weiterHEADmaster

4 files changed, 209 insertions, 77 deletions

diff --git a/bib/stresstensor.bib b/bib/stresstensor.bib
index c2f835d..38066ee 100644
--- a/bib/stresstensor.bib
+++ b/bib/stresstensor.bib
@@ -1,21 +1,21 @@
 @article{Segal1969,
  author    = {Irving Segal},
- title     = {Nonlinear Functions of Weak Processes. I},
+ title     = {Nonlinear Functions of Weak Processes I},
  publisher = {Elsevier Science},
  journal   = {Journal of Functional Analysis},
  issn      = {0022-1236,1096-0783},
- year      = {1969},
+ date      = {1969},
  volume    = {4},
  number     = {3},
  pages     = {404--456},
 }
 @article{Segal1970,
- title     = {Nonlinear Functions of Weak Processes. II},
+ title     = {Nonlinear Functions of Weak Processes II},
  author    = {Irving Segal},
  publisher = {Elsevier Science},
  journal   = {Journal of Functional Analysis},
  issn      = {0022-1236,1096-0783},
- year      = {1970},
+ date      = {1970},
  volume    = {6},
  number     = {1},
  pages     = {29--75},
@@ -28,9 +28,8 @@
   journal = {Pacific Journal of Mathematics},
   publisher = {Mathematical Sciences Publishers},
   author = {Abel Klein},
-  year = {1973},
-  month = mar,
-  pages = {275–-292}
+  date = {1973-03},
+  pages = {275--292}
 }
 @article{Baez1989,
   title = {Wick Products of the Free Bose Field},
@@ -40,9 +39,8 @@
   journal = {Journal of Functional Analysis},
  publisher = {Elsevier Science},
  author    = {John C. Baez},
-  year = {1989},
-  month = oct,
-  pages = {211–-225}
+  date = {1989-10},
+  pages = {211--225}
 }
 @article{Fewster1998,
  title     = {Bounds on negative energy densities in flat spacetime},
@@ -50,7 +48,7 @@
  publisher = {The American Physical Society},
  journal   = {Physical Review D},
  issn      = {1550-7998,1089-4918},
- year      = {1998},
+ date      = {1998},
  volume    = {58},
  number     = {8},
  pages     = {084010},
diff --git a/commutatortheorem.tex b/commutatortheorem.tex
new file mode 100644
index 0000000..9f6384f
--- /dev/null
+++ b/commutatortheorem.tex
@@ -0,0 +1,21 @@
+%\chapter[Scales of Hilbert Spaces and Nelsons Commutator Theorem]{Scales of Hilbert Spaces\\ and Nelsons Commutator Theorem}
+\chapter{Scales of Hilbert Spaces and~Nelsons~Commutator~Theorem}
+
+\begin{theorem}{Nelson Commutator Theorem}{nelson-commutator-theorem}
+  Let $N$ be a selfadjoint operator with $N \ge I$,
+  and let $q$ be a quadratic form with the same domain.
+  Suppose that there exist constants $c_1,c_2$ such that
+  \begin{align*}
+    \abs{q(\psi',\psi)} &\le c_1 \norm{N^{1/2}\psi} \norm{N^{1/2}\psi'} & \forall \psi,\psi' \in D(N^{1/2}) \\
+    \abs{q(N\psi',\psi)-q(\psi',N\psi)} &\le c_2 \norm{N^{1/2}\psi} \norm{N^{1/2}\psi'} & \forall \psi,\psi' \in D(N^{3/2})
+  \end{align*}
+    Then the operator $q_{\mathrm{op}}$ associated to $q$ is defined on the domain of $N$
+      and is essentially selfadjoint on any core for $N$.
+\end{theorem}
+
+\cite{ReedSimon2} 
+\cite{Nelson1972} 
+
+\chapterbib
+
+%vim: syntax=mytex
diff --git a/main.tex b/main.tex
index d09ac92..fc44017 100644
--- a/main.tex
+++ b/main.tex
@@ -1,5 +1,5 @@
 \input{preamble}
-\includeonly{stresstensor,fewstereveson}
+\includeonly{stresstensor,fewstereveson,commutatortheorem,index}
 \begin{document}
 \frontmatter
 \include{titlepage}
@@ -15,6 +15,7 @@
 \include{samplesection}
 \appendix
 \include{sampleappendix}
+\include{commutatortheorem}
 \backmatter
 \include{bibliography}
 \include{symbols}
diff --git a/stresstensor.tex b/stresstensor.tex
index 39c45cd..a4bb6fb 100644
--- a/stresstensor.tex
+++ b/stresstensor.tex
@@ -1,4 +1,4 @@
-\chapter{Construction of the Stress Tensor of a Free Scalar Quantum Field}
+\chapter{Construction of the Stress Tensor of~a~Free~Scalar~Quantum~Field}
 \label{chapter:stress-tensor}
 
 \begin{center}
@@ -88,46 +88,14 @@ as a service to the reader.
     \begin{equation*}
       \realschwartz{M} \ni f \mapsto \Phi(f) = \Phi_{\mathrm{S}}(Ef) = \frac{1}{\sqrt{2}} \parens*{a(Ef) + a(Ef)^\dagger}
     \end{equation*}
-  \item annihilation and creation operators, $f \in \schwartz{M}$, $\psi \in \BosonFock{\hilb{H}}$
+  \item
+    annihilation and creation operators, $f \in \schwartz{M}$, $\psi \in \BosonFock{\hilb{H}}$
     \begin{align*}
       \parens[\big]{a(f) \psi} {}_n (k_1, \ldots, k_n)
       &= \sqrt{n+1} \int_M \! \overline{Ef(p)} \, \psi_{n+1} (p,k_1, \ldots, k_n) \, d\Omega_m(p) \\
       \parens[\big]{a(f)^\dagger \psi} {}_n (k_1, \ldots, k_n)
       &= \frac{1}{\sqrt{n}} \sum_{i=1}^n Ef(k_i) \, \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n)
     \end{align*}
-    annihilation operator a point $p$ in momentum space.
-    \begin{equation*}
-      \parens[\big]{a(p) \psi} {}_n (k_1, \ldots, k_n)
-      = \sqrt{n+1} \, \psi_{n+1} (p,k_1, \ldots, k_n)
-    \end{equation*}
-    creation \enquote{operator} a point $p$ in momentum space.
-    \begin{equation*}
-      \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*}
-    \begin{gather*}
-      a(p)^\dagger : F \times F \longrightarrow \CC \\
-      \innerp[\big]{\psi'}{a(p)^\dagger \psi}
-      \defequal
-      \begin{multlined}[t]
-        \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \sum_{i=1}^n
-        \int_M \! d\Omega_m(k_1) \cdots \widehat{d\Omega_m(k_i)} \cdots d\Omega_m(k_n) \\
-        \cdot \overline{\psi'_{n} (k_1, \ldots, \underset{i}{p}, \ldots, k_n)}
-        \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n)
-      \end{multlined}
-    \end{gather*}
-    \begin{equation*}
-      \innerp[\big]{\psi'}{a(p)^\dagger \psi} =
-      \innerp[\big]{a(p) \psi'}{\psi}
-    \end{equation*}
-    TODO(Explain why creation op at a point is actually not an op but a QF)
-    \begin{equation*}
-      a(p_1)^\dagger \cdots a(p_s)^\dagger a(p_{s+1}) \cdots a(p_r)
-    \end{equation*}
-    \begin{equation*}
-      \innerp[\big]{\psi'}{a(p_1)^\dagger \cdots a(p_s)^\dagger a(p_{s+1}) \cdots a(p_r) \psi}
-      = \innerp[\big]{a(p_1) \cdots a(p_s) \psi'}{a(p_{s+1}) \cdots a(p_r) \psi}
-    \end{equation*}
 \end{itemize}
 
 \begin{equation*}
@@ -180,6 +148,122 @@ into annihilation operators on the left.
   \end{multlined}
 \end{align*}
 
+\section{Quadratic Forms}
+
+In a typical physics literature treatment of second quantization,
+the annihilation and creation operators and the quantized field are treated as
+point-dependent operator valued functions $a(p)$, $a^\dagger(p)$, $\varphi(x)$,
+disregarding the fact that these may not be operators, in a strict sense, and without smearing with a test function.
+Nonetheless, this notational fiction is useful, and we can uphold it with little effort by giving
+the pointwise \enquote{operators} rigorous meaning as quadratic forms.
+
+Given a point $p$ in momentum space,
+we define the annihilation operator $a(p)$ with domain TODO by
+\begin{equation*}
+  \parens[\big]{a(p) \psi} {}_n (k_1, \ldots, k_n)
+  = \sqrt{n+1} \, \psi_{n+1} (k_1, \ldots, k_n,p)
+\end{equation*}
+The issue arises when one looks for an adjoint to this operator.
+A formal calculation based on the adjoint identity
+\begin{equation}
+  \label{equation:adjoint-identity}
+  \innerp[\big]{\psi'}{a(p)^\dagger \psi} =
+  \innerp[\big]{a(p) \psi'}{\psi}
+\end{equation}
+leads to
+\begin{equation}
+  \label{equation:creation-operator-at-point}
+  \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
+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
+formally calculate the inner product of $\psi'$ with~\eqref{equation:creation-operator-at-point} 
+and we use the result to define the $a^\dagger(p)$
+as a mapping that assigns a number to each \emph{pair} of states.
+That is, we define the creation \enquote{operator} $a^\dagger(p)$
+to be the quadratic form
+\begin{gather*}
+  a(p)^\dagger : F \times F \longrightarrow \CC \\
+  \innerp[\big]{\psi'}{a(p)^\dagger \psi}
+  \defequal
+  \begin{multlined}[t]
+    \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \sum_{i=1}^n
+    \int_M \! d\Omega_m(k_1) \cdots \widehat{d\Omega_m(k_i)} \cdots d\Omega_m(k_n) \\
+    \cdot \overline{\psi'_{n} (k_1, \ldots, \underset{i}{p}, \ldots, k_n)}
+    \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n).
+  \end{multlined}
+\end{gather*}
+One can verify directly that
+with this definition the adjoint identity~\eqref{equation:adjoint-identity} 
+holds for all $\psi,\psi' \in F$.
+For completeness, we give a precise definition of quadratic form.
+
+\begin{definition}{Quadratic Form}{}
+  A \emph{quadratic form}\index{quadratic form} $q$ on a complex Hilbert space $\hilb{H}$ is a mapping
+  \begin{equation*}
+    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
+  and linear in its second argument (i.e.\ sesquilinear).
+  We say that $q$ is \emph{densely defined}
+  if $D(q)$ is dense in $\hilb{H}$.
+\end{definition}
+
+Any linear operator on a complex Hilbert space $\hilb{H}$ has an
+obvious interpretation as a quadratic form on $\hilb{H}$,
+and the form domain agrees with the domain of the operator.
+The reverse construction is always possible for densely defined quadratic forms,
+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 $q_{\mathrm{op}}$,
+  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}},
+  \end{equation*}
+  and maps $\psi \in D(q_{\mathrm{op}})$ to the vector $q_{\mathrm{op}}\psi$ in $\hilb{H}$ satisfying
+  $q(\psi',\psi) = \innerp{\psi'}{q_{\mathrm{op}}\psi}$,
+  which exists and is unique by Riesz’s Representation Theorem.
+\end{definition}
+
+We will use the symbol $\QFequal$ between quadratic forms or operators
+to indicate their equality as quadratic forms.
+TODO(statement about domains?)
+
+
+A natural question is how the smeared operators relate to the pointwise ones.
+
+\begin{equation*}
+  a(g) \QFequal \int \overline{g(p)} a(p) \, d\Omega_m(p)
+\end{equation*}
+
+\begin{equation*}
+  a^\dagger(g) \QFequal \int g(p) a^\dagger(p) \, d\Omega_m(p)
+\end{equation*}
+We have to explain what is meant by the integral on the right hand side.
+Suppose $q(p)$ is a quadratic form on $\BosonFock{\hilb{H}}$ for each $q \in \RR^4$,
+that share a common domain $D \subset D(q(p))$,
+and $g$ is in $\hilb{H} = L^2(\RR^4,\Omega_m)$. Then we define
+a quadratic form by
+\begin{equation*}
+  \parens{\int g(p) q(p) \, d\Omega_m(p)}(\psi',\psi)
+  = \int g(p) \parens{q(p)}(\psi',\psi) \, d \Omega_m(p)
+\end{equation*}
+for all $\psi,\psi' \in D$.
+
+\begin{equation*}
+  a(p_1)^\dagger \cdots a(p_s)^\dagger a(p_{s+1}) \cdots a(p_r)
+\end{equation*}
+\begin{equation*}
+  \innerp[\big]{\psi'}{a(p_1)^\dagger \cdots a(p_s)^\dagger a(p_{s+1}) \cdots a(p_r) \psi}
+  = \innerp[\big]{a(p_1) \cdots a(p_s) \psi'}{a(p_{s+1}) \cdots a(p_r) \psi}
+\end{equation*}
+
 \section{Normal Ordering}
 % The Renormalization Map?
 
@@ -229,7 +313,7 @@ where $z,z' \in \hilb{H}$.
 
 \begin{definition}{Infinitesimal Weyl Algebra}{}
   Let $\hilb{H}$ be a complex Hilbert space.
-  The \emph{infinitesimal Weyl algebra} $\WeylAlg(\hilb{H})$ over $\hilb{H}$
+  The \emph{infinitesimal Weyl algebra}\index{infinitesimal Weyl algebra} $\WeylAlg(\hilb{H})$ over $\hilb{H}$
   is the noncommutative associative algebra over $\CC$
   generated by the elements of $\hilb{H}$, with the relations
   \begin{equation*}
@@ -275,7 +359,7 @@ The set of all monomials in $\WeylAlg$ is denoted $\Mon(\WeylAlg)$.
     \prod_{i \in I\vphantom{\lbrace\rbrace}} \weylcreator(z_i)
     \prod_{\mathclap{j \in \braces{1,\ldots,r} \setminus I}} \weylannihilator(z_j),
   \end{gather}
-  is called the \emph{normal} (or \emph{Wick}) \emph{ordering} on $\hilb{H}$.
+  is called the \emph{normal} (or \emph{Wick}) \emph{ordering}\index{normal ordering}\index{Wick ordering} on $\hilb{H}$.
   A monomial $z_1 \cdots z_r \in \Mon(\WeylAlg)$ is said to be in \emph{normal} (or \emph{Wick}) \emph{order},
   if $\normord{z_1 \cdots z_r} = z_1 \cdots z_r$.
 \end{definition}
@@ -414,7 +498,7 @@ Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex)
   \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} a(D^\dagger_{\sigma(j)}f) 
 \end{gather}
 
-\section{Renormalized Products of the Free Field and its Derivatives}
+\section{Renormalized Products of the Free Field and~its~Derivatives}
 
 For any given test function $f \in \schwartz{M}$ the renormalized product $\normord{D_1 \varphi(f) \cdots D_r \varphi(f)}$
 is well defined as a Fock space operator, but the product is \emph{not} an operator-valued distribution, unless $r=1$.
@@ -515,19 +599,19 @@ TODO(Note about the remaining dependence of $K$ on $f$.)
   and combine all terms depending on $\sigma$ into $P_s$.
 \end{myproof}
 
-In the special case that $D_1 = \cdots = D_n = D$ we have
+In the special case that $D_1 = \cdots = D_r = D$ we have
 \begin{equation*}
   P_s(p_1,\ldots,p_r) =
   \frac{1}{\sqrt{2^r}} \parens*{r \atop s\vphantom{y}}
   \ft{D}(p_1) \cdots \ft{D}(p_s)
   \overline{\ft{D}(p_{s+1}) \cdots \ft{D}(p_r)}.
 \end{equation*}
-For squares, that is $r=2$
+In particular, for squares ($r=2$) we have
 \begin{equation*}
   P_s(p_1,p_2) = \begin{cases}
-    \tfrac{1}{2} \, \ft{D}(p_1)\ft{D}(p_2) & s=0 \\
-    \phantom{\tfrac{1}{2}} \, \ft{D}(p_1)\overline{\ft{D}(p_2)} & s=1 \\
-    \tfrac{1}{2} \, \overline{\ft{D}(p_1)\ft{D}(p_2)} & s=2
+    \tfrac{1}{2} \, \overline{\ft{D}(p_1)\ft{D}(p_2)} & s=0 \\
+    \phantom{\tfrac{1}{2}} \, \overline{\ft{D}(p_1)}\ft{D}(p_2) & s=1 \\
+    \tfrac{1}{2} \, \ft{D}(p_1)\ft{D}(p_2) & s=2
   \end{cases}
 \end{equation*}
 
@@ -700,35 +784,24 @@ In the following proof it will we convenient to use the abbreviation
   \end{gather}
   The estimates~\eqref{equation:polynomial-estimate} and~\eqref{equation:one-plus-omega-estimate} entail
   \begin{equation*}
-    \abs{F(k,p',p)} C_s \le
+    \abs{F(k,p',p)} \le C_s
     \prod_{i=1}^{s} \omega(p_i)^{d_i-a/s} 
-    \prod_{j=s}^{r-s} \omega(p_j)^{d_j-a/(r-s)} 
+    \prod_{j=s+1}^{r} \omega(p_j)^{d_j-a/(r-s)} 
   \end{equation*}
 \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 $D^l(H)$.
+  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}
-    = \int dp_1 \!\cdots dp_r
-    \, K_{\psi'\!,\psi}(p_1,\ldots,p_r)
   \end{equation*}
+  exists and depends continously on $x$.
 \end{lemma}
 
-\begin{definition}{Renormalized Product at a Point}{}
-  In the setting of \cref{lemma:renormalized-product-integral-representation},
-  \begin{equation*}
-    \normord{D_1 \varphi \cdots D_r \varphi} \ \vcentcolon \ 
-    M \to \QF{fock}
-  \end{equation*}
-  \begin{equation*}
-    \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(x) \,\psi}
-    = \int dp_1 \!\cdots dp_r
-    \, K_{\psi'\!,\psi}(p_1,\ldots,p_r)
-  \end{equation*}
-\end{definition}
-
 \begin{proof}
   According to \cref{lemma:renormalized-product-integral-representation} we have
   \begin{equation*}
@@ -745,6 +818,20 @@ In the following proof it will we convenient to use the abbreviation
   The Dominated Convergence Theorem implies
 \end{proof}
 
+\begin{definition}{Renormalized Product at a Point}{}
+  In the setting of \cref{lemma:renormalized-product-integral-representation},
+  the mapping defined by
+  \begin{gather*}
+    \normord{D_1 \varphi \cdots D_r \varphi} \ \vcentcolon \ 
+    M \to \QF{fock} \\
+    \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(x) \,\psi}
+    = \lim_{f \to \delta_x}
+    \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi}
+  \end{gather*}
+  is called the xxx
+\end{definition}
+
+
 \begin{lemma}{Renormalized Product as a QF-valued distribution}{}
   In the setting of \cref{lemma:renormalized-product-integral-representation},
   \begin{equation*}
@@ -759,7 +846,8 @@ 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 $\psi,\psi'$
+  Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients.
+  Then we have for all states $\psi,\psi' \in D^l(H)$
   \begin{multline*}
     \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f) \,\psi} = \\
     = \int dp_1 \!\cdots dp_r
@@ -834,13 +922,37 @@ where
   \ \psi_n(k_1,\ldots,k_{n-(r-s)},p_{s+1},\ldots,p_r)
 \end{multline*}
 
-\begin{equation*}
-  A \QFequal B
-\end{equation*}
+\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 $\psi,\psi' \in D^l(H)$
+  \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
+  \begin{multline*}
+    \quad P_s(p_1,\ldots,p_r) =
+    \frac{1}{\sqrt{2^r}}
+    \frac{1}{s!(r-s)!}
+    \sum_{\sigma \in S_r}
+    \ft{D}_{\sigma(1)}(p_1) \cdots \ft{D}_{\sigma(s)}(p_s) \hspace{1.5cm} \\[-1.5ex]
+    \cdot \overline{\ft{D}_{\sigma(s+1)}(p_{s+1}) \cdots \ft{D}_{\sigma(r)}(p_r)}.
+  \end{multline*}  
+\end{theorem}
+
+\begin{definition}{}{}
+  \begin{multline*}
+    \rho(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}
 
 \section{Essential Selfadjointness of Renormalized Products}
 
-TODO
+a
 
 %\nocite{*}