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\begin{document}

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\section*{microcausal polynomial observable}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak
\noindent\hyperlink{review}{Review}\dotfill \pageref*{review} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{microcausal functionals} on the space $C^\infty(X)$ of [[smooth functions]] on a [[globally hyperbolic spacetime]] $(X,e)$ are those which come from [[compactly supported distributions]] on some [[Cartesian product]] of copies of $X$ such that the [[wave front set]] of the distributions excludes those covectors to a point in $X^n$ all whose components are in the [[closed future cone]] or all whose components are in the [[closed past cone]]

These functionals underly the [[Wick algebra]] of [[free field theories]]. The condition on the wave front is such that the [[product of distributions]] with a [[Hadamard distribution]] is well defined, so that the coresponding [[Moyal star product]] is well defined, which gives the [[Wick algebra]]. At the same time the condition includes [[local observables]] and hence in particular the usual ([[adiabatic switching|adiabatically switched]]) point-[[interaction]] terms, such as of [[phi{\tt \symbol{94}}4 theory]].

[[!include perturbative observables -- table]]

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{PolynomialObservable}\hypertarget{PolynomialObservable}{}
\textbf{([[polynomial observable]])}

Let $E \overset{fb}{\to}$ be [[field bundle]] which is a [[vector bundle]]. An [[off-shell]] \emph{[[polynomial observable]]} is a [[smooth function]]

\begin{displaymath}
A 
  \;\colon\;
  \Gamma_\Sigma(E)
    \longrightarrow 
  \mathbb{C}
\end{displaymath}
on the [[on-shell]] [[space of sections]] of the [[field bundle]] $E \overset{fb}{\to} \Sigma$ (space of field histories) which may be expressed as

\begin{displaymath}
A(\Phi)
  \;=\;
  \alpha^{(0)}
  +
  \int_\Sigma
    \alpha^{(1)}_a(x) \Phi^a(x)
   \,
  dvol_\Sigma(x)
  +
  \int_\Sigma \int_\Sigma
   \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \Phi^{a_1}(x_1) \Phi^{a_2}(x_2)
  \,dvol_\Sigma(x_1)
  \, dvol_\Sigma(x_2)
  +
  \cdots
  \,,
\end{displaymath}
where

\begin{displaymath}
\alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)
\end{displaymath}
is a [[compactly supported distribution]] [[distribution of two variables|of k variables]] on the $k$-fold graded-symmetric [[external tensor product of vector bundles]] of the [[field bundle]] with itself.

Write

\begin{displaymath}
PolyObs(E) \hookrightarrow Obs(E)
\end{displaymath}
for the [[subspace]] of off-shell polynomial observables onside all off-shell [[observables]].

Let moreover $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[equations of motion]] are [[Green hyperbolic differential equations]]. Then an \emph{[[on-shell]] polynomial observable} is the [[restriction]] of an off-shell polynomial observable along the inclusion of the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Write

\begin{displaymath}
PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
\end{displaymath}
for the subspace of all on-shell polynomial observables inside all on-shell [[observables]].

By \href{Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions}{this prop.} restriction yields an [[isomorphism]] between polynomial on-shell observables and polynomial off-shell observables modulo the image of the [[differential operator]] $P$:

\begin{displaymath}
PolyObs(E,\mathbf{L})
    \underoverset{\simeq}{\text{restriction}}{\longleftarrow}
  PolyObs(E)/im(P)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{MicrocausalObservable}\hypertarget{MicrocausalObservable}{}
\textbf{([[microcausal observable]])}

For $\Sigma$ a [[spacetime]], hence a [[Lorentzian manifold]] with [[time orientation]], then a \emph{[[microcausal observable]]} is a [[polynomial observable]] (def. \ref{PolynomialObservable}) such that each [[coefficient]] $\alpha^{(k)}$ has [[wave front set]] excluding those points where all $k$ [[wave vectors]] are in the [[closed future cone]] or all in the [[closed past cone]].

\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
\textbf{([[non-singular distribution|non-singular observables]] are [[microcausal observables|microcausal]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]].

Then a [[regular observable]], hence a [[polynomial observable]] (\href{A+first+idea+of+quantum+field+theory+--+Observables#PolynomialObservables}{this def.}) whose [[distribution|distributional]] [[coefficients]] $\alpha_{a_1 \cdots a_k}$ \eqref{RecalledForWickAlgebraExpansionOfPolynomialObservables} are [[non-singular distributions]] is a [[microcausal observable]] (def. \ref{MicrocausalFunctionals}).

This is simply because the [[wave front set]] of [[non-singular distributions]] is [[empty set|empty]] (by definition, via the [[Paley-Wiener-Schwartz theorem]], \href{Schwartz+theorem#DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}{this prop.}).

\end{example}
\begin{example}
\label{}\hypertarget{}{}
\textbf{(compactly averaged point evaluations are microcausal)}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]]. Assume the [[field bundle]] $E$ is a [[trivial vector bundle]] with linear [[fiber]] coordinates $(\phi^a)$.

Let $g \in C^\infty_c(X)$ be a [[bump function]], then for $n \in \mathbb{N}$ the polynomial observables (\href{A+first+idea+of+quantum+field+theory+--+Observables#PolynomialObservables}{this def.}) of the form

\begin{displaymath}
\itexarray{
    \Gamma_\Sigma(E) &\overset{}{\longrightarrow}& \mathbb{C}
    \\
    \Phi 
      &\mapsto& 
    \int_X g(x) 
      \tilde \alpha_{a_1 \cdots a_k}(x) 
      \Phi^{a_1}(x) \cdots \Phi^(a_k)\,  dvol_\Sigma(x)
  }
\end{displaymath}
are [[microcausal observable|microcausal]] (def. \ref{MicrocausalFunctionals}).

If here we think of $\phi(x)^n$ as a point-[[interaction]] term (as for instance in [[phi{\tt \symbol{94}}4 theory]]) then $g$ is to be thought of as an ``[[adiabatic switching|adiabatically switched]]'' [[coupling constant]]. These are the relevant [[interaction terms]] to be quantized via [[causal perturbation theory]].

\end{example}
\begin{proof}
For notational convenience, consider the case of the [[scalar field]] with $k = 2$; the general case is directly analogous. Then the [[local observable]] coming from $\phi^2$ (a [[phi{\tt \symbol{94}}n interaction]]-term), has, regarded as a [[polynomial observable]], the [[delta distribution]] $\delta(x_1-x_2)$ as [[coefficient]] in degree 2:

\begin{displaymath}
\begin{aligned}
    A(\Phi)
    & =
    \underset{\Sigma}{\int} g(x) (\Phi(x))^2 \,dvol_\Sigma(x)
    \\
    & =
    \underset{\Sigma \times \Sigma}{\int} 
      \underset{ = \alpha^{(2)}}{
      \underbrace{
        g(x_1)
        \delta(x_1 - x_2)
      }}
      \, 
     \Phi(x_1) \Phi(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
  \end{aligned}
  \,.
\end{displaymath}
Now for $(x_1, x_2) \in \Sigma \times \Sigma$ and $\mathbb{R}^{2n} \simeq U \subset X \times X$ a [[chart]] around this point, the [[Fourier transform of distributions]] of $g \cdot \delta(-,-)$ restricted to this chart is proportional to the Fourier transform $\hat g$ of $g$ evaluated at the sum of the two covectors:

\begin{displaymath}
\begin{aligned}
    (k_1, k_2)
      & \mapsto
      \underset{\mathbb{R}^{2n}}{\int} 
      g(x_1) 
      \delta(x_1, x_2) 
      e^{i (k_1 \cdot x_1 + k_2 \cdot x_2 )} 
      \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
    \\
    & \propto \hat g(k_1 + k_2)
   \end{aligned}
  \,.
\end{displaymath}
Since $g$ is a plain [[bump function]], its [[Fourier transform]] $\hat g$ is quickly decaying (according to \href{compactly+supported+distribution#eq:DecayEstimateForFourierTransformOfNonSingularDistribution}{this inequality}) along $k_1 + k_2$ (\href{wavefront+set#EmptyWaveFronSetCorrespondsToOrdinaryFunction}{this prop.}), as long as $k_1 + k_2 \neq 0$. Only on the cone $k_1 + k_2 = 0$ the Fourier transform is constant, and hence in particular not decaying.

 This means that the wave front set consists of the elements of the form $(x, (k, -k))$ with $k \neq 0$. Since $k$ and $-k$ are both in the [[closed future cone]] or both in the [[closed past cone]] precisely if $k = 0$, this situation is excluded in the wave front set and hence the distribution $g \cdot \delta(-,-)$ is [[microcausal observable|microcausal]].

\begin{quote}%
(graphics grabbed from \hyperlink{KhavkineMoretti14}{Khavkine-Moretti 14, p. 45})


\end{quote}
\end{proof}
This shows that microcausality in this case is related to conservation of momentum in the point interaction.

More generally:

\begin{example}
\label{CompactlySupportedPolynomialLocalDensities}\hypertarget{CompactlySupportedPolynomialLocalDensities}{}
\textbf{(polynomial [[local observables]] are microcausal)}

Write

\begin{displaymath}
\Omega_{poly}^{h,v}(E)
\end{displaymath}
for the space of differential forms on the [[jet bundle]] of the [[field bundle]] $E$ which locally are [[polynomials]] in the field variables.

\begin{displaymath}
\mathcal{F}_{loc}
    \; \subset \;
  C^\infty_c(\Sigma) \underset{\Omega_{poly}^{0,0}(E)}{\otimes} \Omega_{poly}^{d,0}(E)
\end{displaymath}
for the subspace of [[horizontal differential forms]] of degree $d$ on the [[jet bundle]] ([[local Lagrangian densities]]) of those which are [[compact support|compactly supported]] with respect to $\Sigma$ ([[local observables]]) and [[polynomial]] with respect to the field variables.

Every $L \in \mathcal{F}_{loc}$ induces a functional

\begin{displaymath}
\Gamma_\Sigma(E) \longrightarrow \mathbb{R}
\end{displaymath}
by [[integration of differential forms|integration]] of the [[pullback of differential forms|pullback]] of $L$ along the [[jet prolongation]] of a given [[section]]:

\begin{displaymath}
\phi \mapsto \int_{\Sigma} j^\infty(\phi)^\ast L 
  \,.
\end{displaymath}
These functionals happen to be [[microcausal functional|microcausal]], so that there is an inclusion

\begin{displaymath}
\mathcal{F}_{loc} \hookrightarrow \mathcal{F}_{mc}
\end{displaymath}
into the space of microcausal functionals (e.g. \hyperlink{FredenhagenRejzner12}{Fredenhagen-Rejzner 12, p. 21}). In fact this is a [[dense subspace]] inclusion (e.g. \hyperlink{FredenhagenRejzner12}{Fredenhagen-Rejzner 12, p. 23})

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
Write $\mathcal{F}_{reg} \subset \mathcal{F}_{mc}$ for that subalgebra of the algebra of microcausal functionals whose [[coefficients]] are [[non-singular distributions]].

Let

\begin{displaymath}
\langle -\rangle 
  \;\colon\;
  \mathcal{F}_{reg} 
  \longrightarrow 
  \mathbb{C}
\end{displaymath}
be a [[state on a star-algebra|state]] on regular observables which is [[quasi-free Hadamard state|quasi-free Hadamard]]. Then this uniquely [[extensions|extends]] to a state on microcausal functionsal

\begin{displaymath}
\itexarray{
    \mathcal{F}_{reg} 
      &\overset{\langle -\rangle}{\longrightarrow}& 
    \mathbb{C}
    \\
    \downarrow & \nearrow_{\mathrlap{\exists ! \langle -\rangle}}
    \\
    \mathcal{F}_{mc}
  }
\end{displaymath}
\end{prop}
(\hyperlink{HollandRuan01}{Hollands-Ruan 01, remark 1 on p. 12}, implied by \hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00}, \hyperlink{HollandsWald01}{Hollands-Wald 01}, a special case of \hyperlink{HollandRuan01}{Hollands-Ruan 01, theorem III.1 (ii)})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Wick algebra]]


\item [[time-ordered products]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{original_articles}{}\subsubsection*{{Original articles}}\label{original_articles}

\begin{itemize}%
\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], \emph{Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds}, Commun. Math. Phys. 208 : 623-661, 2000 (\href{https://arxiv.org/abs/math-ph/9903028}{math-ph/9903028})


\item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion}, Commun.Math.Phys. 219 (2001) 5-30 (\href{https://arxiv.org/abs/hep-th/0001129}{arXiv:hep-th/0001129})


\item [[Stefan Hollands]], [[Robert Wald]], \emph{Local Wick polynomials and time ordered products of quantum fields in curved spacetime}, Commun. Math. Phys., Commun.Math.Phys.223:289-326,2001 (\href{https://arxiv.org/abs/gr-qc/0103074}{arXiv:gr-qc/0103074})


\item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Perturbative algebraic field theory, and deformation quantization}, in [[Roberto Longo]] (ed.), \emph{Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects}, volume 30 of Fields Institute Communications, pages 151--160. American Mathematical Society, 2001 (\href{https://arxiv.org/abs/hep-th/0101079}{arXiv:hep-th/0101079})


\item [[Stefan Hollands]], Weihua Ruan, \emph{The State Space of Perturbative Quantum Field Theory in Curved Spacetimes}, Annales Henri Poincare 3 (2002) 635-657 (\href{https://arxiv.org/abs/gr-qc/0108032}{arXiv:gr-qc/0108032})


\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Pedro Lauridsen Ribeiro]], section 4.1 of \emph{Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields} (\href{https://arxiv.org/abs/1209.2148}{arXiv:1209.2148})



\end{itemize}
\hypertarget{review}{}\subsubsection*{{Review}}\label{review}

\begin{itemize}%
\item [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], \emph{Perturbative algebraic quantum field theory}, In \emph{Mathematical Aspects of Quantum Field Theories}, Springer 2016 (\href{https://arxiv.org/abs/1208.1428}{arXiv:1208.1428})


\item [[Igor Khavkine]], [[Valter Moretti]], \emph{Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction}, Chapter 5 in [[Romeo Brunetti]] et al. (eds.) \emph{Advances in Algebraic Quantum Field Theory}, Springer, 2015 (\href{https://arxiv.org/abs/1412.5945}{arXiv:1412.5945})


\item [[Katarzyna Rejzner]], section 4.4.1 of \emph{Perturbative Algebraic Quantum Field Theory}, Mathematical Physics Studies, Springer 2016 (\href{https://link.springer.com/book/10.1007%2F978-3-319-25901-7}{web})


\item [[Michael Dütsch]], def. 1.2 and equation (2.47) in \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018



\end{itemize}
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