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Diffstat (limited to 'much.tex')
-rw-r--r-- | much.tex | 117 |
1 files changed, 83 insertions, 34 deletions
@@ -50,6 +50,10 @@ Poincaré covariance \section{Basic Concepts of Modular Theory} \index{modular!theory} +It is a distinctive feature of the quantum energy inequality \todo{ref}, which is at the center of our investigation, +that the modular operator $\Delta$ associated to a local algebra of observables and the vacuum vector appears in its lower bound. +In this short section we will review the the definition of $\Delta$ and other basic concepts of the Tomita--Takesaki modular theory of von Neumann algebras. + If $\hilb{H}$ is a Hilbert space we shall denote the $C^*$-algebra of all bounded linear operators on $\hilb{H}$ by $\BoundedLinearOperators{\hilb{H}}$. @@ -159,7 +163,7 @@ as this will be our only use case. Let $T$ be an arbitrary closed anti-linear operator in a Hilbert space $\hilb{H}$. Then there exist a positive selfadjoint linear operator $\abs{T}$ and - a partial anti-linear isometry $U$ +a anti-linear partial isometry $U$ such that \begin{equation*} T = U \abs{T} \qquad \bracks[\big]{\text{in particular, $\Domain{T} = \Domain{\abs{T}}$}}. @@ -194,7 +198,12 @@ Now we are able to introduce the fundamental objects of modular theory. the pair $(\vNa{M},\Omega)$. \end{definition} -\todo{clarify why $J$ is anti-unitary} +The anti-linear partial isometry $J$ satisfies +$(\ker J)^\perp = (\ker S)^\perp$ and $\ran J = \overline{\ran S}$ +by \cref{theorem:polar-decomposition}. +Since $S@@$ is injective and has dense range, +it follows that $J@@$ has $\hilb{H}$ as both initial and final space, +and thus is in fact anti-unitary. \begin{definition}{Modular Group}{} Adopt the notation of the foregoing definition. @@ -522,7 +531,7 @@ there exists a unique projection-valued measure $E$ on $\RR^4$ such that \label{equation:spectral-resolution-translation} U(a) = \int_{\RR^4} \exp(ia \cdot k) \, dE(k) \qquad a \in \RR^4. \end{equation} -Then one can define a vector $P$ of unbounded selfadjoint operators +Then one can define a vector $P = (P_0,P_1,P_2,P_3)$ of unbounded selfadjoint operators \begin{equation*} P_i = \int_{\RR^4} k_i \, dE(k) \qquad i=0,\ldots,3 \end{equation*} @@ -639,7 +648,7 @@ Next we consider an operator-valued tempered distribution $u$ that is \emph{cova in the sense that it obeys the relativistic transformation law \begin{equation} \label{equation:covariance-distribution} - U(g) u(f) U(g)^* = u(f_g) \qquad g \in \RestrictedPoincareGroup, f \in \schwartz{\RR^4}, + U(g) u(f) U(g)^* = u(f_g) \qquad g \in \RestrictedPoincareGroup, f \in \SchwartzFunctions{\RR^4}, \end{equation} where $f_g(x) = f(g^{-1} x)$ for all $x \in M$. In particular, if $g=(a,I)$ is the translation by a vector $a \in \RR^4$, @@ -658,12 +667,12 @@ However, thanks to the spectrum condition we may modify this function outside of \begin{lemma}{}{depends-only-on-restriction} Let $u$ be a covariant operator-valued tempered distribution. - Then the vector $u(f) \FockVacuum$, where $f \in \schwartz{\RR^4}$, + Then the vector $u(f) \FockVacuum$, where $f \in \SchwartzFunctions{\RR^4}$, depends only on the restriction of $\ft{f}$ to $\ClosedForwardCone$. \end{lemma} \begin{proof} - We consider a Schwartz function $g \in \schwartz{\RR^4}$ and + We consider a Schwartz function $g \in \SchwartzFunctions{\RR^4}$ and the operator $G = \int g(k) dE(k)$, where $E$ is the unique projection-valued measure on $\RR^4$ such that $U(a) = \int \exp(ik \cdot a) dE(k)$ for all $a \in \RR^4$. @@ -675,7 +684,7 @@ However, thanks to the spectrum condition we may modify this function outside of \end{multline*} \question{Darf ich hier wirklich die Integrationsreihenfolge vertauschen?} - Recall that the Fourier transform of $u$ is defined by $\ft{u}(f) = u(\ft{f}@@)$ for $f \in \schwartz{\RR^4}$. + Recall that the Fourier transform of $u$ is defined by $\ft{u}(f) = u(\ft{f}@@)$ for $f \in \SchwartzFunctions{\RR^4}$. We obtain the action of the translation group on $\ft{u}(\ft{f}@@)\FockVacuum$ by definition chasing and~\eqref{equation:real-translation-law}: \begin{equation*} U(a) \ft{u}(\ft{f}@@)\FockVacuum @@ -706,13 +715,13 @@ Such a function does exist \todo{elaborate, smooth cutoff}. Then $f_z$ will be S Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$, by~\cref{lemma:depends-only-on-restriction}. %\begin{lemma}{}{} - %For every $z \in \ClosedForwardTube$ there exists a Schwartz function $d_z \in \schwartz{\RR^4}$ - %such that $\ft{e_z} \in \schwartz{\RR^4}$ and $\ft{e_z}(p) = \exp(iz \cdot p)$ for $p \in \ClosedForwardCone$. + %For every $z \in \ClosedForwardTube$ there exists a Schwartz function $d_z \in \SchwartzFunctions{\RR^4}$ + %such that $\ft{e_z} \in \SchwartzFunctions{\RR^4}$ and $\ft{e_z}(p) = \exp(iz \cdot p)$ for $p \in \ClosedForwardCone$. %\end{lemma} \begin{proposition}{}{prp} Let $u$ be a covariant operator-valued tempered distribution, - and let $f \in \schwartz{\RR^4}$ be a test function. Then we have, + and let $f \in \SchwartzFunctions{\RR^4}$ be a test function. Then we have, in generalization of~\eqref{equation:real-translation-law}, \begin{equation*} U(z) u(f) \FockVacuum = u(f_z) \FockVacuum \qquad \forall z \in T_+. @@ -726,14 +735,17 @@ Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$, Dann folgt die Behauptung wohl mit Edge of the Wedge~\cite[Theorem 2-17]{Streater1964}} \end{proof} -\begin{corollary}{}{} +\begin{corollary}{}{convolution2} Let $u$ be a covariant operator-valued tempered distribution, - and let $f \in \schwartz{\RR^4}$ be a test function. Then we have, + and let $f \in \SchwartzFunctions{\RR^4}$ be a test function. Then we have, \begin{equation*} U(z) u(f) \FockVacuum = \int dx \, f(x) \, u(d_{z+x}) \FockVacuum \qquad \forall z \in T_+. \end{equation*} \end{corollary} +As discussed in \cref{chapter:convolution}, +the vector-valued integral on the right-hand side exists in the strong sense of Bochner. + \begin{proof} The convolution formula \cref{proposition:vector-valued-convolution-formula} applied to the vector-valued distribution defined by $f \mapsto \alpha(f) = u(f) \FockVacuum$ yields \begin{equation*} @@ -786,9 +798,7 @@ For later use, we give the action of $\Lambda(is)$ on a complex four-vector $x+i y^3 \end{pmatrix} \end{equation} - -We - +The purely imaginary Lorentz boost matrix \begin{equation*} \mathcal{J} \defequal \Lambda(i/2) = \begin{pmatrix} -1 & 0 & \; 0 \; & \; 0 \; \\ @@ -797,22 +807,23 @@ We 0 & 0 & 0 & 1 \\ \end{pmatrix} \end{equation*} +will play a special role because it maps the right and left wedges onto each other. -\begin{equation*} - \mathcal{J}_{\pm} \defequal \Lambda(\pm i/4) = \begin{pmatrix} - 0 & \pm i & \; 0 \; & \; 0 \; \\ - \pm i & 0 & 0 & 0 \\ - 0 & 0 & 1 & 0 \\ - 0 & 0 & 0 & 1 \\ - \end{pmatrix} -\end{equation*} +%\begin{equation*} + %\mathcal{J}_{\pm} \defequal \Lambda(\pm i/4) = \begin{pmatrix} + %0 & \pm i & \; 0 \; & \; 0 \; \\ + %\pm i & 0 & 0 & 0 \\ + %0 & 0 & 1 & 0 \\ + %0 & 0 & 0 & 1 \\ + %\end{pmatrix} +%\end{equation*} We now turn to the unitary representation of (real) Lorentz boosts \begin{equation*} V(t) \defequal U \parens[\big]{0,\Lambda(t)} \qquad t \in \RR \end{equation*} on Fock space and aim for an analytic extension similar to the previous section. -By Stone's theorem theorem there exists a unique selfadjoint operator $K$ such that +By Stone's Theorem there exists a unique selfadjoint operator $K$ such that \begin{equation*} V(t) = \exp(itK) = \int_{\RR} \exp(it \lambda) \,dE_K(\lambda), \end{equation*} @@ -825,10 +836,10 @@ Now we define \emph{complex Lorentz boosts} to be the operators In contrast to the previous section, we -\begin{lemma}{}{} +\begin{lemma}{}{complex-lorentz-boosts} Suppose $A$ is a selfadjoint unbounded operator on some Hilbert space $\hilb{H}$. For each complex number $z$ define the closed normal operator $V(z) = e^{izA}$ by means of functional calculus. - Let $g \in \schwartz{\RR}$ be a Schwartz function. + Let $g \in \SchwartzFunctions{\RR}$ be a Schwartz function. \begin{enumerate} \item $V(z) V(w) = V(z + w)$ for all $z,w \in \CC$. \item The operator $g(A)$ is bounded, and its range is contained in the domain of $V(z)$ for all $z \in \CC$. @@ -850,7 +861,7 @@ but a dense subspace of $\Domain{T}$ need not be a core for $T$. \begin{lemma}{A Common Core for All Complex Lorentz Boosts}{common-core-for-complex-lorentz-boots} Adopt the notation of the foregoing lemma. The linear subspace \begin{equation*} - \mathcal{D}_0 = \Span \Set{\ran g(K) \given g \in \schwartz{\RR}} + \mathcal{D}_0 = \Span \Set{\ran g(K) \given g \in \SchwartzFunctions{\RR}} \end{equation*} is a core for $V(z)$ for every $z \in \CC$. \end{lemma} @@ -971,18 +982,56 @@ Remember that $\mathcal{J} = \Lambda(i/2) = \diag(-1,-1,1,1)$. \begin{lemma}{}{biso3} Suppose that $u$ is a covariant operator-valued tempered distribution. - Let $f \in \schwartz{M}$ with $\supp f \subset \rightwedge$, and - let $g \in \schwartz{M}$ be arbitrary. Then - \begin{equation*} + Let $f \in \SchwartzFunctions{M}$ with $\supp f \subset \rightwedge$, and + let $g \in \SchwartzFunctions{M}$ be arbitrary. Then + \begin{equation} + \label{equation:biso3-claim} V(i/2) g(K) u(f) \FockVacuum = g(K) u(f_{\mathcal{J}}) \FockVacuum - \end{equation*} + \end{equation} \end{lemma} Here, $K$ is the infinitesimal generator of the group $t \mapsto V(t)$ of real Lorentz boosts, $\FockVacuum$ is the Fock vacuum, and $\mathcal{J}$ is the Lorentz transformation given by the diagonal matrix $\diag(-1,-1,1,1)$. \begin{proof} - a + Instead of~\eqref{equation:biso3-claim} we prove + \begin{equation} + \label{equation:modified-claim} + V(i/4) g(K) u(f) \FockVacuum = V(-i/4) g(K) u(f_{\mathcal{J}}) \FockVacuum. + \end{equation} + This is equivalent due to \cref{lemma:complex-lorentz-boosts}(i). + As before, we write $e_0 = (1,0,0,0)$ for the positive time-like unit vector, + and introduce a complex translation as follows: + \begin{equation} + \label{equation:step1} + u(f) \FockVacuum = \stronglim_{\epsilon \downarrow 0} U(i \epsilon e_0) u(f) \FockVacuum. + \end{equation} + \cref{corollary:convolution2} + \begin{equation} + \label{equation:step2} + U(i \epsilon e_0) u(f) \FockVacuum = + \int dx f(x) u(d_{x + i \epsilon e_0}) \FockVacuum + \end{equation} + The operator $V(i/4) g(K)$ is bounded by Lemma xxx, + and therefore, + when applied to ~\eqref{equation:step1} + can be moved inside the strong limit, + and when applied to ~\eqref{equation:step2} + can be moved inside the integrand using \cref{theorem:integral-commutes-with-operator}. + Taken together, we obtain + \begin{equation} + \label{equation:step3} + V(i/4) g(K) u(f) \FockVacuum = + \stronglim_{\epsilon \downarrow 0} \int dx f(x) V(i/4) g(K) U(i \epsilon e_0) u(d_x) \FockVacuum + \end{equation} + Next we aim to bring the right hand side of~\eqref{equation:step3} into a form where \cref{lemma:biso2} is applicable. + The strong limit commutes with the strong integral (\todo{Why?}). + Moreoverk + \begin{equation*} + V(i/4) g(K) u(f) \FockVacuum = + \int dx f_{\delta}(x) V(i/4) g(K) V(-i/4) \stronglim_{\epsilon \downarrow 0} V(i/4) U(x + i \epsilon e_0) u(d_{\delta}) \FockVacuum + \end{equation*} + Now, performing all transformations in reverse yields~\eqref{equation:modified-claim}, as desired. \end{proof} \begin{equation*} @@ -993,8 +1042,8 @@ Die Anwendung auf die Energiedichte $\energydensity$: \begin{proposition}{}{main-result} Suppose $W \subset M$ is any wedge domain, with associated modular operator $\Delta_W$ and modular Hamiltonian $K_W$. - Let $f \in \schwartz{M}$ with $\supp f \subset W$, and - let $h \in \schwartz{M}$ be arbitrary. Then + Let $f \in \SchwartzFunctions{M}$ with $\supp f \subset W$, and + let $h \in \SchwartzFunctions{M}$ be arbitrary. Then \begin{equation*} \norm{\Delta_W^{-1/2} h(K_W) \energydensity(f) \FockVacuum} = \norm{h(K) \energydensity(f_{\mathcal{J}g}) \FockVacuum}, |