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Diffstat (limited to 'much.tex')
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1 files changed, 109 insertions, 41 deletions
@@ -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 |