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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{*} |