schwartz functions on $\RR^n$ +Recall that the class $\SchwartzFunctions{\RR^n}$ of complex-valued Schwartz functions on $\RR^n$ is closed under convolution, a operation that assigns to functions $f$ and $g$ a third one, $f * g$, given by \begin{equation*} (f*g)(x) = \int f(x-y) g(y) \, dy \qquad x \in \RR^n. \end{equation*} - -\begin{definition}{Convolution of a Distribution with a Test Function}{} - Let $u \in \tempdistrib{\RR^n}$ be tempered distribution and - let $f \in \schwartz{\RR^n}$ be a Schwartz test function. +\begin{definition}{Convolution of a Distribution with a Test Function}{convolution-distribution-test-function} + Let $u \in \TemperedDistributions{\RR^n}$ be a tempered distribution and + let $f \in \SchwartzFunctions{\RR^n}$ be a Schwartz test function. Then the \emph{convolution} of $u$ with $f$ is - the tempered distribution $u * f \in \tempdistrib{\RR^n}$ defined by + the tempered distribution $u * f \in \TemperedDistributions{\RR^n}$ defined by \begin{equation*} - (u * f)(g) \defequal u(\tilde{f} * g) \qquad g \in \schwartz{\RR^n}, + (u * f)(g) \defequal u(\tilde{f} * g) \qquad g \in \SchwartzFunctions{\RR^n}, \end{equation*} where $\tilde{f}(x) = f(-x)$ for all $x \in \RR^n$. \end{definition} +The motivation and justification for this definition is provided by the adjoint identity +\begin{equation*} + \int (h * f)(x) \, g(x) \, dx = + \int h(x) \, (\tilde{f} * g)(x) \, dx +\end{equation*} +holding for all $f,g,h \in \SchwartzFunctions{\RR^n}$. + It is well-known that the convolution can be expressed by the integral \begin{equation*} (u * f)(g) = \int u(\tau_x \tilde{f}@@) g(x) \, dx @@ -30,34 +35,269 @@ emphasizing its character of a smoothing operation. The purpose of this appendix is to state and prove a vector-valued version of this formula. -Let $X$ be a complex Banach space. -Denote by $C^{\infty}(\RR^n,X)$ the vector space of all functions $f : \RR^n \to X$ +We proceed to develop a generalization of the Bochner integral +for functions valued in a separable Fréchet space, +as this will facilitate our proof of the convolution formula. + +We consider a $\sigma$-finite measure space $(X,\SigmaAlgebra{A},\mu)$, +a separable Fréchet space $Y$ (over $\CC$) and the task is +to define the integral of functions $f \vcentcolon X \to Y$. +Recall that a measure space is said to be \emph{$\sigma$-finite} +if it can be exhausted by a countable number of measurable subsets of finite measure. +By \emph{Fréchet space} we mean a complete Hausdorff locally convex (topological vector) space +which possesses countable neighborhood bases. +We will make use of a countable family $P@@$ of seminorms that generates the topology of $@@Y$. +A topological space is called \emph{separable} if it contains a countable dense subset. + +A function $f \vcentcolon X \to Y$ will be called \emph{simple} +if it is of the form $\sum_{i=1}^n \chi_{A_i} y_i$ +where $n \in \NN$, $A_i \in \SigmaAlgebra{A}$ with $\mu(A_i) < \infty$, and $y_i \in Y$. +Naturally, the \emph{integral} of $f$ is defined to be the vector $\int f = \sum_{i=1}^n \mu(A_i) y_i \in Y$. +We say that a function $f \vcentcolon X \to Y$ is \emph{strongly measurable} +if it is the $\mu$-almost everywhere pointwise limit of simple functions. + +\begin{definition}{Generalized Bochner Integral}{} + Suppose $(X,\SigmaAlgebra{A},\mu)$ is a $\sigma$-finite measure space, + and $Y@@$ is a separable Fréchet space + whose topology is generated by a family $P@@$ of seminorms. +A strongly measurable function $f \vcentcolon X \to Y$ is called \emph{(generalized Bochner) integrable} +if there exists a sequence $(f_n)$ of simple functions such that +\begin{equation} + \label{equation:bochner-integrable} + \lim_{n \to \infty} \int_X p \circ (f_n - f) \, d\mu = 0 + \qquad \forall p \in P. +\end{equation} +In this case, the \emph{(generalized Bochner) integral} of $f$ is defined by +\begin{equation} + \label{equation:bochner-integral} + \int_X f \ d\mu \defequal + \lim_{n \to \infty} \int_X f_n \, d\mu. +\end{equation} +\end{definition} +This definition needs justification. +First, for the integral in~\eqref{equation:bochner-integrable} to be meaningful, +the functions $p \circ (f_n - f)$ must be $\mu$-measurable. +Since $f$ is strongly measurable, there exists simple functions $s_k$ such that $f (x) = \lim_{k \to \infty} s_k(x)$ for almost all $x \in X$. +The continuity of $p$ implies that $p \circ (f_n - f)$ is the almost everywhere limit simple scalar functions, namely $p \circ (f_n - s_k)$, +and as such must be measurable. +%We will defer this question for now. +Second, we have to verify that the limit in~\eqref{equation:bochner-integral} exists +and is independent of the particular sequence $(f_n)$. +Remember that the sets $U_{F,\epsilon} = \braces{y \in Y \vcentcolon p(y) < \epsilon \forall p \in F}$, +where $F \subset P$ is finite and $\epsilon > 0$, +form a neighborhood basis for $0 \in Y$. +Consider any such $U_{F,\epsilon}$. +Then, for all $p \in F$ and $m,n \in \NN$ +\begin{equation*} + p \parens*{\smallint f_n - \smallint f_m} + %= p \parens*{\smallint (f_n - f_m)} + \le \smallint p \circ (f_n - f_m) + \le \smallint p \circ (f - f_n) + \smallint p \circ (f - f_m). +\end{equation*} +By~\eqref{equation:bochner-integrable} there exists $N_p \in \NN$ such that +$p \parens*{\int f_n - \int f_m} < \epsilon$ for all $m,n \ge N_p$. +If we set $N = \max \braces{N_p \vcentcolon p \in F}$, then $\int f_n - \int f_m \in U_{F,\epsilon}$ for all $m,n \ge N$. +This shows that $(\int f_n)$ is a Cauchy sequence in the topological vector space $Y$. +Now the existence of a limit point follows from the completeness of $Y$. +It is unique because the topology is Hausdorff. + + +\begin{theorem}{Generalized Bochner Integrability Criterion}{generalized-bochner} + Suppose $X$ is a $\sigma$-finite measure space, + and $Y@@$ is a separable Fréchet space + whose topology is generated by a countable family $P@@$ of seminorms. + A function $f \vcentcolon X \to Y@@$ is generalized Bochner integrable if and only if it is strongly measurable and + \begin{equation*} + \int_X p \circ f \ d\mu < \infty + \qquad \forall p \in P. + \end{equation*} +\end{theorem} + +\begin{proof} + Since $X$ is $\sigma$-finite, + $X = \bigcup_{m=1}^{\infty} X_m$ with $\mu(X_m) < \infty$ and $X_m \subset X_{m+1}$. + Clearly, $f$ is the pointwise limit of + the functions $f_m = f \chi_{X_m}$, as $m \to \infty$. + Let $(p_i)_{i \in \NN}$ be an enumeration of the countable family $P$ of seminorms + generating the locally convex topology on $Y$. + Since $Y$ is separable, + there is a dense sequence $(y_j)_{j \in \NN}$ of vectors in $Y$. + For $n,j \in \NN$ let + \begin{gather*} + C_{nj} = y_j + U_{\braces{p_1, \ldots, p_n},1/n} + = \braces[\big]{y \in Y \vcentcolon p_i(y - y_j) \le \tfrac{1}{n} \forall i=1,\ldots,n} \\ + B_{nj} = f^{-1} C_{nj} \qquad + A_{nj} = B_{nj} \setminus \bigcup_{k=1}^{j-1} B_{nk} + \end{gather*} + Observe that for each fixed $n$ the sets $C_{nj}$ cover $Y$, + the sets $B_{nj}$ cover $X$ and + the sets $A_{nj}$ partition $X$. + Moreover, the sets $B_{nj}$, and consequently $A_{nj}$, are $\mu$-measurable + because the functions $x \mapsto p_i \parens[\big]{f(x) - y_j}$ are $\mu$-measurable. + Then, the functions + \begin{equation*} + f_{mn} = \sum_{j=1}^{\infty} \chi_{X_m \cap A_{nj}} y_j + \end{equation*} + satisfy $p_i(f(x) - f_{mn}(x)) \le \frac{1}{n}$ for all $x \in X$ when $i \le n$. + Hence, $p_i \circ f_{mn} \le p_i \circ f + \frac{1}{n}$. + Since $f_{mn}$ is supported in $X_m$, a set of finite measure, and $\int p_i \circ f < \infty$, + we conclude $\int p_i \circ f_{mn} < \infty$ for all $i \le n$. + For each $(m,n) \in \NN^2$ choose $J(m,n)$ so large that + \begin{equation*} + \int_{\bigcup_{j=J(m,n)+1}^{\infty} X_m \cap A_{nj}} p_i \circ f_{mn} < \frac{\mu(X_m)}{n} + \qquad \forall i=1,\ldots,n. + \end{equation*} + The functions + \begin{equation*} + s_{mn} = \sum_{j=1}^{J(m,n)} \chi_{X_m \cap A_{nj}} y_j + \end{equation*} + are simple and satisfy + \begin{equation*} + \int p_i \circ (f_m - s_{mn}) + \le \int p_i \circ (f_m - f_{mn}) + \int p_i \circ (f_{mn} - s_{mn}) + < \frac{2\mu(X_m)}{n} + \end{equation*} + for $n \ge i$. + It follows that + \begin{equation*} + \lim_{n \to \infty} \int p_i \circ (f_m - s_{mn}) = 0 + \qquad \forall i \in \NN. + \end{equation*} + For each $m \in \NN$ choose $N(m)$ so large that + \begin{equation*} + \int p_i \circ (f_m - s_{mN(m)}) < \frac{1}{m} + \qquad \forall i=1,\ldots,m. + \end{equation*} + and therefore + \begin{equation*} + \int p_i \circ (f - s_{m N(m)}) + \le \frac{1}{m} + \int p_i \circ (f - f_m) + \end{equation*} + by the triangle inequality. + %This implies + %\begin{equation*} + %\lim_{n \to \infty} \int p_i \circ (f_m - s_{mN(m)}) = 0 + %\qquad \forall i \in \NN. + %\end{equation*} + For each $i \in \NN$ the increasing sequence $(p_i \circ f_{m})_m$ of positive real-valued measurable functions + converges pointwise to the function $p_i \circ f$, + which is by hypothesis is integrable. + By Dominated Convergence, $\int p_i \circ (f-f_m) \to 0$, as $m \to \infty$. + \begin{equation*} + \lim_{m \to \infty} \int p_i \circ (f - s_{m N(m)}) = 0 + \qquad \forall i \in \NN. + \end{equation*} + This proves that $f$ is generalized Bochner integrable. +\end{proof} + +\begin{theorem}{}{integral-commutes-with-operator} + Suppose $X$ is a $\sigma$-finite measure space. + Let $Y$ and $Z$ be separable Fréchet spaces, + and let $T \vcentcolon Y \to Z$ be a continuous linear operator. + If $f \vcentcolon X \to Y$ is generalized Bochner integrable, + then $T \circ f \vcentcolon X \to Z$ is generalized Bochner integrable, and + \begin{equation*} + \int T \circ f = + T \! \int \! f. + \end{equation*} +\end{theorem} + +\begin{proof} + Clearly, the composition $T \circ f$ is strongly measurable + because $T$ is continuous and $f$ is strongly measurable. + Suppose that the locally convex topologies on $Y$ and $Z$ + are generated by the seminorm families $P$ and $Q$, respectively. + If $q \in Q$, then the fact that $T$ is continuous and linear implies that + there exists a finite subset $F \subset P$ and a constant $M \ge 0$ + such that $q \circ T \le M \max_{p \in F} p$. + If $(f_n)$ is a sequence of simple functions such that $\int p \circ (f - f_n) \to 0$, + then $\int q \circ T \circ (f-f_n) \to 0$. + This shows that $T \circ f$ is generalized Bochner integrable, and + \begin{equation*} + \int T \circ f = \lim_{n \to \infty} \int T \circ f_n + = T \lim_{n \to \infty} \int f_n = T \int f.\qedhere + \end{equation*} + %By \cref{theorem:generalized-bochner}, + %it follows that $\int q \circ T \circ f < \infty$, +\end{proof} + +We now return to tempered distributions. +Denote by $\TestFunctions{\RR^n}$ the vector space of all functions $f \vcentcolon \RR^n \to \CC$ such that the derivatives $\partial^{\alpha} f$ exist and are continuous for all multi-indices $\alpha \in \NN^n$. -We define the space $\schwartz{\RR^n,X}$ of \emph{$X$-valued Schwartz functions} to be the vector space +Recall that the space $\SchwartzFunctions{\RR^n}$ of \emph{Schwartz functions} is defined to be the vector space \begin{equation*} - \schwartz{\RR^n,X} \defequal \braces{f \in C^{\infty}(\RR^n,X) \vcentcolon \norm{f}_{\alpha,\beta} < \infty \forall \alpha,\beta \in \NN^n} + \SchwartzFunctions{\RR^n,X} \defequal \braces{f \in \TestFunctions{\RR^n} \vcentcolon \norm{f}_{\alpha,\beta} < \infty \ \forall \alpha,\beta \in \NN^n} \end{equation*} equipped with the locally convex topology induced by the family of seminorms \begin{equation*} - \norm{f}_{\alpha,\beta} = \sup_{x \in \RR^n} \abs{x^{\alpha}} \norm{\partial^{\beta} f(x)}_X. + \norm{f}_{\alpha,\beta} = \sup_{x \in \RR^n} \abs{x^{\alpha}} \abs{\partial^{\beta} f(x)}. \end{equation*} -We define the space $\tempdistrib{\RR^n,X}$ of \emph{$X$-valued tempered distributions} to be the vector space +It is well known that the Schwartz space is a separable Fréchet space. +Now let $X$ be any separable Fréchet space. +We define the space $\TemperedDistributions{\RR^n\!,X}$ of \emph{$X$-valued tempered distributions} to be the vector space \begin{equation*} - \tempdistrib{\RR^n,X} \defequal \BoundedLinearOperators[\big]{\schwartz{\RR^n},X}. + \TemperedDistributions{\RR^n\!,X} \defequal \ContinousLinearOperators[\big]{\SchwartzFunctions{\RR^n},X}. \end{equation*} +of all continuous linear operators $\SchwartzFunctions{\RR^n} \to X$ equipped with the bounded convergence topology. +The convolution of a $X$-valued tempered distribution $v$ with a Schwartz function $f$ +is defined in the same way as in \cref{definition:convolution-distribution-test-function}, that is by + \begin{equation*} + (v * f)(g) \defequal v(\tilde{f} * g) \qquad g \in \SchwartzFunctions{\RR^n}. + \end{equation*} -\begin{proposition}{Vector-Valued Convolution Formula}{} - Let $v \in \tempdistrib{\RR^n,X}$ be tempered distribution with values in a Banach space $X$, and - let $f \in \schwartz{\RR^n}$ be a Schwartz test function. Then one has +\begin{proposition}{Vector-Valued Convolution Formula}{vector-valued-convolution-formula} + Let $v \in \TemperedDistributions{\RR^n\!,X}$ be a tempered distribution with values in a separable Fréchet space $X$, and + let $f \in \SchwartzFunctions{\RR^n}$ be a Schwartz test function. Then one has \begin{equation*} - (v * f)(g) = \int v(\tau_x \tilde{f}@@) g(x) \, dx \qquad g \in \schwartz{\RR^n}. + (v * f)(g) = \int v(\tau_x \tilde{f}@@) g(x) \, dx \qquad g \in \SchwartzFunctions{\RR^n}. \end{equation*} \end{proposition} -Der Beweis ist in Arbeit ;) +\begin{proof} + We fix a Schwartz function $g$, and consider the finite measure $\mu = \abs{g} \lambda$ on $\RR^n$, + where $\lambda(x) = dx$ is the Lebesgue measure. + We show that the mapping $x \mapsto \tau_x \tilde{f}$ is a generalized Bochner $\mu$-integrable function $\RR^n \to \SchwartzFunctions{\RR^n}$ + using \cref{theorem:generalized-bochner}. + For all $\alpha,\beta \in \NN^n$ we see by substituting $x+y$ for $y$ that + \begin{equation*} + \norm{\tau_x \tilde{f}}_{\alpha,\beta} = + \sup_{y} \abs{y^{\alpha} \partial^{\beta} (\tau_x \tilde{f})(y)} = + \sup_{y} \abs{(x+y)^{\alpha} \partial^{\beta} \tilde{f}(y)}. + \end{equation*} + There exists constants $c_{\gamma \delta}$ with + $\abs{(x+y)^{\alpha}} \le \sum_{\gamma + \delta = \alpha} c_{\gamma \delta} \abs{x^{\gamma} y^{\delta}}$, + and it follows that + \begin{equation*} + \int \norm{\tau_x \tilde{f}}_{\alpha,\beta} \, d \mu(x) + \le \sum_{\gamma + \delta = \alpha} c_{\gamma \delta} \norm{\tilde{f}}_{\delta,\beta} \int \abs{x^{\gamma}} g(x) \, dx < \infty + \end{equation*} + because $g$ is Schwartz class. + Hence, $x \mapsto \tau_x \tilde{f}$ defines an integrable function. + + The mapping $v \vcentcolon \SchwartzFunctions{\RR^n} \to X$ is linear and continuous by definition. + By \cref{theorem:integral-commutes-with-operator}, + the composite mapping $x \mapsto v(\tau_x \tilde{f})$ is a $\mu$-integrable function $\RR^n \to X$, and + \begin{equation} + \label{equation:general-bochner-appears} + \int v(\tau_x \tilde{f}) \, d\mu(x) = v \parens[\bigg]{\int \tau_x \tilde{f} \, d\mu(x)} + \end{equation} + For every fixed $y \in \RR^4$ the evaluation mapping $\ev_{\! @@y} \vcentcolon \SchwartzFunctions{\RR^4} \to \CC$, $h \mapsto h(y)$, clearly is continuous. + A second invocation of \cref{theorem:integral-commutes-with-operator} delivers + \begin{equation*} + \ev_{\! @@y} \parens[\bigg]{\int \tau_x \tilde{f} \, d\mu(x)} = + \int \ev_{\! @@y}(\tau_x \tilde{f}) \, d\mu(x) = + \int \tilde{f}(y-x) g(x) \, dx = + (\tilde{f} * g)(y) + \end{equation*} + and the proof is complete. +\end{proof} + +Let us point out that even in the special case that $X$ is a Banach space +the integral on the right hand side of~\eqref{equation:general-bochner-appears} +only has meaning as a generalized Bochner integral, +since the integrand takes values in $\SchwartzFunctions{\RR^n}$, +which is not a Banach space. +We could not have performed this step with the ordinary Bochner integral. %\nomenclature[B]{$\BoundedLinearOperators{X,Y}$}{bounded linear operators from $X$ to $Y$\nomnorefpage} - -\chapterbib -\cleardoublepage diff --git a/distributions.tex b/distributions.tex index fad319b..6bd290d 100644 --- a/distributions.tex +++ b/distributions.tex @@ -24,7 +24,3 @@ by \nocite{Friedlander1999} \nocite{Hoskins2005} \nocite{Hoskins2009} - - -\chapterbib -\cleardoublepage diff --git a/fewstereveson.tex b/fewstereveson.tex index cb1918b..27da7f2 100644 --- a/fewstereveson.tex +++ b/fewstereveson.tex @@ -5,7 +5,4 @@ Goal: make~\cite{Fewster1998} rigorous based on the previous chapter -\chapterbib -\cleardoublepage - % vim: syntax=mytex diff --git a/much.tex b/much.tex index a7f9bbf..249dc73 100644 --- a/much.tex +++ b/much.tex @@ -43,7 +43,7 @@ Poincaré covariance \begin{definition}{Von Neumann Algebra of Local Observables}{} \begin{equation*} - \localalg{\spacetimeregion{O}} = \braces{b(\varphi(f)) \mid \text{$b$ bounded}, f \in \realschwartz{M}, \supp f \subset \spacetimeregion{O}}'' + \localalg{\spacetimeregion{O}} = \Set{b(\varphi(f)) \mid \text{$b$ bounded}, f \in \realschwartz{M}, \supp f \subset \spacetimeregion{O}}'' \end{equation*} \end{definition} @@ -272,17 +272,17 @@ Thus, modular theory \section{The Geometric Action of the Modular Operator Associated With a Wedge Domain} -\begin{definition}{Right and Left Wedge, General Wedges}{} +\begin{definition}{Right and Left Wedge, General Wedges}{wedge} The \emph{right wedge}\index{wedge!right}\nomenclature[WR]{$\rightwedge$}{right wedge} and \emph{left wedge}\index{wedge!left}\nomenclature[WL]{$\leftwedge$}{left wedge} in Minkowski space $M$ are the open subsets \begin{equation*} - \rightwedge \defequal \braces[\big]{x \in M \vcentcolon x^1 > \abs{x^0}} + \rightwedge \defequal \Set[\big]{x \in M \given x^1 > \abs{x^0}} \quad \text{and} \quad - \leftwedge \defequal \braces[\big]{x \in M \vcentcolon x^1 < -\abs{x^0}}. + \leftwedge \defequal \Set[\big]{x \in M \given x^1 < -\abs{x^0}}. \end{equation*} We say that a spacetime region $W \subset M$ is a \emph{wedge}\index{wedge} - if there exists an element $g$ of the Poincaré group + if there exists an element $g$ of the full Poincaré group such that $W = g \rightwedge$. \end{definition} @@ -443,7 +443,7 @@ That this is generally true is the statement of the following Lemma. \frac{1}{2\pi i} \int_{\alpha}^{\beta} \bracks{R_A(\lambda + i \varepsilon) - R_A(\lambda - i \varepsilon)} d\lambda = E_A \parens[\big]{\bracks{a,b}} \end{equation*} - for all $a \in \RR \cup \braces{-\infty}$, $b \in \RR \cup \braces{\infty}$. + for all $a \in \RR \cup \Set{-\infty}$, $b \in \RR \cup \Set{\infty}$. Observe that $\rho(A) = \rho(U\! @AU^*)$ and that for each (common) regular value $\lambda$ we have \begin{equation*} R_{U\! @AU^*}(\lambda) = U R_A(\lambda) @ U^*\!. @@ -492,6 +492,8 @@ In other words, $K_{\spacetimeregion{O}}$ is the unique selfadjoint operator suc \end{proposition} \todo{domain, proof} +\bluetext{Maybe this is simpler in Rindler coordinates...} + \section{Complex Lorentz Transformations} The main result of this section is \cref{proposition:main-result}. @@ -504,9 +506,9 @@ of complex Minkowski space $M+iM \cong \CC^4$ with respect to the inner product The \emph{complex Poincaré group}\index{Poincaré group!complex}\nomenclature[PC]{$\ComplexPoincareGroup$}{complex Poincaré group} is the semidirect product $\ComplexPoincareGroup \defequal \CC^4 \ltimes \ComplexLorentzGroup$. The action of $\ComplexPoincareGroup$ on $M+iM$ is defined in the obvious way. The complex Poincaré group has just two connected components, the subgroup $\ProperComplexPoincareGroup$ and the subset $\ImproperComplexPoincareTransformations$, -differentiated by the sign of $\det \Lambda \in \braces{\pm 1}$ for its elements $(z,\Lambda)$. +discriminated by the sign of $\det \Lambda \in \Set{\pm 1}$ for its elements $(z,\Lambda)$. The (real) proper orthochronous Poincaré group $\ProperOrthochronousPoincareGroup$ is a subgroup of $\ProperComplexPoincareGroup$. -Each of the two following sections deals with a subgroup $G$ of $\ProperOrthochronousPoincareGroup$, +Each of the two following sections deals with a subgroup $G$ of $\smash{\ProperOrthochronousPoincareGroup}$, and the possibility of extending a unitary representation of $G$ to a larger set within $\ProperComplexPoincareGroup$. \subsection{Analytic Continuation of the Space-Time Translation Group} @@ -540,7 +542,7 @@ and we impose the so-called \emph{spectrum condition} \forall \psi \in D \; \forall a \in \ClosedForwardCone, \end{equation*} -where $\ClosedForwardCone \defequal \braces{a \in \RR^4 \vcentcolon a \cdot a \ge 0, a^0 \ge 0}$ is the \emph{closed forward cone}\index{cone!closed forward}\nomenclature[V]{$\ClosedForwardCone$}{closed forward cone}. +where $\ClosedForwardCone \defequal \Set{a \in \RR^4 \given a \cdot a \ge 0, a^0 \ge 0}$ is the \emph{closed forward cone}\index{cone!closed forward}\nomenclature[V]{$\ClosedForwardCone$}{closed forward cone}. It can be shown \cite{Uhlmann1961} that the spectrum condition is equivalent to the statement that the support of the spectral measure is contained in the closed forward cone, i.e.\ $\supp(E) \subset \ClosedForwardCone$. @@ -570,7 +572,7 @@ Observe that the set $\ClosedForwardTube$ is closed under vector addition and th Since $z$ lies in the closed forward tube, $z=x+iy$ with $x \in \RR^4$ and $y \in \ClosedForwardCone$. Now $\abs{f(k)} = \exp(-y \cdot k)$, and on $\ClosedForwardCone$ this is bounded by $1$ because $y \cdot k \ge 0$ for all $k \in \ClosedForwardCone$. - The identity $U(w+z) = U(w) U(z)$ follows from $\exp(i(w+z) \cdot k) = \exp(iw \cdot k) \exp(iz \cdot k)$ + The identity $U(w+z) = U(w) U(z)$ follows from $\exp(i(w+z) \cdot k) = \exp(iw \cdot k)\*\exp(iz \cdot k)$ and the boundedness of the operators, see~\cite[Proposition 4.16(iii) and (v)]{Schmüdgen2012}. \end{proof} @@ -620,6 +622,19 @@ Observe that the set $\ClosedForwardTube$ is closed under vector addition and th \todo{Explain what it means for an operator-valued function of several complex variables to be analytic.} +\begin{lemma}{}{complex-translation} + Let $g = (b,\Lambda)$ be a proper orthochronous Poincaré transform with $b \in \OpenForwardCone$. + Then, for all $z \in \OpenForwardTube$ + \begin{equation*} + gz \in \OpenForwardTube \qquad + U(g) U(z) = U(gz). + \end{equation*} +\end{lemma} + +\begin{proof} + \bluetext{Edge of the Wedge} +\end{proof} + Next we consider an operator-valued tempered distribution $u$ that is \emph{covariant} in the sense that it obeys the relativistic transformation law \begin{equation} @@ -695,7 +710,7 @@ Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$, %such that $\ft{e_z} \in \schwartz{\RR^4}$ and $\ft{e_z}(p) = \exp(iz \cdot p)$ for $p \in \ClosedForwardCone$. %\end{lemma} -\begin{proposition}{}{} +\begin{proposition}{}{prp} Let $u$ be a covariant operator-valued tempered distribution, and let $f \in \schwartz{\RR^4}$ be a test function. Then we have, in generalization of~\eqref{equation:real-translation-law}, @@ -711,6 +726,27 @@ Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$, Dann folgt die Behauptung wohl mit Edge of the Wedge~\cite[Theorem 2-17]{Streater1964}} \end{proof} +\begin{corollary}{}{} + Let $u$ be a covariant operator-valued tempered distribution, + and let $f \in \schwartz{\RR^4}$ be a test function. Then we have, + \begin{equation*} + U(z) u(f) \FockVacuum = \int dx \, f(x) \, u(d_{z+x}) \FockVacuum \qquad \forall z \in T_+. + \end{equation*} +\end{corollary} + +\begin{proof} + The convolution formula \cref{proposition:vector-valued-convolution-formula} applied to the vector-valued distribution defined by $f \mapsto \alpha(f) = u(f) \FockVacuum$ yields + \begin{equation*} + (\alpha * \tilde{d}_z)(f) = \int dx \, f(x) \, \alpha(\tau_x d_z) + \end{equation*} + Using \cref{proposition:prp}, we calculate + \begin{equation*} + (\alpha * \tilde{d}_z)(f) = \alpha(d_z * f) = \alpha(f_z) = u(f_z) \FockVacuum = U(z) u(f) \FockVacuum. + \end{equation*} + It is easily seen by Fourier transformation that $\tau_x d_z = d_{x+z}$. + Hence, $\alpha(\tau_x d_z) = u(d_{x+z}) \FockVacuum$. +\end{proof} + \subsection{Complex Lorentz Boosts} The Lorentz boosts $\Lambda(t)$ given by @@ -723,7 +759,6 @@ In particular, the vector-valued function $\CC \ni w \mapsto \Lambda(w) z$ is en We are particularly interested in the case of a purely imaginary parameter. The relations $\cosh iz = \cos z$ and $\sinh iz = i \sin z$ between the complex hyperbolic and trigonometric functions imply - \begin{equation*} \Lambda(is) = \begin{pmatrix} \phantom{i}\cos(2 \pi @ s) & i\sin(2 \pi @ s) & \; 0 \; & \; 0 \; \\ @@ -733,8 +768,9 @@ between the complex hyperbolic and trigonometric functions imply \end{pmatrix} \qquad \forall s \in \RR. \end{equation*} - -\begin{equation*} +For later use, we give the action of $\Lambda(is)$ on a complex four-vector $x+iy$: +\begin{equation} + \label{equation:pure-imaginary-lorentz-boost} \Lambda(is) (x+iy) = \begin{pmatrix} \cos(2 \pi @ s) x^0 - \sin(2 \pi @ s) y^1 \\ @@ -749,7 +785,7 @@ between the complex hyperbolic and trigonometric functions imply y^2 \\ y^3 \end{pmatrix} -\end{equation*} +\end{equation} We @@ -814,49 +850,75 @@ but a dense subspace of $\Domain{T}$ need not be a core for $T$. \begin{lemma}{A Common Core for All Complex Lorentz Boosts}{common-core-for-complex-lorentz-boots} Adopt the notation of the foregoing lemma. The linear subspace \begin{equation*} - \mathcal{D}_0 = \Span \braces{\ran g(K) \vcentcolon g \in \schwartz{\RR}} + \mathcal{D}_0 = \Span \Set{\ran g(K) \given g \in \schwartz{\RR}} \end{equation*} is a core for $V(z)$ for every $z \in \CC$. \end{lemma} \begin{proof} - xxx + $M_n = \bracks{-n,n}$ \end{proof} \subsection{Application to the Energy Density} -\bluetext{Achtung: Dieser Abschnitt ist noch roh, lückenhaft und enthält inkonsistente Notation und falsche Aussagen.} -The following three Lemmas are variations of the arguments +The following three lemmas are variations of the arguments brought forward by~\citeauthor{Bisognano1975} in their proof of \cref{theorem:bisognano-wichmann}. -The main difference is that we state xxx and xxx as operator identities without reference to a field operator, -and proof xxx for arbitrary Lorentz-covariant operator-valued distributions +The main difference is that we state \cref{lemma:biso1} and \cref{lemma:biso2} as operator identities without reference to a field operator, +and proof \cref{lemma:biso3} for arbitrary Lorentz-covariant operator-valued distributions rather than products of field operators. This generalization is necessary for the application to the energy density. In addition, we provide in \cref{chapter:convolution} a complete proof of the convolution formula for vector-valued distributions. -Roughly speaking, the following Lemma asserts that a translation by a complex vector +Roughly speaking, the following lemma asserts that a translation by a complex vector followed by a suitable imaginary boost is again a complex translation. -\begin{lemma}{}{} -Let $z = x + iy \in \OpenForwardTube$ with $x,y$ real, and suppose $y^3=y^4=0$. +\begin{lemma}{}{biso1} +Let $z = x + iy \in \OpenForwardTube$ with $x,y$ real, and suppose $y^2=y^3=0$. \begin{enumerate} - \item If $x \in \rightwedge$, then for all $s \in [0,1/4]$ + \item If $x \in \rightwedge$, then for all $s \in [0,\tfrac{1}{4}]$ \begin{equation*} \Lambda(is) z \in \OpenForwardTube, \qquad \ran U(z) \subset \dom V(is), \qquad V(is) U(z) = U \parens[\big]{\Lambda(is) z}. \end{equation*} - \item If $x \in \leftwedge$, then the above holds for all $s \in [0,-1/4]$. + \item If $x \in \leftwedge$, then the above holds for all $s \in [0,-\tfrac{1}{4}]$. \end{enumerate} \end{lemma} \nomenclature[dom]{$\dom T$}{domain of the operator $T$\nomnorefpage} \nomenclature[ran]{$\ran T$}{range of the operator $T$\nomnorefpage} \begin{proof} - xxx + The result of applying $\Lambda(is)$ for some $s \in \RR$ to a complex four-vector $x+iy$ + has been given in~\eqref{equation:pure-imaginary-lorentz-boost}. + To show that this vector lies in the open forward tube, + we need to verify that its imaginary part lies in the open forward cone $\OpenForwardCone$. + By definition, a real four-vector $a$ lies in $\OpenForwardCone$ if and only if + $a^0 > 0$ and $a \cdot a > 0$. If $a^2 = a^3 = 0$, then + these conditions are easily seen to be equivalent to the conditions + $a^0 > 0$ and $a^0 \mp a^1 > 0$ (for both sign choices). + + Since the second and third components of the imaginary part of~\eqref{equation:pure-imaginary-lorentz-boost}, + $y^2$ and $y^3$, vanish by assumption, it is sufficient to prove + \begin{equation} + \label{equation:inequalities} + \begin{aligned} + \sin(2 \pi s) x^1 + \cos(2 \pi s) y^0 &> 0 \ \text{and} \\ + \sin(2 \pi s) \parens[\big]{x^1 \mp x^0} + \cos(2 \pi s) \parens[\big]{y^0 \mp y^1} &> 0. + \end{aligned} + \end{equation} + The assumption $x \in \rightwedge$ implies that + $x^1 > 0$ and $x^1 \mp x^0 > 0$, + by \cref{definition:wedge}. + The assumptions $y \in \OpenForwardCone$ and $y^2 = y^3 = 0$ imply that + $y^0 > 0$ and $y^0 \mp y^1 > 0$, + by the argument in the foregoing paragraph. + So, all we need to do to ensure~\eqref{equation:inequalities} holds, + is choose $s$ such that both $\sin(2 \pi s)$ and $\cos(2 \pi s)$ are nonnegative. + (Then, at least one of these will be positive.) + Clearly, this is true for all $s \in \bracks{0,\tfrac{1}{4}}$. \noindent\begin{minipage}{0.5\textwidth} - The vector-valued function $\CC \ni s \mapsto \Lambda(is) z$ is entire analytic. + \hspace{\parindent} The vector-valued function $\CC \ni s \mapsto \Lambda(is) z$ is entire analytic. In particular it is continuous, and we have shown that it maps the compact subset $[0,1/4]$ into the open set $\OpenForwardTube$. This implies that there exists a connected open neighborhood $N \subset \CC$ of $[0,1/4]$ such that $\Lambda(is) z \in \OpenForwardTube$ for all $s \in N$. @@ -878,26 +940,36 @@ Let $z = x + iy \in \OpenForwardTube$ with $x,y$ real, and suppose $y^3=y^4=0$. Let $\xi \in \hilb{F}$ be arbitrary, and let $\eta$ be in the common dense domain $\mathcal{D}_0$ of the operators $V(is)$ from \cref{lemma:common-core-for-complex-lorentz-boots}. Then the function $f_1(s) = \innerp{V(is)^* \eta}{U(z) \xi}$ is well-defined, and entire analytic by Lemma xxx. The function $f_2(s) = \innerp{\eta}{U(\Lambda(is) z) \xi}$ is analytic on $N$, by \cref{proposition:analyticity-complex-translations}. - By Lemma xxx, $f_1$ and $f_2$ agree in an open real neighborhood $is$. Since $N$ is an open neighborhood of $0$, there is an $\epsilon >0$ such that $i(-\epsilon,\epsilon) \subset N$. - It follows that $f_1 \equiv f_2$ on $N$. - core \ldots + By \cref{lemma:complex-translation}, $V(is) U(z) = U(\Lambda(is) z)$ if $is \in \RR$. + Hence, $f_1$ and $f_2$ agree in an open real neighborhood of $is$. + Now the Identity Principle implies $f_1 \equiv f_2$ on $N$. + Since this holds for all $\eta$ in $\mathcal{D}_0$, + which is a core for $V(is)^{**} = V(is)$, + we conclude that $U(z) \xi$ lies in the domain of $V(is)$, + and $V(is) U(z) \xi = U(\Lambda(is) z) \xi$. + As $\xi$ was arbitary, the proof is complete. \end{proof} -\begin{lemma}{}{} - Let $x \in \rightwedge$ +Remember that $\mathcal{J} = \Lambda(i/2) = \diag(-1,-1,1,1)$. + +\begin{lemma}{}{biso2} + Let $x \in \rightwedge$, and let $e_0 = (1,0,0,0)$ be the forward timelike unit vector. Then \begin{equation*} \stronglim_{\varepsilon \downarrow 0} V(i/4) U(x+i \varepsilon e_0) - = U \parens[\big]{V(i/4)x} + = U \parens[\big]{\Lambda(i/4)x} = \stronglim_{\varepsilon \downarrow 0} V(-i/4) U(\mathcal{J}x+i \varepsilon e_0) \end{equation*} \end{lemma} \begin{proof} - xxx + The first identity follows from \cref{lemma:biso2}(i) + and the strong continuity of $z \mapsto U(z)$ on $\ClosedForwardTube$ (\cref{proposition:analyticity-complex-translations}). + For the second identity, note that $\mathcal{J} x \in \leftwedge$ and + apply \cref{lemma:biso2}(ii), then use $\Lambda(-i/4) \mathcal{J} = \Lambda(i/4)$. \end{proof} -\begin{lemma}{}{} +\begin{lemma}{}{biso3} Suppose that $u$ is a covariant operator-valued tempered distribution. Let $f \in \schwartz{M}$ with $\supp f \subset \rightwedge$, and let $g \in \schwartz{M}$ be arbitrary. Then @@ -909,9 +981,8 @@ Let $z = x + iy \in \OpenForwardTube$ with $x,y$ real, and suppose $y^3=y^4=0$. Here, $K$ is the infinitesimal generator of the group $t \mapsto V(t)$ of real Lorentz boosts, $\FockVacuum$ is the Fock vacuum, and $\mathcal{J}$ is the Lorentz transformation given by the diagonal matrix $\diag(-1,-1,1,1)$. - \begin{proof} - xxx + a \end{proof} \begin{equation*} @@ -938,7 +1009,4 @@ In der Ungleichung aus~\cite{Much2022} ist $h$ eine Gauß-Funktion. coming soon\ldots -\chapterbib -\cleardoublepage - % vim: syntax=mytex diff --git a/preamble.tex b/preamble.tex index 8479b31..fb67d0a 100644 --- a/preamble.tex +++ b/preamble.tex @@ -16,16 +16,15 @@ %\usepackage{graphicx} \usepackage{tikz} \usepackage{tcolorbox} -%\usepackage{wrapfig} \usepackage[style=ext-alphabetic]{biblatex} \usepackage[intoc,refpage]{nomencl} \usepackage{makeidx} -\usepackage{idxlayout} \usepackage{hyperref} \usepackage{bookmark} \usepackage{hypdestopt} \usepackage[capitalize,nameinlink]{cleveref} %\usepackage{refcheck} +%\usepackage{nag} % ---------- fontspec \setfontfamily\fausansoffice{FAUSansOffice} @@ -88,8 +87,10 @@ \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\ran}{ran} \DeclareMathOperator{\Span}{span} +\DeclareMathOperator{\ev}{ev} % extend amsmath's proof environment -\NewDocumentEnvironment{myproof}{Ob}{\IfNoValueTF{#1}{\begin{proof}}{\begin{proof}[\proofname\ of \Cref{#1}]}}{\end{proof}} +\newenvironment{myproof}[1]{\proof[\proofname\ of \Cref{#1}]}{\endproof} +%\NewDocumentEnvironment{myproof}{Ob}{\IfNoValueTF{#1}{\begin{proof}}{\begin{proof}[\proofname\ of \Cref{#1}]}}{\end{proof}} % ---------- mathtools \DeclarePairedDelimiter\abs{\lvert}{\rvert} @@ -99,6 +100,8 @@ \DeclarePairedDelimiter\braces{\lbrace}{\rbrace} \DeclarePairedDelimiter\angles{\langle}{\rangle} % TODO set macro with proper spacing +\newcommand\given{\vcentcolon} +\DeclarePairedDelimiterX\Set[1]{\lbrace}{\rbrace}{#1} \DeclarePairedDelimiter\bra{\lvert}{\rangle} \DeclarePairedDelimiter\ket{\langle}{\rvert} \DeclarePairedDelimiterX\innerp[2]{\langle}{\rangle}{#1,#2} @@ -111,7 +114,7 @@ \usetikzlibrary{arrows.meta} % ---------- tcolorbox -\tcbuselibrary{skins,theorems,breakable} % add breakable library? +\tcbuselibrary{skins,theorems,breakable} \tcbset{% beforeafter skip balanced=0.4\baselineskip, mythmstyle/.style={% @@ -151,16 +154,10 @@ \newenunciation{example}{gray} \newenunciation{remark}{gray} -%\theoremstyle{definition} -%\newtheorem{defin} -%\renewenvironment{definition}[2]{\begin{defin}[#1]\label{#2}}{\end{defin}} - % ---------- biblatex \addbibresource[glob]{bib/*.bib} \ExecuteBibliographyOptions{% refsegment=chapter, - sorting=none, - defernumbers, giveninits, backref, } @@ -170,6 +167,7 @@ \bookmark[level=section,italic,dest=refbm:\arabic{refsegment}]{#1} } \newcommand*{\chapterbib}{\printbibliography[segment=\therefsegment,heading=chapterbib]} +\AddToHook{include/end}{\chapterbib} \DefineBibliographyStrings{english}{% backrefpage={ref.\ on \pno}, %TODO use \addspace ? backrefpages={ref.\ on \ppno} @@ -275,7 +273,7 @@ % -------------- \newcommand*{\hilb}[1]{\mathcal{#1}} \newcommand*{\Hilb}[1]{\mathcal{#1}} -% algebraic dircet sum +% algebraic direct sum \newcommand{\AlgebraicDirectSum}[1]{\sideset{}{_{\!\ts{alg}}}\bigoplus#1} % Hilbert space direct sum %\newcommand{\AlgebraicDirectSum}[1]{\sideset{}{_{\!\ts{Hilb}}}\bigoplus #1} @@ -283,10 +281,14 @@ % Test functions and distributions % -------------- \newcommand*{\testfun}[1]{\mathcal{D}(#1)} -\newcommand*{\distrib}[1]{\mathcal{D}'(#1)} -\newcommand*{\schwartz}[1]{\mathcal{S}(#1)} +\newcommand*{\TestFunctions}[2][]{\mathcal{D}\parens[#1]{#2}} +\newcommand*{\distrib}[1]{\mathcal{D}'(#1)} %todo replace by command below +\newcommand*{\Distributions}[2][]{\mathcal{D}\parens[#1]{#2}} +\newcommand*{\schwartz}[1]{\mathcal{S}(#1)} %todo replace by command below +\newcommand*{\SchwartzFunctions}[2][]{\mathcal{S}\parens[#1]{#2}} \newcommand*{\realschwartz}[1]{\mathcal{S}_{\RR}(#1)} \newcommand*{\tempdistrib}[1]{\mathcal{S}'(#1)} +\newcommand*{\TemperedDistributions}[2][]{\mathcal{S}'\parens[#1]{#2}} \newcommand*{\tempdistribnoarg}{\mathcal{S}'} % Fock spaces @@ -314,7 +316,7 @@ % Standard Subspaces % ------------------ -% real scalarproduct +% real scalar product \DeclarePairedDelimiterXPP\realscalarp[2]{\Re}{\langle}{\rangle}{}{#1,#2} \DeclarePairedDelimiterXPP\symplecticp[2]{\Im}{\langle}{\rangle}{}{#1,#2} % symplectic complement @@ -340,8 +342,9 @@ \newcommand*{\vNa}[1]{\mathcal{#1}} \newcommand*{\localalg}[1]{\vNa{R}(#1)} -% Measure Theroy -\newcommand*{\BorelSigmaAlgebra}[2][]{\mathfrak{B}\parens[#1]{#2}} +% Measure Theory +\newcommand*{\SigmaAlgebra}[1]{\mathfrak{#1}} +\newcommand*{\BorelSigmaAlgebra}[2][]{\SigmaAlgebra{B}\parens[#1]{#2}} % Lorentz and Poincaré groups, subgroups and connected components \newcommand*{\LorentzGroup}{\mathcal{L}} @@ -378,6 +381,7 @@ % Functional Analysis \newcommand*{\BoundedLinearOperators}[2][]{B\parens[#1]{#2}} +\newcommand*{\ContinousLinearOperators}[2][]{L\parens[#1]{#2}} \DeclareMathOperator*{\stronglim}{s-lim} \DeclareMathOperator*{\weaklim}{w-lim} diff --git a/sampleappendix.tex b/sampleappendix.tex index 09d68f4..6cdf6cb 100644 --- a/sampleappendix.tex +++ b/sampleappendix.tex @@ -15,5 +15,3 @@ $x \equiv y$ \nocite{*} \cref{lemma:xxx} - -\chapterbib diff --git a/samplesection.tex b/samplesection.tex index 35f1c38..205fa6f 100644 --- a/samplesection.tex +++ b/samplesection.tex @@ -6,5 +6,3 @@ Just some \index{sample text}sample text. \section{Subsection} \section{Another Subsection} - -\chapterbib diff --git a/second.tex b/second.tex index 241d4c1..d7e1a38 100644 --- a/second.tex +++ b/second.tex @@ -4,6 +4,3 @@ First quantization is a mystery, but second quantization is a functor. } Just more text. - -\chapterbib -\cleardoublepage diff --git a/standard.tex b/standard.tex index 93ccdcc..86bd1ab 100644 --- a/standard.tex +++ b/standard.tex @@ -19,6 +19,3 @@ test \begin{definition}{cyclic, separating}{} test \end{definition} - -\chapterbib -\cleardoublepage diff --git a/stresstensor.tex b/stresstensor.tex index 79c0930..70a79ea 100644 --- a/stresstensor.tex +++ b/stresstensor.tex @@ -1087,10 +1087,6 @@ where and $\psi_n \equiv 0$ for $n \ne 2$. \end{proposition} -\begin{equation*} - \energydensity(f) \Omega = ? -\end{equation*} - \section{Essential Selfadjointness of Renormalized Products} \begin{lemma}{H-Bounds for the Renormalized Product}{} @@ -1148,7 +1144,4 @@ where \end{equation*} \end{theorem} -\chapterbib -\cleardoublepage - % vim: syntax=mytex -- cgit v1.2.3-54-g00ecf