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\begin{document}

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\section*{causal additivity}

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

\hypertarget{algebraic_qunantum_field_theory}{}\paragraph*{{Algebraic Qunantum Field Theory}}\label{algebraic_qunantum_field_theory}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{causal_additivity}{Causal additivity}\dotfill \pageref*{causal_additivity} \linebreak
\noindent\hyperlink{causal_factorization}{Causal factorization}\dotfill \pageref*{causal_factorization} \linebreak
\noindent\hyperlink{causal_locality_of_quantum_observables}{Causal locality of quantum observables}\dotfill \pageref*{causal_locality_of_quantum_observables} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The formalization of [[perturbative quantum field theory]] via [[causal perturbation theory]] states the key properties that the [[scattering matrix]] of the theory is supposed to have as [[axioms]] (instead of imagining that the scattering matrix is defined by a [[path integral]]). The key such axiom is the statement that the [[S-matrix]] be \emph{causally additive} (which gives \emph{[[causal perturbation theory]]} its name). This essentially encodes the idea that effects in the quantum theory propagate causally, hence within the [[future cone]] and [[past cone]] of the region which causes the effect.

\hypertarget{causal_additivity}{}\subsection*{{Causal additivity}}\label{causal_additivity}

The [[S-matrix]] (as discussed there) is a [[functional]] of the form

\begin{displaymath}
S \;\colon\; \mathcal{F}_{loc} \longrightarrow \mathcal{W}[ [ \tfrac{g}{\hbar} ] ]
\end{displaymath}
sending \hyperlink{Wick+algebra#CompactlySupportedPolynomialLocalDensities}{local observables} with [[compact support]] in [[spacetime]] (i.e. [[adiabatic switching|adiabatically switched]] [[interaction]] [[Lagrangian densities]] $L_{int}$ and [[source fields]] $J$) to [[formal power series]] in the ratio of a [[coupling constant]] $g$ over [[Planck's constant]] $\hbar$ with [[coefficients]] in the [[Wick algebra]] $\mathcal{W}$ of the underlying [[free field theory]].

Causal additivity (\hyperlink{EpsteinGlaser73}{Epstein-Glaser 73}) is the statement/requirement that

\begin{displaymath}
\left( supp(J_1) \geq supp(J_2) \right)
  \;\; \Rightarrow \;\;
  \left(
    \underset{L \in \mathcal{F}_{loc}}{\forall}
    \left(
      S(L + J_1 + J_2)
      =
      S(L + J_1) S(L)^{-1} S(L + J_2)
    \right)
  \right)
  \,.
\end{displaymath}
Here $supp(-)$ denotes the [[spacetime]] [[support]] of the given \hyperlink{Wick+algebra#CompactlySupportedPolynomialLocalDensities}{local observables} and $supp(J_1) \geq supp(J_2)$ means that $supp(J_1)$ does not intersect the [[causal past]] of $supp(J_2)$. The product on the right is the product in the [[Wick algebra]] (hence the [[normal ordered product]]).

\hypertarget{causal_factorization}{}\subsection*{{Causal factorization}}\label{causal_factorization}

As a special case, causal additivity immediately implies \textbf{causal factorization}:

\begin{displaymath}
\left( supp(L_1) \geq supp(L_2) \right)
  \;\; \Rightarrow \;\;
  \left(
    S(L_1 + L_2)
    =
    S(L_1) S(L_2)
  \right)
  \,.
\end{displaymath}
This in turn directly implies that the perturbative expansion of the [[S-matrix]] is given by [[time-ordered products]].

\hypertarget{causal_locality_of_quantum_observables}{}\subsection*{{Causal locality of quantum observables}}\label{causal_locality_of_quantum_observables}

But the reason why the full condition is called \emph{causal additivity} is that it is equivalent to a simpler-looking condition on the [[generating function]]

\begin{displaymath}
Z_{L}(J)
  \;\colon\;
  S(L)^{-1} S(L + J)
\end{displaymath}
that is induced by the [[S-matrix]]. In terms of these, causal additivity is equivalently (by \href{S-matrix#CausalLocalityOfThePerturbativeSMatrix}{this lemma}) the condition

\begin{displaymath}
\left( supp(J_1) \geq supp(J_2) \right)
  \;\; \Rightarrow \;\;
  \left(
    \underset{L \in \mathcal{F}_{loc}}{\forall}
    \left(
      Z_{L}(J_1 + J_2)
      =
      Z_L(J_1) Z_L(J_2)
    \right)
  \right)
  \,.
\end{displaymath}
Notice that what these generating functions generate is the perturbative [[quantum observables]]

\begin{displaymath}
\hat A
  \coloneqq
  \frac{d}{d \epsilon}
  T_{\tfrac{g}{\hbar}L_{int}}( \epsilon A )
  \vert|_{\epsilon = 0}
\end{displaymath}
and causal additivity implies that as long as $L_{int}$ is [[adiabatic switching|adiabatically switched]] only outside the [[causal closure]] of $supp(A)$, then the system of quantum observables satisfies \textbf{[[causal locality]]} and hence forms a [[local net of observables]] (\hyperlink{S-matrix#PerturbativeQuantumObservablesIsLocalnet}{this prop.}).

This is how [[causal perturbation theory]] gives rise to \emph{[[perturbative AQFT]]}.

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

The concept goes back to

\begin{itemize}%
\item [[Henri Epstein]], [[Vladimir Glaser]], \emph{[[The Role of locality in perturbation theory]]}, Annales Poincar\'e{} Phys. Theor. A 19 (1973) 211 (\href{http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0}{Numdam})

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[geometry of physics -- perturbative quantum field theory]]} (around \href{geometry+of+physics+--+perturbative+quantum+field+theory#eq:CausalAdditivity}{this equation})

\end{itemize}
See also the references at \emph{[[causal perturbation theory]]}, \emph{[[perturbative AQFT]]} and \emph{[[S-matrix]]}.

[[!redirects causally additive]]

[[!redirects causal factorization]] [[!redirects causal factorizations]]



\end{document}