nLab Bogoliubov's formula

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 below) and mathematically justified by the fact that it does yield a causally local net of observables in causal perturbation theory (prop. 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

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

be an adiabatically switched interaction-functional.

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. 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. 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.

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

For proof see this prop. at S-matrix.

Related concepts

• quantum Møller operator

• interacting field algebra

product in perturbative QFT	$\,\,$ induces
normal-ordered product	Wick algebra (free field quantum observables)
time-ordered product	S-matrix (scattering amplitudes)
retarded product	interacting quantum observables

References

The formula originates in

• Nikolay Bogoliubov, Dmitry Shirkov, Introduction to the Theory of Quantized Fields, New York (1959)

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

• Henri Epstein, Vladimir Glaser, The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211 (Numdam)

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

• Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)

and made more explicit in

• Michael Dütsch, Klaus Fredenhagen, Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion, Commun.Math.Phys. 219 (2001) 5-30 (arXiv:hep-th/0001129)

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

• Giovanni Collini, section 2.2 of Fedosov Quantization and Perturbative Quantum Field Theory (arXiv:1603.09626)

• Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory (arXiv:1612.09157)

Review includes

• Michael Dütsch, Klaus Fredenhagen, Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion, Commun.Math.Phys. 219 (2001) 5-30 (arXiv:hep-th/0001129)

• Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (pdf)