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\section*{Bogoliubov&#39;s formula}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_the_wouldbe_path_integral}{Relation to the would-be path integral}\dotfill \pageref*{relation_to_the_wouldbe_path_integral} \linebreak
\noindent\hyperlink{causal_locality_of_interacting_field_quantum_observables}{Causal locality of interacting field quantum observables}\dotfill \pageref*{causal_locality_of_interacting_field_quantum_observables} \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 ([[perturbative quantum field theory|perturbative]]) [[quantum field theory]], what is called \emph{Bogoliubov's formula}, originally due to (\hyperlink{BogoliubovShirkov59}{Bogoliubov-Shirkov 59}) is an expression for the [[interacting quantum observables]] as the [[derivatives]] with respect to a [[source field]] of the [[generating function]] corresponding to a given [[S-matrix]].

Originally this is an ad-hoc formula, motivated by the would-be [[path integral]] picture of the [[Feynman perturbation series]] (remark \ref{InterpretationOfPerturbativeSMatrix} below) and mathematically justified by the fact that it does yield a [[causally local net of observables]] in [[causal perturbation theory]] (prop. \ref{CausalLocalityOfBogoliubovFormula} below).

Much more recently it was shown that Bogoliubov's formula indeed expresses quantum observables as systematically obtained by [[Fedosov deformation quantization]] of [[field theory]] (\hyperlink{Collini16}{Collini 16}, \hyperlink{HawkinsRejzner16}{Hawkins-Rejzner 16}).

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

Let $\Sigma$ be a [[spacetime]] of [[dimension]] $p + 1$ and let $E \overset{fb}{\longrightarrow} \Sigma$ be a [[field bundle]]. Let $\mathbf{L}_{free}\in \Omega^{p+1,0}_\Sigma(E)$ be a [[local Lagrangian density]] for a [[free field theory]] with [[field (physics)|fields]] of type $E$. Let $\mathcal{W}$ be the corresponding [[Wick algebra]] of [[quantum observables]] of the free field, with

\begin{displaymath}
LocObs(E_{\text{BV-BRST}}) 
    \overset{:(-):}{\longrightarrow} 
  PolyObs(E_{\text{BV-BRST}})
\end{displaymath}
the corresponding quantization map from [[local observables]] (``[[normal ordering]]'').

Let then

\begin{displaymath}
\mathcal{S}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
    \longrightarrow 
  PolyObs(E_{\text{BV-BRST}})((\hbar))[ [g,j] ]
\end{displaymath}
be a [[perturbative S-matrix]] scheme. Moreover let

\begin{displaymath}
g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g] ]
\end{displaymath}
be an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]].

For $A \in \mathcal{F}_{loc}$ a [[local observable]] and $j \in C^\infty_{cp}(\Sigma)$, write

\begin{displaymath}
\mathcal{Z}_{g S_{int}}(j A)
  \; \coloneqq \;
  \mathcal{S}(g S_{int})^{-1} \mathcal{S}( g S_{int} + j A )
\end{displaymath}
for the [[generating function]] induced by the perturbative [[S-matrix]] (where the product shown by juxtaposition is that in the [[Wick algebra]], hence the [[star product]] induced by the [[Wightman propagator]]).

\begin{defn}
\label{GeneratingFunctionsForCorrelationFunctions}\hypertarget{GeneratingFunctionsForCorrelationFunctions}{}
\textbf{([[Bogoliubov's formula]])}

The perturbative [[interacting field observable]]

\begin{displaymath}
A_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
\end{displaymath}
corresponding to a [[free field]] [[local observable]] $A \in LocObs(E_{\text{BV-BRST}})$ is the [[coefficient]] in the [[generating function]] $\mathcal{S}$ (\href{S-matrix#SchemeGeneratingFunction}{this def.}) of the term linear in the [[source field]] strenght $j$:

\begin{displaymath}
A_{int}
    \;\coloneqq\;
  i \hbar \frac{d}{d  j } \mathcal{Z}_{ g S_{int}}( j  A)\vert_{j = 0}
  \,.
\end{displaymath}
\end{defn}
This is due to \hyperlink{BogoliubovShirkov59}{Bogoliubov-Shirkov 59}, later named \emph{Bogoliubov's formula} (e.g. \hyperlink{Rejzner16}{Rejzner 16 (6.12)}). Based on this [[causal perturbation theory]] was formulated in (\hyperlink{EpsteinGlaser73}{Epstein-Glaser 73 around (74)}). Review includes (\hyperlink{DuetschFredenhagen00}{D\"u{}tsch-Fredenhagen 00, around (17)}).

The assignment $A \mapsto A_{int}$ is also called the \emph{[[quantum Møller operator]]}.

The [[coefficients]] of $A_{int}$ as a [[formal power series]] in the [[coupling constant]] and [[Planck's constant]] are called the \emph{[[retarded products]]}.

\begin{remark}
\label{PowersInPlancksConstant}\hypertarget{PowersInPlancksConstant}{}
\textbf{(powers in [[Planck's constant]])}

That the observables as defined in def. \ref{GeneratingFunctionsForCorrelationFunctions} indeed are [[formal power series]] in $\hbar$ as opposed to more general [[Laurent series]] requires a little argument.

The explicit $\hbar$-dependence of the perturbative [[S-matrix]] is

\begin{displaymath}
\mathcal{S}(g S_{int} + j A)
  = 
  T \exp\left(
    \tfrac{1}{i \hbar}
     \left(
       g S_{int} + j A
     \right)
  \right)
  \,,
\end{displaymath}
where $T(-)$ denotes [[time-ordered products]]. The generating function

\begin{displaymath}
\mathcal{Z}_{S_{int}}(j A)
  \;\coloneqq\;
  \mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(g S_{int} + j A)
\end{displaymath}
involves the [[star product]] of the free theory (the [[normal-ordered product]] of the [[Wick algebra]]). This is a [[formal deformation quantization]] of the [[Peierls-Poisson bracket]], and therefore the [[commutator]] in this algebra is a [[formal power series]] in $\hbar$ that however has no constant term in $\hbar$ (but starts out with $\hbar$ times the [[Poisson bracket]], followed by possibly higher order terms in $\hbar$):

\begin{displaymath}
[L_{int},A] \;=\; \hbar(\cdots)
  \,.
\end{displaymath}
Now schematically the derivative of the generating function is of the form

\begin{displaymath}
\begin{aligned}
    A_{int}
    & \coloneqq
    i \hbar \frac{d}{d j }
    \mathcal{Z}_{ S_{int}}(j A)\vert_{j  = 0}
    \\
    & =
    \exp\left(
      \tfrac{1}{i \hbar}[g S_{int}, -]
    \right)
    (A)
  \end{aligned}
  \,.
\end{displaymath}
(The precise expression is given by the \emph{[[retarded products]]}, see \hyperlink{DuetschFredenhagen00}{Dütsch-Fredenhagen 00, prop. 2 (ii)} \hyperlink{Rejzner16}{Rejzner 16, prop. 6.1}, \hyperlink{HawkinsRejzner16}{Hawkins-Rejzner 16, cor. 5.2}.) By the above, the exponent here yields a [[formal power series]] in $\hbar$, and hence so does the full exponential.

That the quantum observables takes values in formal power series of $\hbar$ is the hallmark of [[formal deformation quantization]]. While [[Bogoliubov's formula]] proceeds from an [[S-matrix]] which is axiomatized by [[causal perturbation theory]], this suggests that it actually computes the quantum observables in a [[formal deformation quantization]] of the interacting field theory. This is indeed the case (\hyperlink{Collini16}{Collini 16}, \hyperlink{HawkinsRejzner16}{Hawkins-Rejzner 16}).

For the analogous analysis of powers of $\hbar$ in the [[S-matrix]] itself, see at \emph{[[loop order]]} the section \emph{\href{loop%20order#RelationToPowersInPlancksConstant}{Relation to powers in Planck's constant}}.

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

\hypertarget{relation_to_the_wouldbe_path_integral}{}\subsubsection*{{Relation to the would-be path integral}}\label{relation_to_the_wouldbe_path_integral}

\begin{remark}
\label{InterpretationOfPerturbativeSMatrix}\hypertarget{InterpretationOfPerturbativeSMatrix}{}
\textbf{(interpretation of Bogoliubov's formula via a ``[[path integral]]'')}

In informal heuristic discussion of [[perturbative quantum field theory]] the [[S-matrix]] is thought of as a [[path integral]], written

\begin{displaymath}
S\left(
    \tfrac{g}{\hbar} L_{int} + j
  \right)
  \;\overset{\text{not really!}}{=}\;
  \underset{\Phi \in \Gamma_\Sigma(E)_{asmpt}}{\int}
  \exp\left(
    \int_X
    \left(
      \tfrac{g}{i \hbar} L_{int}(\Phi) + j A(\Phi)
    \right)
  \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]
\end{displaymath}
where the integration is thought to be over the [[configuration space]] $\Gamma_\Sigma(E)_{asmpt}$ of [[field (physics)|fields]] $\Phi$ (the [[space of sections]] of the given [[field bundle]]) which satisfy given asymptotic conditions at $x^0 \to \pm \infty$; and as these boundary conditions vary the above is regarded as an integral kernel that defines the required operator in $\mathcal{W}$.

With the perturbative S-matrix informally thought of as a path integral this way, the the Bogoliubov formula in def. \ref{GeneratingFunctionsForCorrelationFunctions} similarly would have the following interpretation:

\begin{displaymath}
A_{int}
  \;\overset{\text{not really!}}{=}\;
  \frac{
    \int
    j A(\Phi)
    \exp\left(
      \int_X
      \left(
        \tfrac{g}{i \hbar} L_{int}(\Phi)
      \right)
    \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]
  }
  {
    \int
    \exp\left(
      \int_X
      \left(
        \tfrac{g}{i \hbar} L_{int}(\Phi)
      \right)
    \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]
  }
\end{displaymath}
If here we were to regard the expression

\begin{displaymath}
\mu(\Phi)
  \;\overset{\text{not really}}{\coloneqq}\;
  \frac{
    \exp\left(
      \int_X
      \left(
        \tfrac{g}{i \hbar} L_{int}(\Phi)
      \right)
    \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]
  }
  {
    \int
    \exp\left(
      \int_X
      \left(
        \tfrac{g}{i \hbar} L_{int}(\Phi)
      \right)
    \right)  e^{\tfrac{1}{i \hbar}\int_X L_{free}(\phi) }D[\phi]
  }
\end{displaymath}
as a ``complex probability measure'' on the the configuration space of fields, then this formula would express the [[expectation value]] of the functional $A$ under this measure:

\begin{displaymath}
A_{int} 
    \overset{\text{not really!}}{=} 
   [A]_{\mu} = \int  A(\Phi) \, \mu(\Phi)
  \,.
\end{displaymath}
\end{remark}
\hypertarget{causal_locality_of_interacting_field_quantum_observables}{}\subsubsection*{{Causal locality of interacting field quantum observables}}\label{causal_locality_of_interacting_field_quantum_observables}

\begin{prop}
\label{CausalLocalityOfBogoliubovFormula}\hypertarget{CausalLocalityOfBogoliubovFormula}{}
\textbf{([[causal locality]])}

As the spacetime support varies, the algebras of [[interacting field quantum observables]] spanned via the Bogoliubov formula consistitute a [[causally local net of observables]], hence an instance of [[perturbative AQFT]].

\end{prop}
(\hyperlink{DuetschFredenhagen00}{D\"u{}tsch-Fredenhagen 00, section 3}, following \hyperlink{BrunettiFredenhagen99}{Brunetti-Fredenhagen 99, section 8}, \hyperlink{IlinSlavnov78}{Il'in-Slavnov 78})

For \textbf{[[proof]]} see \href{S-matrix#PerturbativeQuantumObservablesIsLocalnet}{this prop.} at \emph{[[S-matrix]]}.

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

\begin{itemize}%
\item [[quantum Møller operator]]


\item [[interacting field algebra]]



\end{itemize}
[[!include products in pQFT -- table]]

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

The formula originates in

\begin{itemize}%
\item [[Nikolay Bogoliubov]], [[Dmitry Shirkov]], \emph{Introduction to the Theory of Quantized Fields}, New York (1959)

\end{itemize}
Its rigorous discussion in terms of [[causal perturbation theory]] is due 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}
The observation that the Bogoliubov formula yields a [[causally local net of quantum observables]] is due to

\begin{itemize}%
\item V. A. Il'in and D. S. Slavnov, \emph{Observable algebras in the S-matrix approach}, Theor. Math. Phys. 36 (1978) 32. (\href{http://inspirehep.net/record/135575}{spire}, \href{http://dx.doi.org/10.1007/BF01035870}{doi})

\end{itemize}
then rediscovered in

\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})

\end{itemize}
and made more explicit in

\begin{itemize}%
\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})

\end{itemize}
The recognition of the Bogoliubov formula as the result [[formal deformation quantization]] and specifically of [[Fedosov deformation quantization]] is due to

\begin{itemize}%
\item [[Giovanni Collini]], section 2.2 of \emph{Fedosov Quantization and Perturbative Quantum Field Theory} (\href{https://arxiv.org/abs/1603.09626}{arXiv:1603.09626})


\item [[Eli Hawkins]], [[Kasia Rejzner]], \emph{The Star Product in Interacting Quantum Field Theory} (\href{https://arxiv.org/abs/1612.09157}{arXiv:1612.09157})



\end{itemize}
Review includes

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



\end{itemize}
[[!redirects Bogoliubov formula]]



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