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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-04-10 12:06:37 +0200 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-04-10 12:06:37 +0200 |
commit | 0c4c33d879709ad8625d63267ae23a2ac0155ba4 (patch) | |
tree | 710e46501547187209d408a469ea5043cc5100d0 /stresstensor.tex | |
parent | 80e764a067b96b4766a5ce28e0d12758bdbf5b58 (diff) | |
download | master-0c4c33d879709ad8625d63267ae23a2ac0155ba4.tar.zst |
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-rw-r--r-- | stresstensor.tex | 466 |
1 files changed, 358 insertions, 108 deletions
diff --git a/stresstensor.tex b/stresstensor.tex index 4cdc321..39c45cd 100644 --- a/stresstensor.tex +++ b/stresstensor.tex @@ -1,6 +1,10 @@ \chapter{Construction of the Stress Tensor of a Free Scalar Quantum Field} \label{chapter:stress-tensor} +\begin{center} + \emph{Note: Work in Progress} +\end{center} + \begin{equation*} H = \tfrac{1}{2} \parens*{(\partial_t \phi)^2 + \abs{\nabla_{\!\!\symbfit{x}} \phi}^2 + m^2 \phi^2} \end{equation*} @@ -10,7 +14,7 @@ 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. +is realized by essentially selfadjoint operators. \section{Choosing Conventions and Fixing Notation} \label{section:conventions} @@ -67,7 +71,9 @@ as a service to the reader. \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*} + For later use, we also give the action of an $s$-fold product of annihilation operators: \begin{multline} + \label{equation:multiple-annihilation-operators} \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) @@ -114,7 +120,7 @@ as a service to the reader. \innerp[\big]{\psi'}{a(p)^\dagger \psi} = \innerp[\big]{a(p) \psi'}{\psi} \end{equation*} - Define ... + 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*} @@ -122,7 +128,6 @@ as a service to the reader. \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*} @@ -175,25 +180,181 @@ into annihilation operators on the left. \end{multlined} \end{align*} -\begin{proposition}{}{} - asdf -\end{proposition} +\section{Normal Ordering} +% The Renormalization Map? + +%\blockcquote{Wick1950}{% + %\textelp{} we then proceed to rearrange such a product so as to carry all + %creation operators to the left of all destruction operators \textelp{}. The + %main problem to be solved in carrying out this idea is one of algebraic + %technique \textelp{} +%} + +The process of renormalizing a product of field operators +has the purpose of discarding infinite constants +that occur when calculating the vacuum expectation value. +(TODO: present physicists way of introducing normal ordering) + +Now let us extract the algebraic essence of the situation. +The objects of our calculations are the field operators $\Phi(f)$, +but it does not matter that these are realized as linear maps on Fock space. +Forming the product $\Phi(f)\Phi(g)$ might as well be done purely symbolically, +since none of what we want to do depends on this product +having the meaning of operator composition; +and similar for the other two arithmetic operations, +addition and multiplication with a complex scalar. +Thus we should calculate with abstract objects $\Phi(f)$ labeled by Hilbert space vectors $f \in \hilb{H}$. +Considering that here $\Phi$ carries no meaning, we can use the label $f$ itself to represent the object. + +This leads us to consider formal expressions +\begin{equation*} + \alpha^{(0)} e + \sum_{i} \alpha^{(1)}_i z^{(1)}_i + \sum_{j,k} \alpha^{(2)}_{j,k} z^{(2)}_j z^{(2)}_k + \cdots +\end{equation*} +where the $z^{(1)}_i,z^{(2)}_j,z^{(2)}_k,\ldots$ are in $\hilb{H}$, +the $\alpha^{(0)},\alpha^{(1)}_i,\alpha^{(2)}_{j,k},\ldots$ are complex numbers, +of which only finitely many are nonzero, +and $e$ is a special object representing an empty product of $z$'s. +To make this mathematically precise: +we are speaking of the noncommutative associative algebra over $\CC$ +freely generated by the elements of $\hilb{H}$. +The unit of the algebra is $e$. + +This in not quite what we want +TODO(explain need for commutation relations) +By abstract algebra, this is viable +by forming the quotient of the free algebra +with respect to the two-sided ideal +generated by all elements $zz' - z'z = i \Imag \innerp{z}{z'} \, e$, +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 $\hilb{H}$ + is the noncommutative associative algebra over $\CC$ + generated by the elements of $\hilb{H}$, with the relations + \begin{equation*} + zz' - z'z = i \Imag \innerp{z}{z'} \, e \qquad z,z' \in \hilb{H}, + \end{equation*} + where $e$ is the unit of the algebra. +\end{definition} + +TODO(introduce $\Phi$ as representation of $\WeylAlg$) + +\begin{definition}{Annihilator and Creator}{} + Suppose $\WeylAlg$ is the infinitesimal Weyl algebra + over some complex Hilbert space $\hilb{H}$. + For all $z \in \hilb{H}$, + we define, as elements of $\WeylAlg$, the \emph{annihilator} + \begin{equation*} + \weylannihilator(z) = \frac{1}{\sqrt{2}} \parens{z+iz}, + \end{equation*} + and the \emph{creator} + \begin{equation*} + \weylcreator(z) = \frac{1}{\sqrt{2}} \parens{z-iz}. + \end{equation*} +\end{definition} + +\begin{equation*} + z = \frac{1}{\sqrt{2}} \parens[\big]{\weylannihilator(z) + \weylcreator(z)} +\end{equation*} + +A \emph{monomial} in the Weyl algebra $\WeylAlg$ over a complex Hilbert space $\hilb{H}$ is an element of the form +$z_1 \cdots z_r \in \WeylAlg$, where $r \ge 0$ and $z_1,\ldots,z_r$ are in $\hilb{H}$. +We allow $r=0$, meaning that the unit $e$ is a monomial. +The set of all monomials in $\WeylAlg$ is denoted $\Mon(\WeylAlg)$. + +\begin{definition}{Normal Ordering}{} + Let $\hilb{H}$ be a complex Hilbert space and $\WeylAlg$ its associated infinitesimal Weyl algebra. + The mapping $\normord{\,\,}$, defined by + \begin{gather} + \Mon(\WeylAlg) \longrightarrow \WeylAlg \nonumber\\ + \label{equation:normal-ordering} + \normord{z_1 \!\cdots z_r} = + \frac{1}{\sqrt{2^r}} + \sum_{I \subset \braces{1,\ldots,r}} \, + \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 $\hilb{H}$. + 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} + +The products in~\eqref{equation:normal-ordering} are well defined +because creators commute with creators and annihilators commute with annihilators. +Since an empty product equals, per convention, the neutral element of multiplication, which here is the unit $e$, +the formula makes sense even for $r=0$ and asserts that $\normord{e} = e$. +The cases $r=1$ and $r=2$ read +\begin{align*} + \normord{z} &= + \frac{1}{\sqrt{2}} \parens[\big]{\weylannihilator(z) + \weylcreator(z)} = z \\ + \normord{z_1 z_2} &= \frac{1}{2} + \parens[\big]{\weylannihilator(z_1) \weylannihilator(z_2) + \weylannihilator(z_1) \weylcreator(z_2) + + \weylcreator(z_1) \weylannihilator(z_2) + \weylcreator(z_1) \weylcreator(z_2) } +\end{align*} +This suggests that the normally ordered product $\normord{z_1 \!\cdots z_r}$ +is symmetric in $z_1,\ldots,z_n$. This is in fact true, and becomes evident +if one brings~\eqref{equation:normal-ordering} into the equivalent form +\begin{gather} + \label{equation:normal-ordering-symmetric} + \normord{z_1 \!\cdots z_r} = + \frac{1}{\sqrt{2^r}} + \sum_{\sigma \in S_r} + \sum_{s=0\vphantom{S}}^{r} + \frac{1}{s!(r-s)!} + \prod_{i=1\vphantom{S}}^{s} \weylcreator(z_{\sigma(i)}) + \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} \weylannihilator(z_{\sigma(j)}) +\end{gather} +by basic combinatorial arguments (TODO: further explanation?). +In~\cite{Klein1973}, the factor $\frac{1}{s!(r-s)!}$ is erroneously missing. + + +\begin{equation*} + E(\normord{z_1 \!\cdots z_r}) = 0 \qquad \forall z_1,\ldots,z_r \in \hilb{H}, r \ge 1 +\end{equation*} +\begin{equation*} + E\parens[\Big]{\prod_{i=1\vphantom{S}}^{s} \weylcreator(z_i) + \prod_{\mathclap{j=s+1\vphantom{S}}}^{r} \weylannihilator(z_j)} = 0 + \qquad \forall z_1,\ldots,z_r \in \hilb{H}, r \ge 1, 1 \le s \le r +\end{equation*} + -\subsubsection{Linear Differential Operators and their Formal Adjoint} +The normal ordered product is supposed to represent the identical quantity as before ordering, +except that we have adjusted our point of reference, such that measurements yield finite results. +It is therefore \emph{physically reasonable} that the commutation relations +of the normal ordered product with the field are analogous. +As it turns out, this additional property makes the construction of normal ordering +\emph{mathematically unique}. + +\begin{theorem}{Uniqueness of the Normal Order}{} + Normal ordering is the unique mapping $N : \Mon(\WeylAlg) \to \WeylAlg$ such that + \begin{gather*} + E\parens[\big]{N(z_1 \!\cdots z_r)} = 0 \\ + \bracks{N(z_1 \!\cdots z_r), z'} = + \sum_{s=1}^{r} \bracks{z_s,z'} N(z_1 \!\cdots \widehat{z_i} \cdots z_r) + \end{gather*} + for all $z_1,\ldots,z_r,z' \in \hilb{H}$ and all $r \ge 1$. +\end{theorem} + +%\begin{theorem}{}{} + %The normal ordering is the renormalization with respect to the normal vacuum. +%\end{theorem} + +\section{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 $\varphi(f)$ +that is a mapping that assigns to each (Schwartz 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. +is a (scalar-valued) tempered distribution on $M$. +It is well known that tempered distributions 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 @@ -242,17 +403,37 @@ The operator corresponding to $D$ in Fourier space is the multiplication operato 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*} +\begin{gather} + \label{equation:renormalized-product} + \normord{D_1 \varphi(f) \cdots D_r \varphi(f)} = + \frac{1}{\sqrt{2^r}} + \sum_{\sigma \in S_r} + \sum_{s=0\vphantom{S}}^{r} + \frac{1}{s!(r-s)!} + \prod_{i=1\vphantom{S}}^{s} a^\dagger(D^\dagger_{\sigma(i)}f) + \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} - +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 $r=1$. +This is because it has a multi-linear dependence on the test function. +Conceptually, one wishes to treat any physical quantity derived from the quantum field +on the same footing as the field itself. +One construction to obtain an operator-valued distribution, +is described in~\cite{Segal1969}, \cite{Segal1970} and \cite{Klein1973}. +The idea is to take the limit $f \to \delta_x$, where $\delta_x$ is the delta distribution supported at a point $x \in M$, +resulting in the renormalized product at the point $x$, now just a quadratic form, +which is then smeared with \emph{one} instance of $f$. +As we shall see, +this approach incurs significant technical difficulties. \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 $\psi,\psi'$ + Let $\varphi$ be the free scalar quantum field with mass parameter $m > 0$. + Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) coefficients. + Then, for arbitrary Schwartz functions $f \in \schwartz{M}$ and Fock states $\psi,\psi' \in \BosonFock{L^2(X_m^+,d\Omega_m)}$, + we have \begin{equation*} \innerp{\psi'\!}{\normord{D_1 \varphi(f) \cdots D_r \varphi(f)} \,\psi} = \int dp_1 \!\cdots dp_r @@ -271,41 +452,44 @@ Let $D_1, \ldots, D_r$ be linear differential operators with constant (complex) \ \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)}. + \text{where} \quad \chi(p) = \begin{cases*} + \overline{\ft{f}(p)} / \ft{f}(p) & if $\ft{f}(p) \ne 0$, \\ + 1 & otherwise, + \end{cases*} \end{equation*} + \begin{multline*} + \text{and} \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{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*} +TODO(Note about the remaining dependence of $K$ on $f$.) \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*} + From equation~\eqref{equation:renormalized-product}, + applying the definition of the Fock space inner product, + and moving all creation operators to the left hand side, + we get + \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} \frac{1}{s!(r-s)!} \\ + \cdot \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*} + Notice that the inner product in the second line + can only be nonzero if the particle numbers match up + after the application of the annihilation operators in each argument, + that is if $m-s=n-(r-s)$. + With~\eqref{equation:multiple-annihilation-operators} + this expression may be further expanded into \begin{gather*} \sqrt{m(m-1) \cdots (m-s+1)} \sqrt{n(n-1) \cdots (n-(r-s)+1)} @@ -317,14 +501,42 @@ For squares, that is $r=2$ \ \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*} + Now recall that $E$ stands for Fourier transformation (followed by restriction to the mass shell) + and that in Fourier space the linear differential operator $D^\dagger$ corresponds to a + multiplication with the function $\hat{D}$, so that + \begin{equation*} + ED_{\sigma(i)}^{\dagger}f(p_i) = \hat{D}_{\sigma(i)}(p_i) \cdot \ft{f} (p_i) + \qquad \forall i + \end{equation*} + By Fubini’s Theorem, the we may interchange the integrals with respect to the variables $p_i$ + with the $k$-integrals. + This allows us to move all factors involving $\ft{f}$ in front of the $k$-integrals. + Finally, we introduce the $\chi$s through the substitution $\overline{\ft{f}} = \chi \ft{f}$, + and combine all terms depending on $\sigma$ into $P_s$. \end{myproof} -The following assertion is key +In the special case that $D_1 = \cdots = D_n = D$ 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 $r=2$ +\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 + \end{cases} +\end{equation*} + +The following assertion is key to realizing the idea of taking the limit $f \to \delta_x$. \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 $\psi,\psi' \in xxx$, + such that for arbitrary states $\psi,\psi' \in \BosonFock{\hilb{H}}$, and test functions $f \in \schwartz{M}$, the function $K_{\psi'\!,\psi}$ is integrable (that is, $L^1$) and satisfies the $H$-bound @@ -334,7 +546,14 @@ The following assertion is key \end{equation*} \end{lemma} -\begin{myproof} +The Hamilton operator $H$ acts on $n$-particle states $\psi_n$ +by multiplication with $\omega(p_1)$ +In the following proof it will we convenient to use the abbreviation +\begin{equation*} + \omega(p_1,\ldots,p_s) = \omega(p_1) + \cdots + \omega(p_s) +\end{equation*} + +\begin{myproof}[lemma:integral-kernel-h-bound] We have to find an estimate for \begin{equation*} \norm{K_{\psi'\!,\psi}}_1 = @@ -349,16 +568,16 @@ The following assertion is key \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 + and finally reorder the integration with Fubini’s Theorem to obtain \begin{equation} - \label{first-estimate} + \label{equation: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)}, + \hspace{2.5cm} \cdot \int \!dk \int \!dp'\! \int \!dp \, + \abs*{P_s(p',p) \, \psi'_m(k,p') \, \psi_n(k,p)}, \end{multlined} \end{equation} where we have used the abbreviations @@ -369,6 +588,8 @@ The following assertion is key dk &= dk_1 \cdots dk_{m-s} \quad \text{and so on.} \end{align*} + For following discussion we will assume $n-(r-s)=m-s$. + 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 @@ -377,54 +598,77 @@ The following assertion is key \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 + There is no reason to expect arbitrary 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} + Thus, the integral in~\eqref{equation:first-estimate} will not converge, in general. + However, if $\psi$ lies in the domain of $H^a$ for some positive integer $a$, + then we can be sure that $(1+H)^a \psi$ is square integrable, and we have + \begin{align*} + \psi_n(k,p) &= \parens[\big]{1+\omega(k,p)} {}^{-a} (1+H)^a \psi_n(k,p) \\ + \psi'_m(k,p') &= \parens[\big]{1+\omega(k,p')} {}^{-a} (1+H)^a \psi'_m(k,p') + \end{align*} + We use this to rewrite the integral part of~\eqref{equation:first-estimate} as follows: + \begin{equation} + \label{equation:rewritten-integral} + \int \!dk \int \!dp'\! \int \!dp \, \abs*{F(k,p',p) \, G'(k,p') \, G(k,p)}, + \end{equation} + where we have introduced the functions + \begin{align*} + F(k,p',p) &= \parens[\big]{1+\omega(k,p')} {}^{-a} \parens[\big]{1+\omega(k,p)} {}^{-a} P_s(p',p) \\ + G'(k,p') &= \sqrt{m^r} (1+H)^a \psi'_m(k,p') \\ + G(k,p) &= \sqrt{n^r} (1+H)^a \psi_n(k,p). + \end{align*} + Next, we derive an estimate of~\eqref{equation:rewritten-integral}. + By Cauchy-Schwarz, we have \begin{equation*} - \abs*{\int dp G(k,p) F(k,p',p)}^2 + \abs*{\int dp \, \abs{F(k,p',p)G(k,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 + and this implies + \begin{equation} + \label{equation:estimate1} + \int dp' \abs*{\int \!dp \, \abs{F(k,p',p)G(k,p)}}^2 \le \int dp \abs{G(k,p)}^2 - \sup_{k} \norm{F(k,\cdot,\cdot)}_2^2 - \end{equation*} + \cdot \int dp'\! \int dp \abs{F(k,p',p)}^2. + \end{equation} + Notice that the second factor is the $L^2$ norm of $F$ with its first argument held fixed: + \begin{equation} + \label{equation:norm} + \norm{F(k,\cdot,\cdot)}_2^2 + = \int dp'\! \int dp \abs{F(k,p',p)}^2 + \end{equation} + By another application of Cauchy-Schwarz, and using~\eqref{equation:estimate1} and~\eqref{equation:norm}, we obtain \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} \\ + \text{\eqref{equation:rewritten-integral}} \, + &= \int \!dk \int \!dp' \abs{G'(k,p')} \int \!dp \, \abs*{F(k,p',p) \, G(k,p)} \\ + &\le \norm{G'}_2 \parens[\bigg]{\int dk \int dp' \abs*{\int \!dp \, \abs*{F(k,p',p)G(k,p)}}^2}^{\frac{1}{2}} \\ + &\le \norm{G'}_2 \parens[\bigg]{\int dk \, \norm{F(k,\cdot,\cdot)}_2^2 \int dp \abs{G(k,p)}^2}^{\frac{1}{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 + \norm{G}_2 \le C_1 \norm{(1+H)^{a+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 $M$ - - $H \psi_n(k,p) = (1+\omega(k,p) \psi_n(k,p)$ - - $1+\omega(k,p) \ge n \epsilon$ + and similarly for $G'$. + Since $\omega$ has a positive lower bound on the mass shell, + there exists a constant $\epsilon > 0$ independent of $n$ such that + \begin{equation*} + 1+\omega(k,p) = 1 + \underbrace{\omega(k_1) + \cdots + \omega(k_{m-s}) + \omega(p_{s+1}) + \cdots + \omega(p_r)}_{\text{$(m-s)+(r-s)=n$ terms}} \ge n \epsilon + \end{equation*} + Combine this with the fact that + $N \psi_n = n\psi_n$, where $N$ is the number operator, + to see + \begin{equation*} + \norm{G}_2 = \norm{N^{r/2} (1+H)^a \psi_n}_2 \le \epsilon^{-r/2} \norm{(1+H)^{a+r/2} \psi_n}_2. + \end{equation*} + Now set $C_1 = \epsilon^{-r/2}$. + The proof for $G'$ is analogous. + In particular, we have shown that both $\norm{G}_{2}$ and $\norm{G'}_{2}$ are finite, + provided that $\psi_n$ and $\psi'_m$ lie in the domain of $H^l$, where $l=a+r/2$. - $\norm{(1+H)\psi_n}_2 \ge n \epsilon \norm{\psi_n} = \epsilon \norm{N \psi_n}$ + %$H \psi_n(k,p) = (1+\omega(k,p)) \psi_n(k,p)$ + %$\norm{(1+H)\psi_n}_2 \ge n \epsilon \norm{\psi_n} = \epsilon \norm{N \psi_n}$ 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 $\omega(p_1),\ldots,\omega(p_r)$. @@ -435,27 +679,30 @@ The following assertion is key 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 + Moreover, $\omega(q)$ has a positive lower bound on $X_m^+$, 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*} + \begin{equation} + \label{equation:polynomial-estimate} + \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} + and $C_s$ is a constant independent of $m$ and $n$. + By the Arithmetic Mean-Geometric Mean Inequality, we have + \begin{gather} \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*} + \le \omega(p') \le 1 + \omega(k,p'), \nonumber\\ + \shortintertext{hence} + \label{equation:one-plus-omega-estimate} + \parens[\big]{1+\omega(k,p')} {}^{-a} + \le \parens[\big]{\omega(p_1) \cdots \omega(p_s)} {}^{-a/s}. + \end{gather} + The estimates~\eqref{equation:polynomial-estimate} and~\eqref{equation:one-plus-omega-estimate} entail \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)} + \abs{F(k,p',p)} C_s \le + \prod_{i=1}^{s} \omega(p_i)^{d_i-a/s} + \prod_{j=s}^{r-s} \omega(p_j)^{d_j-a/(r-s)} \end{equation*} \end{myproof} @@ -494,6 +741,7 @@ The following assertion is key 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 $f \to \delta_x$. + TODO(With of choice of FT constants, $\ft{f} \to 1/(2\pi)^2$. Change here or change def?) The Dominated Convergence Theorem implies \end{proof} @@ -533,10 +781,10 @@ The following assertion is key and $P_s(p_1,\ldots,p_r)$ is defined as before. \end{lemma} -\[ - f(T), f\left( T \right), - \int_{a}^{b} f\left( x \right) d x, \frac{1}{T}, -\] +%\[ + %f(T), f\left( T \right), + %\int_{a}^{b} f\left( x \right) d x, \frac{1}{T}, +%\] In the theory of a real scalar field $\phi$ of mass $m$, the Lagrangian density of the Klein-Gordon action is given by @@ -590,9 +838,11 @@ where A \QFequal B \end{equation*} -\section{Essential Self-Adjointness of Renormalized Products} +\section{Essential Selfadjointness of Renormalized Products} + +TODO -\nocite{*} +%\nocite{*} \chapterbib \cleardoublepage |