\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

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