# Contents

## Idea

In (perturbative) quantum field theory, what is called Bogoliubov’s formula, originally due to (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 2 below) and mathematically justified by the fact that it does yield a causally local net of observables in causal perturbation theory (prop. 1 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 (Collini 16, Hawkins-Rejzner 16).

## 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 fields of type $E$. Let $\mathcal{W}$ be the corresponding Wick algebra of quantum observables of the free field, with

$LocObs(E_{\text{BV-BRST}}) \overset{:(-):}{\longrightarrow} PolyObs(E_{\text{BV-BRST}})$

the corresponding quantization map from local observables (“normal ordering”).

Let then

$\mathcal{S} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})((\hbar))[ [g,j] ]$

be a perturbative S-matrix scheme. Moreover let

$g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g] ]$

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

$\mathcal{Z}_{g S_{int}}(j A) \; \coloneqq \; \mathcal{S}(g S_{int})^{-1} \mathcal{S}( g S_{int} + j A )$

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).

###### Definition

(Bogoliubov's formula)

The perturbative interacting field observable

$A_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]$

corresponding to a free field local observable $A \in LocObs(E_{\text{BV-BRST}})$ is the coefficient in the generating function $\mathcal{S}$ (this def.) of the term linear in the source field strenght $j$:

$A_{int} \;\coloneqq\; i \hbar \frac{d}{d j } \mathcal{Z}_{ g S_{int}}( j A)\vert_{j = 0} \,.$

This is due to Bogoliubov-Shirkov 59, later named Bogoliubov’s formula (e.g. Rejzner 16 (6.12)). Based on this causal perturbation theory was formulated in (Epstein-Glaser 73 around (74)). Review includes (Dütsch-Fredenhagen 00, around (17)).

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

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

###### Remark

(powers in Planck's constant)

That the observables as defined in def. 1 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

$\mathcal{S}(g S_{int} + j A) = T \exp\left( \tfrac{1}{i \hbar} \left( g S_{int} + j A \right) \right) \,,$

where $T(-)$ denotes time-ordered products. The generating function

$\mathcal{Z}_{S_{int}}(j A) \;\coloneqq\; \mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(g S_{int} + j A)$

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 $\hbar$):

$[L_{int},A] \;=\; \hbar(\cdots) \,.$

Now schematically the derivative of the generating function is of the form

\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} \,.

(The precise expression is given by the retarded products, see Dütsch-Fredenhagen 00, prop. 2 (ii) Rejzner 16, prop. 6.1, 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 (Collini 16, Hawkins-Rejzner 16).

For the analogous analysis of powers of $\hbar$ in the S-matrix itself, see at loop order the section Relation to powers in Planck’s constant.

## Properties

### Relation to the would-be path integral

###### Remark

(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

$S\left( \tfrac{g}{\hbar} L_{int} + j \right) \;\overset{\text{not really!}}{=}\; \underset{\Phi \in \Gamma_\Sigma(E)_{asmpt}}{\int} \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) + j A(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]$

where the integration is thought to be over the configuration space $\Gamma_\Sigma(E)_{asmpt}$ of 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. 1 similarly would have the following interpretation:

$A_{int} \;\overset{\text{not really!}}{=}\; \frac{ \int j A(\Phi) \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi] } { \int \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi] }$

If here we were to regard the expression

$\mu(\Phi) \;\overset{\text{not really}}{\coloneqq}\; \frac{ \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi] } { \int \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\phi) }D[\phi] }$

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:

$A_{int} \overset{\text{not really!}}{=} [A]_{\mu} = \int A(\Phi) \, \mu(\Phi) \,.$

### Causal locality of interacting field quantum observables

###### Proposition

(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.

For proof see this prop. at S-matrix.

product in perturbative QFT$\,\,$ induces
normal-ordered productWick algebra (free field quantum observables)
time-ordered productS-matrix (scattering amplitudes)
retarded productinteracting quantum observables

## References

The formula originates in

Its rigorous discussion in terms of causal perturbation theory is due to

The observation that the Bogoliubov formula yields a causally local net of quantum observables is due to

• V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32. (spire, doi)

then rediscovered in