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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-04-01 13:00:04 +0200 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-04-01 13:00:04 +0200 |
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diff --git a/stresstensor.tex b/stresstensor.tex new file mode 100644 index 0000000..4cdc321 --- /dev/null +++ b/stresstensor.tex @@ -0,0 +1,600 @@ +\chapter{Construction of the Stress Tensor of a Free Scalar Quantum Field} +\label{chapter:stress-tensor} + +\begin{equation*} + H = \tfrac{1}{2} \parens*{(\partial_t \phi)^2 + \abs{\nabla_{\!\!\symbfit{x}} \phi}^2 + m^2 \phi^2} +\end{equation*} + +At the end of this \namecref{chapter:stress-tensor} +we will have gained the ability to rigorously define +arbitrary renormalized products of the free field and its derivatives +as a densely defined quadratic-form valued tempered distribution, +which on the dense subspace of the smooth vectors of the Hamiltonian +is realized by essentially self-adjoint operators. + +\section{Choosing Conventions and Fixing Notation} +\label{section:conventions} + +It is an unfortunate reality of quantum physics literature that +there is a great deal of variation in notation and choice of signs and constants. +While this does not affect the physical or mathematical content, +it is a hindrance when working with formulas from multiple sources. +In the present \namecref{section:conventions} we detail our choices +as a service to the reader. + +\begin{itemize} + \item \emph{Minkowski space} $M=\RR^4$ equipped with the \emph{Lorentz bilinear form} (or metric) + \begin{equation*} + x \cdot y = g_{\mu \nu} x^{\mu} y^{\nu} = x^0y^0 - x^1 y^1 - x^2 y^2 - x^3 y^3 + \end{equation*} + points $x = (x^0,x^1,x^2,x^3) \in M$ are sometimes written $x = (x^0,\symbfit{x})$ with separated time and space coordinates + \item Given a complex-valued function $f$ on $M$, we define its \emph{Fourier transform} $\ft{f}\,$ by + \begin{equation} + \label{fourier-transform} + \hat{f}(p) = \frac{1}{(2 \pi)^2} \int_{M} e^{i p \cdot x} f(x) \, dx + \end{equation} + whenever the integral converges. The \emph{inverse Fourier transform} is TODO + \item To a mathematician $\overline{\phantom{z}}$ usually means complex conjugation and ${}^*$ indicates the Hilbert adjoint of an operator, + while a physicist may read ${}^*$ as complex conjugation and + denotes the Hilbert adjoint with ${}^{\dagger}$. + We avoid confusion by using $\overline{\phantom{z}}$ for complex conjugation + and ${}^{\dagger}$ for the Hilbert adjoint exclusively. + \item $\schwartz{M}$ is the space of complex-valued Schwartz functions on $M$ \\ + $\realschwartz{M}$ is the space of real-valued Schwartz functions on $M$ + \item The \emph{Bosonic Fock space} over a Hilbert space $\hilb{H}$ is denoted $\BosonFock{\hilb{H}}$. \\ + Its \emph{finite particle subspace} is denoted $\BosonFockFinite{\hilb{H}}$. + \item Abstract free field: The \emph{Segal quantization} $\Phi$ assigns to every + $g \in \hilb{H}$, a selfadjoint (unbounded) operator $\Phi(g)$ in $\BosonFock{\hilb{H} }$, + which on the the finite particle subspace is given by + \begin{equation*} + \Phi_{\mathrm{S}} (g) = \frac{1}{\sqrt{2}} \parens*{a(g) + a(g)^\dagger} + \end{equation*} + annihilation and creation operators, $g \in \hilb{H}$, $\psi \in \BosonFock{\hilb{H}}$ for $\hilb{H} = L^2(R^4,\Omega_m)$ + \begin{align*} + \parens[\big]{a(g) \psi} {}_n (k_1, \ldots, k_n) + &= \sqrt{n+1} \int_M \! \bar{g}(p) \, \psi_{n+1} (p,k_1, \ldots, k_n) \, d\Omega_m(p) \\ + \parens[\big]{a(g)^\dagger \psi} {}_n (k_1, \ldots, k_n) + &= \frac{1}{\sqrt{n}} \sum_{i=1}^n g(k_i) \, \psi_{n-1} (k_1, \ldots, \widehat{k_i}, \ldots, k_n) + \end{align*} + The symbol $\widehat{\hphantom{k_i}}$ over $k_i$ indicates omission. + \begin{align*} + \parens[\big]{a(g) \psi} {}_{n-1} (k_1, \ldots, k_{n-1}) + &= \sqrt{n} \int_M \! \bar{g}(p) \, \psi_n (p,k_1, \ldots, k_{n-1}) \, d\Omega_m(p) \\ + \parens[\big]{a(g)^\dagger \psi} {}_{n+1} (k_1, \ldots, k_{n+1}) + &= \frac{1}{\sqrt{n+1}} \sum_{i=1}^{n+1} g(k_i) \, \psi_n (k_1, \ldots, \widehat{k_i}, \ldots, k_{n+1}) + \end{align*} + \begin{multline*} + \parens[\big]{a(g) a(g) \psi} {}_{n-2} (k_1, \ldots, k_{n-2}) = \\ + \sqrt{n} \sqrt{n-1} \int_M \int_M \! \bar{g}(p_1) \bar{g}(p_2) \, \psi_n (p_1,p_2,k_1, \ldots, k_{n-s}) \, d\Omega_m(p_1) d\Omega_m(p_2) \\ + \end{multline*} + \begin{multline} + \parens[\big]{a(g_1) \cdots a(g_s) \psi} {}_{n-s} (k_1, \ldots, k_{n-s}) = + \sqrt{n (n-1) \cdots (n-s+1)} \cdot {} \\ + \cdot \int_M \!\! d\Omega_m(p_1) \cdots \!\! \int_M \!\! d\Omega_m(p_s) \ \bar{g_1}(p_1) \cdots \bar{g_s}(p_s) \ \psi_n (k_1, \ldots, k_{n-2},p_1,\ldots,p_s) + \end{multline} + \item We consider the free Hermitian scalar field of mass $m > 0$. \\ + \emph{mass hyperboloid} $X_m^+ = \braces{p \in M \mid p^2 = m^2, p^0 > 0 }$ + with normalized Lorentz invariant measure $\Omega_m$ + \item single particle state space: $\hilb{H} = L^2(X_m^+, \Omega_m)$ + \begin{equation*} + E : \schwartz{M} \to \hilb{H}, \quad f \mapsto Ef = \left.\ft{f}\,\right\vert {X_m^+} + \end{equation*} + \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, $f \in \schwartz{M}$, $\psi \in \BosonFock{\hilb{H}}$ + \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*} + Define ... + \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*} + abc +\end{itemize} + +\begin{equation*} + \normord{\varphi(f)^2} = \tfrac{1}{2} \parens[\big]{a^{\dagger}(Ef) a^{\dagger}(Ef) + a(Ef) a(Ef)} + a^{\dagger}(Ef) a(Ef) +\end{equation*} + +\begin{equation*} + \normord{\varphi(f)^2} = \tfrac{1}{2} \parens[\big]{a^{\dagger} a^{\dagger} + a a} + a^{\dagger} a \quad \text{where} \quad a = a(Ef) +\end{equation*} + +\begin{equation*} + \innerp{\psi'}{\normord{\varphi(f)^2} \,\psi} = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \innerp{\psi'_m}{\normord{\varphi(f)^2} \,\psi_n} +\end{equation*} + +There will be no contribution to the sum +unless either $m=n+2$ or $m=n$ or $m=n-2$. +Now let us calculate those contributions. +To avoid the introduction of sums, +we use the adjoint identity +to transform creation operators on the right +into annihilation operators on the left. + +\begin{align*} + \innerp{\psi'_{n+2}}{\normord{\varphi(f)^2} \,\psi_n} + &= \tfrac{1}{2} \innerp{a(Ef) a(Ef) \psi'_{n+2}}{\psi_n} \\ + &= \begin{multlined}[t][10cm] + \tfrac{1}{2} \sqrt{n+2} \sqrt{n+1} \int dp_1 dp_2 \, \ft{f}(p_1) \ft{f}(p_2) \int dk_1 \cdots dk_n \\ + \overline{\psi'_{n+2}(p_1,p_2,k_1,\ldots,k_n)} \, \psi_n(k_1,\ldots,k_n) + \end{multlined} \\[1ex] + \innerp{\psi'_{n}}{\normord{\varphi(f)^2} \,\psi_n} + &= \innerp{a(Ef) \psi'_{n}}{a(Ef) \psi_n} \\ + &= \begin{multlined}[t][10cm] + \sqrt{n} \sqrt{n} \int dp_1 dp_2 \, \ft{f}(p_1) \overline{\ft{f}(p_2)} \int dk_1 \cdots dk_{n-1} \\ + \overline{\psi'_{n}(p_1,k_1,\ldots,k_{n-1})} \, \psi_n(p_2,k_1,\ldots,k_{n-1}) + \end{multlined} \\[1ex] + \innerp{\psi'_{n-2}}{\normord{\varphi(f)^2} \,\psi_n} + &= \tfrac{1}{2} \innerp{\psi'_{n-2}}{a(Ef) a(Ef) \psi_n} \\ + &= \begin{multlined}[t][10cm] + \tfrac{1}{2} \sqrt{n} \sqrt{n-1} \int dp_1 dp_2 \, \overline{\ft{f}(p_1)} \overline{\ft{f}(p_2)} \int dk_1 \cdots dk_{n-2} \\ + \overline{\psi'_{n-2}(k_1,\ldots,k_{n-2})} \psi_n(p_1,p_2,k_1,\ldots,k_{n-2}) + \end{multlined} +\end{align*} + +\begin{align*} + \innerp{\psi'_{m}}{\normord{\varphi(f)^2} \,\psi_n} + &= \begin{multlined}[t][10cm] + \tfrac{1}{2} \sqrt{n+2} \sqrt{n+1} \int dp_1 dp_2 \, \ft{f}(p_1) \ft{f}(p_2) \int dk_1 \cdots dk_n \\ + \chi(p_{s+1}) \cdots \chi(p_2) \\ + \overline{\psi'_{m}\parens{k_1,\ldots,k_{n-s},p_1,\ldots,p_s}} \, \psi_n\parens{k_1,\ldots,k_{n-(2-s)},p_{s+1},\ldots,p_2} + \end{multlined} +\end{align*} + +\begin{proposition}{}{} + asdf +\end{proposition} + +\subsubsection{Linear Differential Operators and their Formal Adjoint} + +Before we turn to the problem of defining renormalized products of a quantum field and its derivatives +we must clarify what is meant mathematically by the derivative of a field. +For this, we recall that in Wightmans approach to quantum field theory, +a quantum field $\varphi$ on a spacetime manifold $M$ is modeled by an operator valued tempered distribution, +that is a mapping that assigns to each (Schwatz class) test function $f$ on $M$ an unbounded operator $\varphi(f)$ +in the Fock space xxx over some Hilbert space $\hilb{H}$, such that for each fixed pair of states $\psi,\psi'$ +the mapping +\begin{equation*} + \schwartz{M} \to \CC, \quad + f \mapsto \innerp{\psi'}{\varphi(f) \psi} +\end{equation*} +is a (scalar-valued) tempered distibution on $M$. +It is well known that tempered distibutions have partial derivatives of any order. +Suppose we work with $M = \RR^d$ for simplicity, +and let $\partial_i$ denote the partial derivative with respect to the $i$-th coordinate. +Then a general \emph{linear differential operator with constant coefficients} on $M$ looks like +\begin{equation*} + D = \sum_{\alpha} a_{\alpha} \partial^{\alpha}, +\end{equation*} +where the sum runs over all multi-indices $\alpha = (\alpha_1,\ldots,\alpha_d) \in \NN^d$, +the coefficients $a_{\alpha}$ are complex numbers, +and $\partial^{\alpha} = \partial_1^{\alpha_1} \!\cdots \partial_d^{\alpha_d}$. +Then the \emph{distributional derivative} of a tempered distribution $\eta \in \tempdistrib{\RR^d}$ +is defined by +\begin{equation*} + (D\eta)(f) = \eta(D^{\dagger}f) \quad \forall f \in \mathcal{S}, +\end{equation*} +where the \emph{formal adjoint} of $D$ is the linear differential operator with constant coefficients given by +\begin{equation*} + D^\dagger = \sum_{\alpha} (-1)^{\abs{\alpha}} a_{\alpha} \partial^{\alpha}. +\end{equation*} +Here we use the notation $\abs{\alpha} = \alpha_1 + \cdots + \alpha_d$. +The functional $D \eta$ is well defined, because the Schwartz class is stable under the application of +linear differential operators with constant coefficients. +It can be shown that $D \eta$ is again a tempered distribution. +The appearance of $-1$ in $D^{\dagger}$ is justified by the adjoint identity +\begin{equation*} + \int (Df)(x) g(x) dx = \int f(x) (D^{\dagger}g)(x) dx, +\end{equation*} +which holds for all functions $f,g \in \schwartz{\RR^d}$ +and may be obtained via integration by parts. + +Naturally, we now define the \emph{distributional derivative} of the field by +\begin{equation*} + D \varphi(f) = \varphi(D^{\dagger} f) \qquad \forall f \in \schwartz{\RR^d} +\end{equation*} +As one expects, $D\varphi$ is an operator-valued tempered distribution on $M=\RR^d$. +TODO +\begin{equation} + \label{derivative-free-field} + D \varphi(f) = \frac{1}{\sqrt{2}} \parens*{a(ED^{\dagger}f)^{\dagger} + a(ED^{\dagger}f)} +\end{equation} + + +The operator corresponding to $D$ in Fourier space is the multiplication operator +\begin{equation*} + -i \sum_{\alpha} a_{\alpha} p_0^{\alpha_0} (-p_1)^{\alpha_1} (-p_2)^{\alpha_2} (-p_3)^{\alpha_3} +\end{equation*} + + +Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients. +\begin{equation*} + \normord{D_1 \varphi(f) \cdots D_r \varphi(f)} +\end{equation*} + +\section{Renormalized Products of the Free Field and its Derivatives} + + + +\begin{lemma}{Integral Representation of the Renormalized Product}{renormalized-product-integral-representation} + 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 $\psi,\psi'$ + \begin{equation*} + \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} = + \int dp_1 \!\cdots dp_r + \, \ft{f}(p_1) \cdots\! \ft{f}(p_r) + \, K_{\psi'\!,\psi}(p_1,\ldots,p_r) + \end{equation*} + where the \enquote{integral kernel} is given by + \begin{multline*} + K_{\psi'\!,\psi}(p_1,\ldots,p_r) = + \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{s=0}^{r} + \delta_{m-s}^{n-(r-s)} + \ \chi(p_{s+1}) \cdots \chi(p_{r}) + \ P_s(p_1,\ldots,p_r) \\ + \cdot \sqrt{m(m-1) \cdots (m-s+1)} \sqrt{n(n-1) \cdots (n-(r-s)+1)} \\ + \cdot \int dk_1 \cdots dk_{m-s} + \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)} + \ \psi_n(k_1,\ldots,k_{n-(r-s)},p_{s+1},\ldots,p_r) + \end{multline*} + where $\chi(p) = \overline{\ft{f}(p)} / \ft{f}(p)$ and + \begin{equation*} + P_s(p_1,\ldots,p_r) = + \frac{1}{\sqrt{2^r}} \sum_{\sigma \in S_r} + \ft{D}_{\sigma(1)}(p_1) \cdots \ft{D}_{\sigma(s)}(p_s) + \overline{\ft{D}_{\sigma(s+1)}(p_{s+1}) \cdots \ft{D}_{\sigma(r)}(p_r)}. + \end{equation*} +\end{lemma} + +In the special case that $D_1 = \cdots = D_n = D$ we have +\begin{equation*} + P_s(p_1,\ldots,p_r) = + \sqrt{2^r} + \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 $r=2$ +\begin{equation*} + P_s(p_1,p_2) = \begin{cases} + 2 \, \ft{D}(p_1)\ft{D}(p_2) & s=0 \\ + 2 \, \ft{D}(p_1)\overline{\ft{D}(p_2)} & s=1 \\ + 2 \, \overline{\ft{D}(p_1)\ft{D}(p_2)} & s=2 + \end{cases} +\end{equation*} + +\begin{myproof}[lemma:renormalized-product-integral-representation] +\begin{multline*} + \innerp{\psi'}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} = + \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \frac{1}{\sqrt{2^r}} \sum_{s=0}^{r} \sum_{\sigma \in S_r} \\ + \big\langle + a(ED_{\sigma(1)}^{\dagger}f) \cdots a(ED_{\sigma(s)}^{\dagger}f) \psi_m, + a(ED_{\sigma(s+1)}^{\dagger}f) \cdots a(ED_{\sigma(r)}^{\dagger}f) \psi_n + \big\rangle +\end{multline*} + \begin{gather*} + \sqrt{m(m-1) \cdots (m-s+1)} + \sqrt{n(n-1) \cdots (n-(r-s)+1)} + \int dk_1 \cdots dk_{m-s} \\ + \int dp_1 \cdots dp_s + \ ED_{\sigma(1)}^{\dagger}f(p_1) \cdots ED_{\sigma(s)}^{\dagger}f(p_s) + \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)} \\ + \int dp_{s+1} \cdots dp_r + \ \overline{ED_{\sigma(s+1)}^{\dagger}f(p_{s+1}) \cdots ED_{\sigma(r)}^{\dagger}f(p_r)} + \ \psi_n(k_1,\ldots,k_{n-(r-s)},p_{s+1},\ldots,p_r) + \end{gather*} +\end{myproof} + +The following assertion is key + +\begin{lemma}{}{integral-kernel-h-bound} + In the setting of \cref{lemma:renormalized-product-integral-representation}, + there exist a constant $C$, and a positive integer $l$, + such that for arbitary states $\psi,\psi' \in xxx$, + and test functions $f \in \schwartz{M}$, + the function $K_{\psi'\!,\psi}$ is integrable (that is, $L^1$) + and satisfies the $H$-bound + \begin{equation*} + \norm{K_{\psi'\!,\psi}}_1 \le + C \norm{(1+H)^l \psi'} \norm{(1+H)^l \psi}. + \end{equation*} +\end{lemma} + +\begin{myproof} + We have to find an estimate for + \begin{equation*} + \norm{K_{\psi'\!,\psi}}_1 = + \int dp_1 \!\cdots dp_r + \, \abs{K_{\psi'\!,\psi}(p_1,\ldots,p_r)}. + \end{equation*} + We apply the triangle inequalities for sums and integrals + to the expression for $K_{\psi'\!,\psi}$ given in \cref{lemma:renormalized-product-integral-representation}, + use the fact that $\chi(p)$ has modulus one, make the estimates + \begin{equation*} + m(m-1) \cdots (m-s+1) \le m^r + \quad \text{and} \quad + n(n-1) \cdots (n-(r-s)+1) \le n^r, + \end{equation*} + and finally reorder the integration with Fubini’s theorem + to obtain + \begin{equation} + \label{first-estimate} + \begin{multlined}[c] + \norm{K_{\psi'\!,\psi}}_1 \le + \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{s=0}^{r} + \delta_{m-s}^{n-(r-s)} \sqrt{m^r n^r} \\ + \hspace{2.5cm} \cdot \abs*{\int \!dk \int \!dp'\! \int \!dp \, + P_s(p',p) \, \psi'_m(k,p') \, \psi_n(k,p)}, + \end{multlined} + \end{equation} + where we have used the abbreviations + \begin{align*} + k &= (k_1,\ldots,k_{m-s}) + \quad p' = (p_1,\ldots,p_s) + \quad p = (p_{s+1},\ldots,p_r) \\ + dk &= dk_1 \cdots dk_{m-s} + \quad \text{and so on.} + \end{align*} + Observe that $P_s(p_1,\ldots,p_r)$ is a (complex) polynomial + in the $4r$ variables $p_i^\mu$, $i=1,\ldots,r$, $\mu=0,\ldots,3$. + Its degree is given by + \begin{equation*} + \deg P_s = \sum_{i=1}^r \deg \ft{D}_i + \end{equation*} + that is the sum of the highest orders of differentiation + occurring in each of the operators $D_1, \ldots, D_r$. + There is no reason to expect arbitary states $\psi,\psi'$ to temper fast enough + to counteract this polynomial growth. + Thus, the integral in \cref{first-estimate} will not converge, in general. + However, if $\psi$ lies in the domain of $H^l$ for some positive integer $l$, + then we can be sure that $(1+H)^l \psi$ is square integrable, and we have + \begin{equation*} + \psi_n(k,p) = \parens[\big]{1+\omega(k,p)} {}^{-l} (1+H)^l \psi_n(k,p) + \end{equation*} + \begin{equation*} + (1+H)^l psi + \end{equation*} + \begin{equation*} + \abs*{\int \!dk \int \!dp'\! \int \!dp \, F(k,p',p) \, G'(k,p') \, G(k,p)} + \end{equation*} + where + \begin{align} + F(k,p',p) &= \parens[\big]{1+\omega(k,p')} {}^{-l} \parens[\big]{1+\omega(k,p)} {}^{-l} P_s(p',p) \\ + G'(k,p') &= \sqrt{m^r} (1+H)^l \psi_m(k,p') \\ + G(k,p) &= \sqrt{n^r} (1+H)^l \psi_n(k,p) + \end{align} + \begin{equation*} + \abs*{\int dp G(k,p) F(k,p',p)}^2 + \le \int dp \abs{F(k,p',p)}^2 + \cdot \int dp \abs{G(k,p)}^2 + \end{equation*} + \begin{equation*} + \int dp' \abs*{\int dp G(k,p) F(k,p',p)}^2 + \le \int dp \abs{G(k,p)}^2 + \sup_{k} \norm{F(k,\cdot,\cdot)}_2^2 + \end{equation*} + \begin{align*} + &\quad \abs*{\int \!dk \int \!dp'\! \int \!dp \, F(k,p',p) \, G'(k,p') \, G(k,p)} \\ + &\le \int \!dk \int \!dp' \abs{G'(k,p')} \abs*{\int \!dp \, F(k,p',p) \, G(k,p)} \\ + &\le \norm{G'}_2 \parens*{\int dk \int dp' \abs*{\int dp G(k,p) F(k,p',p)}^2} \\ + &\le \norm{G'}_2 \norm{G}_2 \sup_{k} \norm{F(k,\cdot,\cdot)}_2 + \end{align*} + We claim that there exists a positive constant $C_1$ independent of $m$ and $n$ such that + \begin{equation*} + \norm{G}_2 \le C_1 \norm{(1+H)^{l+r/2} \psi_n}_2 + \end{equation*} + and similary for $G'$. + This follows from $N \psi_n = n\psi_n$, where $N$ is the number operator, and the fact that $\omega(q)$ has a positive lower bound $M$ + + $H \psi_n(k,p) = (1+\omega(k,p) \psi_n(k,p)$ + + $1+\omega(k,p) \ge n \epsilon$ + + $\norm{(1+H)\psi_n}_2 \ge n \epsilon \norm{\psi_n} = \epsilon \norm{N \psi_n}$ + + In order to determine conditions for the finiteness of the remaining factor involving $F$, + it is desireable to have an estimate of the growth of $P_s$ in terms of $\omega(p_1),\ldots,\omega(p_r)$. + Notice that it is sufficient to make an estimate that is valid on the support of the measure $\Omega_m$, that is, the mass shell $X_m^+$, + since $F$ appears in an integral with respect to $p_1,\ldots,p_r$. + For an arbitrary point $q$ on the mass shell $X_m^+$ we have + \begin{align*} + q^{0} &= \omega(q) \\ + \abs{q^{\mu}} &\le \omega(q) \quad \mu = 1,2,3. + \end{align*} + Moreover, $\omega(q)$ has a positive lower bound, namely $m$, so that + for all exponents $a,b \in \NN$ with $a < b$ there + exists a constant $c_{a,b}$ such that $\omega(q)^a \le c_{a,b}\, \omega(q)^b$. + This allows us to make the estimate + \begin{equation*} + \abs{P_s(p_1,\ldots,p_r)} \le C_s \prod_{i=1}^r \omega(p_i)^{d_i} \quad \text{where}\ d_i = \deg \ft{D}_i. + \end{equation*} + + \begin{equation*} + \sqrt[s]{\omega(p_1) \cdots \omega(p_s)} + \le \frac{\omega(p_1) + \cdots + \omega(p_s)}{s} + \le \omega(p') \le 1 + \omega(k,p') + \end{equation*} + \begin{equation*} + \parens[\big]{1+\omega(k,p')} {}^{-l} + \le \parens[\big]{\omega(p_1) \cdots \omega(p_s)} {}^{-l/s} + \end{equation*} + \begin{equation*} + \abs{F(k,p',p)} \le + \prod_{i=1}^{s} \omega(p_i)^{d_i-l/s} + \prod_{j=s}^{r-s} \omega(p_j)^{d_j-l/(r-s)} + \end{equation*} +\end{myproof} + +\begin{lemma}{Renormalized Product at a Point}{} + In the setting of \cref{lemma:renormalized-product-integral-representation}, + \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*} +\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*} + \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} + = \int dp_1 \!\cdots dp_r + \, \ft{f}(p_1) \cdots\! \ft{f}(p_r) + \, K_{\psi'\!,\psi}(p_1,\ldots,p_r) + \end{equation*} + The integrand is dominated by the function $\abs{K_{\psi'\!,\psi}(p_1,\ldots,p_r)}$, + which has finite integral as it is $L^1$ + by \cref{lemma:integral-kernel-h-bound}. + Moreover, the integrand converges pointwise to $K_{\psi'\!,\psi}(p_1,\ldots,p_r)$, since $\ft{f} \to 1$ when $f \to \delta_x$. + The Dominated Convergence Theorem implies +\end{proof} + +\begin{lemma}{Renormalized Product as a QF-valued distribution}{} + In the setting of \cref{lemma:renormalized-product-integral-representation}, + \begin{equation*} + \normord{D_1 \varphi \cdots D_r \varphi} \ \vcentcolon \ + \schwartz{M} \to \QF{fock} + \end{equation*} + \begin{equation*} + \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f) \,\psi} = + \int_M \!dx \ f(x) \ \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(x) \,\psi} + \end{equation*} +\end{lemma} + +\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 $\psi,\psi'$ + \begin{multline*} + \innerp{\psi'\!}{\normord{D_1 \varphi \cdots D_r \varphi}(f) \,\psi} = \\ + = \int dp_1 \!\cdots dp_r + \sum_{s=0}^{r} + \, \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r) + \, L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) + \end{multline*} + where + \begin{multline*} + L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) = + \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \delta_{m-s}^{n-(r-s)} + \ P_s(p_1,\ldots,p_r) \\ + \cdot \sqrt{m(m-1) \cdots (m-s+1)} \sqrt{n(n-1) \cdots (n-(r-s)+1)} \\ + \cdot \int dk_1 \cdots dk_{m-s} + \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)} + \ \psi_n(k_1,\ldots,k_{n-(r-s)},p_{s+1},\ldots,p_r) + \end{multline*} + and $P_s(p_1,\ldots,p_r)$ is defined as before. +\end{lemma} + +\[ + f(T), f\left( T \right), + \int_{a}^{b} f\left( x \right) d x, \frac{1}{T}, +\] + +In the theory of a real scalar field $\phi$ of mass $m$, +the Lagrangian density of the Klein-Gordon action is given by +\begin{equation} + \label{lagrangian-density} + \mathcal{L} = \frac{1}{2} \parens{\partial^{\mu} \phi \partial_{\mu} \phi - m^2 \phi^2} +\end{equation} +and the \emph{canonical stress-energy tensor} is defined by +\begin{equation*} + T^{\mu}_{\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \phi)} \partial_{\nu} \phi - \delta^{\mu}_{\nu} \mathcal{L} +\end{equation*} +Raising the index $\nu$ and inserting \cref{lagrangian-density} yields +\begin{equation*} + T^{\mu\nu} = \partial^{\mu}\phi \partial^{\nu}\phi + \frac{1}{2} \eta^{\mu\nu} \parens*{m^2 \phi^2 - \partial_{\lambda}\phi \partial^{\lambda}\phi} +\end{equation*} +The \emph{energy density}: +\begin{equation*} + \rho = T^{00} = \frac{1}{2} \parens*{\sum_{\mu=0}^{3} (\partial^{\mu}\phi)^2 + m^2 \phi^2} +\end{equation*} +The discussion in the previous section enables us to define +the \emph{renormalized stress-energy tensor} of a free scalar field $\varphi$ by +\begin{equation*} + T^{\mu\nu} = \normord{\partial^{\mu}\varphi \partial^{\nu}\varphi + \frac{1}{2} \eta^{\mu\nu} \parens*{m^2 \varphi^2 - \partial_{\lambda}\varphi \partial^{\lambda}\varphi}} +\end{equation*} +and this is a quadratic form. +In particular, the energy density is +\begin{equation*} + \rho = \frac{1}{2} \sum_{\mu=0}^{3} \normord{(\partial^{\mu}\phi)^2} + \frac{1}{2} m^2 \normord{\phi^2} +\end{equation*} + +\begin{multline*} + \innerp{\psi'\!}{\rho(f) \,\psi} = \\ + = \int dp_1 dp_2 + \parens{p_1^{\mu} p_2^{\mu} + m^2} + \sum_{s=0}^{r} (-1)^{s+1} + \, \ft{f}(p_1 + \cdots + p_s - p_{s+1} - \cdots - p_r) + \, L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) +\end{multline*} +where +\begin{multline*} + L_{\psi'\!,\psi}^{s}(p_1,\ldots,p_r) = + \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \delta_{m-s}^{n-(r-s)} \\ + \cdot \sqrt{m(m-1) \cdots (m-s+1)} \sqrt{n(n-1) \cdots (n-(r-s)+1)} \\ + \cdot \int dk_1 \cdots dk_{m-s} + \ \overline{\psi'_m(k_1,\ldots,k_{m-s},p_1,\ldots,p_s)} + \ \psi_n(k_1,\ldots,k_{n-(r-s)},p_{s+1},\ldots,p_r) +\end{multline*} + +\begin{equation*} + A \QFequal B +\end{equation*} + +\section{Essential Self-Adjointness of Renormalized Products} + +\nocite{*} + +\chapterbib +\cleardoublepage + +% vim: syntax=mytex |