weiterHEADmaster

1 files changed, 83 insertions, 34 deletions

diff --git a/much.tex b/much.tex
index 249dc73..d11b04a 100644
--- a/much.tex
+++ b/much.tex
@@ -50,6 +50,10 @@ Poincaré covariance
 \section{Basic Concepts of Modular Theory}
 \index{modular!theory}
 
+It is a distinctive feature of the quantum energy inequality \todo{ref}, which is at the center of our investigation,
+that the modular operator $\Delta$ associated to a local algebra of observables and the vacuum vector appears in its lower bound.
+In this short section we will review the the definition of $\Delta$ and other basic concepts of the Tomita--Takesaki modular theory of von Neumann algebras.
+
 If $\hilb{H}$ is a Hilbert space
 we shall denote the $C^*$-algebra of all bounded linear operators on $\hilb{H}$ by $\BoundedLinearOperators{\hilb{H}}$.
 
@@ -159,7 +163,7 @@ as this will be our only use case.
   Let $T$ be an arbitrary closed anti-linear operator in a Hilbert space $\hilb{H}$.
   Then there exist
   a positive selfadjoint linear operator $\abs{T}$ and
-  a partial anti-linear isometry $U$
+a anti-linear partial isometry $U$
   such that
   \begin{equation*}
     T = U \abs{T} \qquad \bracks[\big]{\text{in particular, $\Domain{T} = \Domain{\abs{T}}$}}.
@@ -194,7 +198,12 @@ Now we are able to introduce the fundamental objects of modular theory.
   the pair $(\vNa{M},\Omega)$.
 \end{definition}
 
-\todo{clarify why $J$ is anti-unitary}
+The anti-linear partial isometry $J$ satisfies
+$(\ker J)^\perp = (\ker S)^\perp$ and $\ran J = \overline{\ran S}$
+by \cref{theorem:polar-decomposition}.
+Since $S@@$ is injective and has dense range,
+it follows that $J@@$ has $\hilb{H}$ as both initial and final space,
+and thus is in fact anti-unitary.
 
 \begin{definition}{Modular Group}{}
   Adopt the notation of the foregoing definition.
@@ -522,7 +531,7 @@ there exists a unique projection-valued measure $E$ on $\RR^4$ such that
   \label{equation:spectral-resolution-translation}
   U(a) = \int_{\RR^4} \exp(ia \cdot k) \, dE(k) \qquad a \in \RR^4.
 \end{equation}
-Then one can define a vector $P$ of unbounded selfadjoint operators
+Then one can define a vector $P = (P_0,P_1,P_2,P_3)$ of unbounded selfadjoint operators
 \begin{equation*}
   P_i = \int_{\RR^4} k_i \, dE(k) \qquad i=0,\ldots,3
 \end{equation*}
@@ -639,7 +648,7 @@ Next we consider an operator-valued tempered distribution $u$ that is \emph{cova
 in the sense that it obeys the relativistic transformation law
 \begin{equation}
   \label{equation:covariance-distribution}
-  U(g) u(f) U(g)^* = u(f_g) \qquad g \in \RestrictedPoincareGroup, f \in \schwartz{\RR^4},
+  U(g) u(f) U(g)^* = u(f_g) \qquad g \in \RestrictedPoincareGroup, f \in \SchwartzFunctions{\RR^4},
 \end{equation}
 where $f_g(x) = f(g^{-1} x)$ for all $x \in M$.
 In particular, if $g=(a,I)$ is the translation by a vector $a \in \RR^4$,
@@ -658,12 +667,12 @@ However, thanks to the spectrum condition we may modify this function outside of
 
 \begin{lemma}{}{depends-only-on-restriction}
   Let $u$ be a covariant operator-valued tempered distribution.
-  Then the vector $u(f) \FockVacuum$, where $f \in \schwartz{\RR^4}$,
+  Then the vector $u(f) \FockVacuum$, where $f \in \SchwartzFunctions{\RR^4}$,
   depends only on the restriction of $\ft{f}$ to $\ClosedForwardCone$.
 \end{lemma}
 
 \begin{proof}
-  We consider a Schwartz function $g \in \schwartz{\RR^4}$ and
+  We consider a Schwartz function $g \in \SchwartzFunctions{\RR^4}$ and
   the operator $G = \int g(k) dE(k)$,
   where $E$ is the unique projection-valued measure on $\RR^4$ such that
   $U(a) = \int \exp(ik \cdot a) dE(k)$ for all $a \in \RR^4$.
@@ -675,7 +684,7 @@ However, thanks to the spectrum condition we may modify this function outside of
   \end{multline*}
   \question{Darf ich hier wirklich die Integrationsreihenfolge vertauschen?}
 
-  Recall that the Fourier transform of $u$ is defined by $\ft{u}(f) = u(\ft{f}@@)$ for $f \in \schwartz{\RR^4}$.
+  Recall that the Fourier transform of $u$ is defined by $\ft{u}(f) = u(\ft{f}@@)$ for $f \in \SchwartzFunctions{\RR^4}$.
   We obtain the action of the translation group on $\ft{u}(\ft{f}@@)\FockVacuum$ by definition chasing and~\eqref{equation:real-translation-law}:
   \begin{equation*}
     U(a) \ft{u}(\ft{f}@@)\FockVacuum
@@ -706,13 +715,13 @@ Such a function does exist \todo{elaborate, smooth cutoff}. Then $f_z$ will be S
 Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$, by~\cref{lemma:depends-only-on-restriction}.
 
 %\begin{lemma}{}{}
-  %For every $z \in \ClosedForwardTube$ there exists a Schwartz function $d_z \in \schwartz{\RR^4}$
-  %such that $\ft{e_z} \in \schwartz{\RR^4}$ and $\ft{e_z}(p) = \exp(iz \cdot p)$ for $p \in \ClosedForwardCone$.
+  %For every $z \in \ClosedForwardTube$ there exists a Schwartz function $d_z \in \SchwartzFunctions{\RR^4}$
+  %such that $\ft{e_z} \in \SchwartzFunctions{\RR^4}$ and $\ft{e_z}(p) = \exp(iz \cdot p)$ for $p \in \ClosedForwardCone$.
 %\end{lemma}
 
 \begin{proposition}{}{prp}
   Let $u$ be a covariant operator-valued tempered distribution,
-  and let $f \in \schwartz{\RR^4}$ be a test function. Then we have,
+  and let $f \in \SchwartzFunctions{\RR^4}$ be a test function. Then we have,
   in generalization of~\eqref{equation:real-translation-law},
   \begin{equation*}
     U(z) u(f) \FockVacuum = u(f_z) \FockVacuum \qquad \forall z \in T_+.
@@ -726,14 +735,17 @@ Moreover, $u(f_z) \FockVacuum$ does not depend on the specific choice of $d_z$,
   Dann folgt die Behauptung wohl mit Edge of the Wedge~\cite[Theorem 2-17]{Streater1964}}
 \end{proof}
 
-\begin{corollary}{}{}
+\begin{corollary}{}{convolution2}
   Let $u$ be a covariant operator-valued tempered distribution,
-  and let $f \in \schwartz{\RR^4}$ be a test function. Then we have,
+  and let $f \in \SchwartzFunctions{\RR^4}$ be a test function. Then we have,
   \begin{equation*}
     U(z) u(f) \FockVacuum = \int dx \, f(x) \, u(d_{z+x}) \FockVacuum \qquad \forall z \in T_+.
   \end{equation*}
 \end{corollary}
 
+As discussed in \cref{chapter:convolution},
+the vector-valued integral on the right-hand side exists in the strong sense of Bochner.
+
 \begin{proof}
   The convolution formula \cref{proposition:vector-valued-convolution-formula} applied to the vector-valued distribution defined by $f \mapsto \alpha(f) = u(f) \FockVacuum$ yields
   \begin{equation*}
@@ -786,9 +798,7 @@ For later use, we give the action of $\Lambda(is)$ on a complex four-vector $x+i
     y^3 
   \end{pmatrix}
 \end{equation}
-
-We
-
+The purely imaginary Lorentz boost matrix
 \begin{equation*}
   \mathcal{J} \defequal \Lambda(i/2) = \begin{pmatrix}
     -1 & 0 & \; 0 \; & \; 0 \; \\
@@ -797,22 +807,23 @@ We
     0 & 0 & 0 & 1 \\
   \end{pmatrix}
 \end{equation*}
+will play a special role because it maps the right and left wedges onto each other.
 
-\begin{equation*}
-  \mathcal{J}_{\pm} \defequal \Lambda(\pm i/4) = \begin{pmatrix}
-    0 & \pm i & \; 0 \; & \; 0 \; \\
-    \pm i & 0 & 0 & 0 \\
-    0 & 0 & 1 & 0 \\
-    0 & 0 & 0 & 1 \\
-  \end{pmatrix}
-\end{equation*}
+%\begin{equation*}
+  %\mathcal{J}_{\pm} \defequal \Lambda(\pm i/4) = \begin{pmatrix}
+    %0 & \pm i & \; 0 \; & \; 0 \; \\
+    %\pm i & 0 & 0 & 0 \\
+    %0 & 0 & 1 & 0 \\
+    %0 & 0 & 0 & 1 \\
+  %\end{pmatrix}
+%\end{equation*}
 
 We now turn to the unitary representation of (real) Lorentz boosts
 \begin{equation*}
   V(t) \defequal U \parens[\big]{0,\Lambda(t)} \qquad t \in \RR
 \end{equation*}
 on Fock space and aim for an analytic extension similar to the previous section.
-By Stone's theorem theorem there exists a unique selfadjoint operator $K$ such that
+By Stone's Theorem there exists a unique selfadjoint operator $K$ such that
 \begin{equation*}
   V(t) = \exp(itK) = \int_{\RR} \exp(it \lambda) \,dE_K(\lambda),
 \end{equation*}
@@ -825,10 +836,10 @@ Now we define \emph{complex Lorentz boosts} to be the operators
 In contrast to the previous section, we
 
 
-\begin{lemma}{}{}
+\begin{lemma}{}{complex-lorentz-boosts}
   Suppose $A$ is a selfadjoint unbounded operator on some Hilbert space $\hilb{H}$.
   For each complex number $z$ define the closed normal operator $V(z) = e^{izA}$ by means of functional calculus.
-  Let $g \in \schwartz{\RR}$ be a Schwartz function.
+  Let $g \in \SchwartzFunctions{\RR}$ be a Schwartz function.
   \begin{enumerate}
     \item $V(z) V(w) = V(z + w)$ for all $z,w \in \CC$.
     \item The operator $g(A)$ is bounded, and its range is contained in the domain of $V(z)$ for all $z \in \CC$.
@@ -850,7 +861,7 @@ but a dense subspace of $\Domain{T}$ need not be a core for $T$.
 \begin{lemma}{A Common Core for All Complex Lorentz Boosts}{common-core-for-complex-lorentz-boots}
   Adopt the notation of the foregoing lemma. The linear subspace
   \begin{equation*}
-    \mathcal{D}_0 = \Span \Set{\ran g(K) \given g \in \schwartz{\RR}} 
+    \mathcal{D}_0 = \Span \Set{\ran g(K) \given g \in \SchwartzFunctions{\RR}} 
   \end{equation*}
   is a core for $V(z)$ for every $z \in \CC$.
 \end{lemma}
@@ -971,18 +982,56 @@ Remember that $\mathcal{J} = \Lambda(i/2) = \diag(-1,-1,1,1)$.
 
 \begin{lemma}{}{biso3}
   Suppose that $u$ is a covariant operator-valued tempered distribution.
-  Let $f \in \schwartz{M}$ with $\supp f \subset \rightwedge$, and
-  let $g \in \schwartz{M}$ be arbitrary. Then
-  \begin{equation*}
+  Let $f \in \SchwartzFunctions{M}$ with $\supp f \subset \rightwedge$, and
+  let $g \in \SchwartzFunctions{M}$ be arbitrary. Then
+  \begin{equation}
+    \label{equation:biso3-claim}
     V(i/2) g(K) u(f) \FockVacuum = g(K) u(f_{\mathcal{J}}) \FockVacuum
-  \end{equation*}
+  \end{equation}
 \end{lemma}
 
 Here, $K$ is the infinitesimal generator of the group $t \mapsto V(t)$ of real Lorentz boosts,
 $\FockVacuum$ is the Fock vacuum, and $\mathcal{J}$ is the Lorentz transformation given by the diagonal matrix $\diag(-1,-1,1,1)$.
 
 \begin{proof}
-  a
+  Instead of~\eqref{equation:biso3-claim} we prove
+  \begin{equation}
+    \label{equation:modified-claim}
+    V(i/4) g(K) u(f) \FockVacuum = V(-i/4) g(K) u(f_{\mathcal{J}}) \FockVacuum.
+  \end{equation}
+  This is equivalent due to \cref{lemma:complex-lorentz-boosts}(i).
+  As before, we write $e_0 = (1,0,0,0)$ for the positive time-like unit vector,
+  and introduce a complex translation as follows:
+  \begin{equation}
+    \label{equation:step1}
+    u(f) \FockVacuum = \stronglim_{\epsilon \downarrow 0} U(i \epsilon e_0) u(f) \FockVacuum.
+  \end{equation}
+  \cref{corollary:convolution2}
+  \begin{equation}
+    \label{equation:step2}
+    U(i \epsilon e_0) u(f) \FockVacuum =
+    \int dx f(x) u(d_{x + i \epsilon e_0}) \FockVacuum
+  \end{equation}
+  The operator $V(i/4) g(K)$ is bounded by Lemma xxx,
+  and therefore,
+  when applied to ~\eqref{equation:step1}
+  can be moved inside the strong limit,
+  and when applied to ~\eqref{equation:step2}
+  can be moved inside the integrand using \cref{theorem:integral-commutes-with-operator}.
+  Taken together, we obtain
+  \begin{equation}
+    \label{equation:step3}
+    V(i/4) g(K) u(f) \FockVacuum = 
+    \stronglim_{\epsilon \downarrow 0} \int dx f(x) V(i/4) g(K) U(i \epsilon e_0) u(d_x) \FockVacuum 
+  \end{equation}
+  Next we aim to bring the right hand side of~\eqref{equation:step3} into a form where \cref{lemma:biso2} is applicable.
+  The strong limit commutes with the strong integral (\todo{Why?}).
+  Moreoverk
+  \begin{equation*}
+    V(i/4) g(K) u(f) \FockVacuum = 
+    \int dx f_{\delta}(x) V(i/4) g(K) V(-i/4) \stronglim_{\epsilon \downarrow 0} V(i/4) U(x + i \epsilon e_0) u(d_{\delta}) \FockVacuum
+  \end{equation*}
+  Now, performing all transformations in reverse yields~\eqref{equation:modified-claim}, as desired.
 \end{proof}
 
 \begin{equation*}
@@ -993,8 +1042,8 @@ Die Anwendung auf die Energiedichte $\energydensity$:
 
 \begin{proposition}{}{main-result}
   Suppose $W \subset M$ is any wedge domain, with associated modular operator $\Delta_W$ and modular Hamiltonian $K_W$.
-  Let $f \in \schwartz{M}$ with $\supp f \subset W$, and
-  let $h \in \schwartz{M}$ be arbitrary. Then
+  Let $f \in \SchwartzFunctions{M}$ with $\supp f \subset W$, and
+  let $h \in \SchwartzFunctions{M}$ be arbitrary. Then
   \begin{equation*}
     \norm{\Delta_W^{-1/2} h(K_W) \energydensity(f) \FockVacuum}
     = \norm{h(K) \energydensity(f_{\mathcal{J}g}) \FockVacuum},