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\section*{time-ordered product}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{OnRegularPolynomialObservables}{On regular polynomial observables}\dotfill \pageref*{OnRegularPolynomialObservables} \linebreak
\noindent\hyperlink{OnLocalObservables}{On local observables}\dotfill \pageref*{OnLocalObservables} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[relativistic field theory|relativistic]] [[perturbative quantum field theory]], the \emph{time-ordered product} is a product on suitably well-behave [[observables]] which re-orders its arguments according to the [[causal ordering]] of their spacetime supports befor multiplying with the [[Wick algebra]] product.

(Analogously reverse [[causal ordering]] this is called the \emph{reverse-time ordered} or \emph{anti-time ordered} prouct.)

For example for point-evaluation [[field observables]] and distinct [[events]] $x,y \in \Sigma$ the time-ordered product is defined by

\begin{displaymath}
T(\mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y))
  \;\coloneqq\;
  \left\{
    \itexarray{
      \mathbf{\Phi}^a(x) \mathbf{\Phi}^a(y) 
        &\vert& 
      x\, \text{not in the past of}\ y
      \\
      \mathbf{\Phi}^a(y) \mathbf{\Phi}^a(x) &\vert& \text{otherwise} 
    }
  \right.
\end{displaymath}
This may be understood as arising from the [[causal additivity]]-[[axiom]] of the [[perturbative S-matrix]]. It generalizes the 1-dimensional time-ordering (path ordering) of the [[Dyson series]] in [[quantum mechanics]].

More precisely, the time-ordere product is a [[commutative algebra]]-[[structure]] on the [[microcausal polynomial observables]] of a [[free field theory|free]] [[Lagrangian field theory]] equipped with a [[vacuum state]] ([[Hadamard state]]) which on [[regular polynomial observables]] given on the [[regular polynomial observables]] by the [[star product]] which is induced (via \href{star+product#PropagatorStarProduct}{this def.}) by the [[Feynman propagator]] and which is extended from there, in the sense of [[extensions of distributions]], to all [[microcausal polynomial observables]]. (This extension is the ``[[renormalization]]'' of the time-ordered product).

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

\hypertarget{OnRegularPolynomialObservables}{}\subsubsection*{{On regular polynomial observables}}\label{OnRegularPolynomialObservables}

\begin{defn}
\label{OnRegularPolynomialObservablesTimeOrderedProduct}\hypertarget{OnRegularPolynomialObservablesTimeOrderedProduct}{}
\textbf{([[time-ordered product]] on [[regular polynomial observables]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H$ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].

Then the \emph{[[time-ordered product]]} on the space of [[off-shell]] [[regular polynomial observable]] $PolyObs(E)_{reg}$ is the [[star product]] induced by the [[Feynman propagator]] (via \href{star+product#PropagatorStarProduct}{this prop.}):

\begin{displaymath}
\itexarray{
    PolyObs(E)_{reg}[ [\hbar] ]
    \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{}{\longrightarrow}&
    PolyObs(E)_{reg}[ [\hbar] ]
    \\
    (A_1, A_2)
    &\mapsto&
    \phantom{\coloneqq} A_1 \star_F A_2
  }
\end{displaymath}
hence

\begin{displaymath}
A_1 \star_F A_2
  \;
  \coloneqq
  \;
  ((-)\cdot(-))
  \circ
  \exp\left(
    \underset{\Sigma \times \Sigma}{\int}
    \Delta_F^{a b}(x,y)
    \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
    \otimes
    \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
    \,
    dvol_\Sigma(x)
    \,
    dvol_\Sigma(y)
  \right)
\end{displaymath}
(Notice that this does not descend to the [[on-shell]] observables, since the [[Feynman propagator]] is not a solution to the \emph{homogeneous} [[equations of motion]].)

\end{defn}
\begin{prop}
\label{CausalOrderingTimeOrderedProductOnRegular}\hypertarget{CausalOrderingTimeOrderedProductOnRegular}{}
\textbf{([[time-ordered product]] is indeed causally ordered [[Wick algebra]] product)}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H$ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].

Then the [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is indeed a time-ordering of the [[Wick algebra]] product $\star_H$ in that for all [[pairs]] of [[regular polynomial observables]]

\begin{displaymath}
A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ]
\end{displaymath}
with [[disjoint subset|disjoint]] [[spacetime]] [[support]] we have

\begin{displaymath}
T(A_1 A_2)
  \;=\;
  \left\{
    \itexarray{
      A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
      \\
      A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
    }
  \right.
  \,.
\end{displaymath}
Here $S_1 {\vee\!\!\!\wedge} S_2$ is the [[causal order]] relation (``$S_1$ does not intersect the [[past cone]] of $S_2$''). Beware that for general [[pairs]] $(S_1, S-2)$ of subsets neither $S_1 {\vee\!\!\!\wedge} S_2$ nor $S_2 {\vee\!\!\!\wedge} S_1$.

\end{prop}
\begin{proof}
Recall the following facts:

\begin{enumerate}%
\item the [[advanced and retarded propagators]] $\Delta_{\pm}$ by definition are [[support|supported]] in the [[future cone]]/[[past cone]], respectively

\begin{displaymath}
supp(\Delta_{\pm}) \subset \overline{V}^{\pm}
\end{displaymath}

\item they turn into each other under exchange of their arguments (\href{causal+propagator#CausalPropagatorIsSkewSymmetric}{this cor.}):

\begin{displaymath}
\Delta_\pm(y,x) = \Delta_{\mp}(x,y)
  \,.
\end{displaymath}

\item the real part $H$ of the [[Feynman propagator]], which by definition is the real part of the [[Wightman propagator]] is symmetric (by definition or else by \href{Hadamard+distribution#SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}{this prop.}):

\begin{displaymath}
H(x,y) = H(y,x)
\end{displaymath}


\end{enumerate}
Using this we compute as follows:

\begin{displaymath}
\begin{aligned}
    A_1 \underset{\Delta_{F}}{\star} A_2
    \;
    & =
    A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2
    \\
    & =
    \left\{
      \itexarray{
        A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)    
      }
    \right.
    \\
    & = 
    \left\{
      \itexarray{
        A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)    
      }
    \right.
    \\
    & =
    \left\{
      \itexarray{
        A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)    
      }
    \right.
    \\
    & =
    \left\{
      \itexarray{
        A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)    
      }
    \right.
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{prop}
\label{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}\hypertarget{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{}
\textbf{([[time-ordered product]] on [[regular polynomial observables]] [[isomorphism|isomorphic]] to pointwise product)}

The [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{CausalOrderingTimeOrderedProductOnRegular}) is [[isomorphism|isomorphism]] to the pointwise product of [[observables]] (\href{A+first+idea+of+quantum+field+theory#Observable}{this def.}) via the [[linear isomorphism]]

\begin{displaymath}
\mathcal{T}
  \;\colon\;
  PolyObs(E)_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{reg}[ [\hbar] ]
\end{displaymath}
given by

\begin{displaymath}
\mathcal{T}A
  \;\coloneqq\;
  \exp\left(
    \tfrac{1}{2}
    \hbar 
    \underset{\Sigma}{\int} 
     \Delta_F(x,y)^{a b} 
     \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
  \right)
  A
\end{displaymath}
in that

\begin{displaymath}
\begin{aligned}
    T(A_1 A_2)
    & \coloneqq
    A_1 \star_{F} A_2
    \\
    & =
    \mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) )
  \end{aligned}
\end{displaymath}
hence

\begin{displaymath}
\itexarray{
    PolyObs(E)_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{(-)\cdot (-)}{\longrightarrow}&   
    PolyObs(E)_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow 
      && 
    \downarrow^{\mathrlap{\mathcal{T}}}_\simeq
    \\
    PolyObs(E)_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{(-) \star_F (-)}{\longrightarrow}&   
    PolyObs(E)_{reg}[ [\hbar] ]
  }
\end{displaymath}
\end{prop}
(\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, (12)-(13)}, \hyperlink{FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b, (14)})

\begin{proof}
Since the [[Feynman propagator]] is symmetric (\href{A+first+idea+of+quantum+field+theory#SymmetricFeynmanPropagator}{this prop.}), the statement is a special case of \href{star+product#SymmetricContribution}{this prop.}).

\end{proof}
\begin{example}
\label{RegularObservablesExponentialTimeOrdered}\hypertarget{RegularObservablesExponentialTimeOrdered}{}
\textbf{([[time-ordered product|time-ordered]] [[exponential]] of [[regular polynomial observables]])}

Let

\begin{displaymath}
V \in PolyObs_{reg, deg = 0}[ [ \hbar ] ]
\end{displaymath}
be a [[regular polynomial observables]] of degree zero, and write

\begin{displaymath}
\exp(V)
  =
  1 + V + \tfrac{1}{2!} V \cdot V + \tfrac{1}{3!} V \cdot V \cdot V + \cdots
\end{displaymath}
for the [[exponential]] of $V$ with respect to the pointwise product.

Then the [[exponential]] $\exp_{\mathcal{T}}(V)$ of $V$ with respect to the [[time-ordered product]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equal to the [[conjugation]] of the exponential with respect to the pointwise product by the time-ordering isomorphism $\mathcal{T}$ from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}:

\begin{displaymath}
\begin{aligned}
    \exp_{\mathcal{T}}(V)
    &
    \coloneqq
    1 + V + \tfrac{1}{2} V \star_F V + \tfrac{1}{3!} V \star_F V \star_F V + \cdots
    \\
    & = 
    \mathcal{T} \circ \exp(-) \circ \mathcal{T}^{-1}(V)
  \end{aligned}
\end{displaymath}
\end{example}
\hypertarget{OnLocalObservables}{}\subsubsection*{{On local observables}}\label{OnLocalObservables}

The time-ordered product on regular polynomial observables from prop. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} extends to a product on [[polynomial observable|polynomial]] [[local observables]], then taking values in [[microcausal observables]]:

\begin{displaymath}
T
  \;\colon\;
  PolyLocObs(E)^{\otimes_n}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{mc}[ [\hbar] ]
  \,.
\end{displaymath}
This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of \emph{[[renormalization scheme]]} for the given [[perturbative quantum field theory]]. Every such choice corresponds to a choice of [[perturbative S-matrix]] for the theory. This construction is called \emph{[[causal perturbation theory]]}.

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

[[!include products in pQFT -- table]]

$\,$

[[!include Wick algebra -- table]]

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

See also the references at \emph{[[S-matrix]]}

The equivalence of the time-ordered product on regular observables to the point-wise product was maybe first highlighted in

\begin{itemize}%
\item [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], p. 6 of \emph{Perturbative Algebraic Quantum Field Theory and the Renormalization Groups}, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (\href{http://arxiv.org/abs/0901.2038}{arXiv:0901.2038})

\end{itemize}
and then further amplified in

\begin{itemize}%
\item [[Klaus Fredenhagen]], [[Kasia Rejzner]], p. 6 of \emph{Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory}, Commun. Math. Phys. 317(3), 697--725 (2012) (\href{https://arxiv.org/abs/1110.5232}{arXiv:1110.5232})

\end{itemize}
[[!redirects time-ordered products]]

[[!redirects time ordered product]] [[!redirects time ordered products]]

[[!redirects reverse-time ordered product]] [[!redirects reverse-time ordered products]]

[[!redirects anti-time ordered product]] [[!redirects anti-time ordered products]]



\end{document}