Bogoliubov's formula


Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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).


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

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

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

Let then

𝒮:LocObs(E BV-BRST)[[,g,j]]PolyObs(E BV-BRST)(())[[g,j]] \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

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

be an adiabatically switched interaction-functional.

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

𝒵 gS int(jA)𝒮(gS int) 1𝒮(gS int+jA) \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).


(Bogoliubov's formula)

The perturbative interacting field observable

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

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

A intiddj𝒵 gS int(jA)| j=0. 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 AA intA \mapsto A_{int} is also called the quantum Møller operator.

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


(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

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

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

𝒵 S int(jA)𝒮(S int) 1 H𝒮(gS int+jA) \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]=(). [L_{int},A] \;=\; \hbar(\cdots) \,.

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

A int iddj𝒵 S int(jA)| j=0 =exp(1i[gS int,])(A). \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.


Relation to the would-be path integral


(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(gL int+j)=not really!ΦΓ Σ(E) asmptexp( X(giL int(Φ)+jA(Φ)))e 1i XL free(Φ)D[Φ] 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 Γ Σ(E) asmpt\Gamma_\Sigma(E)_{asmpt} of fields Φ\Phi (the space of sections of the given field bundle) which satisfy given asymptotic conditions at x 0±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 𝒲\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=not really!jA(Φ)exp( X(giL int(Φ)))e 1i XL free(Φ)D[Φ]exp( X(giL int(Φ)))e 1i XL free(Φ)D[Φ] 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

μ(Φ)not reallyexp( X(giL int(Φ)))e 1i XL free(Φ)D[Φ]exp( X(giL int(Φ)))e 1i XL free(ϕ)D[ϕ] \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 AA under this measure:

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

Causal locality of interacting field quantum observables


(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.

(Dütsch-Fredenhagen 00, section 3, following Brunetti-Fredenhagen 99, section 8, Il’in-Slavnov 78)

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


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

and made more explicit in

The recognition of the Bogoliubov formula as the result formal deformation quantization and specifically of Fedosov deformation quantization is due to

Review includes

Revised on January 15, 2018 08:24:16 by Urs Schreiber (