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Communications in Theoretical Physics

Wick’s theorem for superoperators linear in bosonic operators and its applications

Yi Yang, Rui Zhang, Jian Ma, Zhaoyu Fei, X G 晓光 Wang 王

Published May 13, 2026• 0 citations
Publisher

Abstract

Abstract For the superoperators that are linear in bosonic operators, we prove Wick’s theorem rigorously by applying stable partition theory. Higher-order time-ordered superoperator terms can be decomposed into sums of products of second-order time-ordered terms. By applying the theorem, explicit expressions for all even-order terms of the reduced density matrix are derived, with vanishing odd-order terms. We obtain the set of equations for the reduced dynamics of a system interacting with a bosonic bath initially in thermal equilibrium.

DOI
10.1088/1572-9494/ae6ccb

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Chapter 3

Perturbation Theory

In this chapter we derive the perturbation series for the partition function. Since it is given by a trace of an exponential of an operator H0 +λV , we first expand the exponential with respect to λ. The result is a power series with operator valued coefficients. Then we compute the trace of the n’th order coefficient. This can be done with Wick’s theorem which lies at the bottom of any diagrammatic expansion. It states that the expectation value, with respect to exp[−βH0], of a product of 2n annihilation and creation operators is given by a sum of terms where each term is a product of expectation values of only two annihilation and creation operators. Depending on whether the model is fermionic or bosonic or complex or scalar, this sum is given by a determinant, a pfaffian, a permanent or simply the sum over all pairings which is the bosonic analog of a pfaffian. For the many-electron system, one obtains an n × n determinant. The final result is summarized in Theorem 3.2.4 below.

Chapman & Hall/CRC eBooks•Aug 30, 2004•0 citations
Abstract For the superoperators that are linear in bosonic operators, we prove Wick’s theorem rigorously by applying stable partition theory. Higher-order time-ordered superoperator terms can be decomposed into sums of products of second-order time-ordered terms. By applying the theorem, explicit expressions for all even-order terms of the reduced density matrix are derived, with vanishing odd-order terms. We obtain the set of equations for the reduced dynamics of a system interacting with a bosonic bath initially in thermal equilibrium.
Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the general Wick's theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e., operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independent of the type of operator involved. By means of a few examples, we show how the GWT helps treat demanding problems by reducing the amount of calculations required.
The new ccc-number calculus for functions of noncommuting operators, developed in Paper I and employed in Paper II to formulate a general phase-space description of boson systems, deals with situations involving equal-time operators only. In the present paper extensions are presented for the treatment of problems involving boson operators at two or more instants of time. The mapping of time-ordered products onto ccc-number functions is studied in detail. The results make it possible to evaluate time-ordered products of boson operators by phase-space techniques. The usual Wick theorem for boson systems is obtained as a special case of a much more general theorem on time ordering. Our method of derivation appears to provide the first direct proof of Wick's theorem as well as a clear insight into its true meaning. A closed expression is also obtained for the time-evolution operator in terms of the solution of the ccc-number differential equation for the phase-space equivalent of this operator. The new calculus is also applied to the problem of evaluating normally ordered time-ordered, and also the antinormally ordered time-ordered, correlation functions.
This chapter considers a general form of the Wick’s theorem, which leads to a perturbation expansion of the (contour) single- and two-particle Green’s functions, which are expressed in terms of the contour time-ordering operator. The strategy for proving the Wick’s theorem is similar to that adopted within the Matsubara formalism for the Green’s functions at finite temperature and relies on the Gibbs form of the statistical operator in the interaction picture. An extension to superfluid Bose and Fermi systems is also considered.
We study perturbation expansions for quantum systems which are nonlinear in the sense that they may contain derivative couplings. The characteristic feature of these systems is the occurrence of products of noncommuting operators in the interaction Hamiltonian. In order to perform perturbation expansions in the Dyson-Wick manner it is in general necessary to order these operators according to some ordering scheme. We determine the most general scheme of this sort which allows a Dyson-Wick expansion with pair contractions. It involves a set $\ensuremath{\lambda}$ of three dimensionless ordering parameters which are related to the three aaa priori undetermined equal-time contractions. Different $\ensuremath{\lambda}$-ordering schemes are related via an "equal-time" or "static" Wick theorem. In order to be able to contract operators which may be taken either at equal or at different times we introduce a time-ordering operation ${T}_{\ensuremath{\lambda}}$ which extends Dyson's time-ordering operation to equal times in a fashion dependent on the $\ensuremath{\lambda}$ ordering. We arrive at a "dynamic" Wick theorem which enables us to decompose the ${T}_{\ensuremath{\lambda}}$ product of a set of operators taken at arbitrary times into a sum of ${\ensuremath{\lambda}}^{\ensuremath{'}}$ products, where the most important case is that in which ${\ensuremath{\lambda}}^{\ensuremath{'}}$ represents normal ordering. Contrary to the usual Wick theorem ours works also in the case when two or more times coincide. Applying this theorem to the scattering operator expressed as a ${T}_{\ensuremath{\lambda}}$ product we obtain a class of Dyson-Wick expansions for nonlinear systems. Different expansions correspond to different choices of the ordering $\ensuremath{\lambda}$ and lead to different Feynman rules. The resulting diagrammatic technique for nonlinear systems is seen to have the following special features: (i) The interaction vertices include terms which arise from "quantum corrections," which are dependent on the ordering \ensuremath{\lambda} chosen for the noncommuting factors in the interaction Hamiltonian, and (ii) the equal-time contractions which occur in the closed-loop diagrams must be assigned corresponding $\ensuremath{\lambda}$-dependent values. In this way individual diagrams are $\ensuremath{\lambda}$ dependent. Nevertheless their sum is $\ensuremath{\lambda}$ independent, confirming the fact that the complete scattering operator is independent of the particular ordering chosen. The ordering multiplicity is also reflected in functional versions of the Dyson-Wick expansion. In the path-integral version this multiplicity lurks in the different possible discretization prescriptions. We discuss the relevance of our considerations to the question of the invariance of a quantum theory with respect to nonlinear point transformations. For the special case of a reducible system we present explicit calculations which show how this invariance is maintained in perturbation theory, for any ordering scheme belonging to the $\ensuremath{\lambda}$ class. In any particular scheme the invariance is ensured by a more or less intricate cancellation mechanism between the individual diagrams.
Computing response functions by following the time evolution of superoperators in Liouville space (whose vectors are ordinary Hilbert space operators) offers an attractive alternative to the diagrammatic perturbative expansion of many-body equilibrium and nonequilibrium Green's functions. The bookkeeping of time ordering is naturally maintained in real (physical) time, allowing the formulation of Wick's theorem for superoperators, giving a factorization of higher order response functions in terms of two fundamental Green's functions. Backward propagations and analytic continuations using artificial times (Keldysh loops and Matsubara contours) are avoided. A generating functional for nonlinear response functions unifies quantum field theory and the classical mode coupling formalism of nonlinear hydrodynamics and may be used for semiclassical expansions. Classical response functions are obtained without the explicit computation of stability matrices.
In this chapter we derive the perturbation series for the partition function. Since it is given by a trace of an exponential of an operator H0 +λV , we first expand the exponential with respect to λ. The result is a power series with operator valued coefficients. Then we compute the trace of the n’th order coefficient. This can be done with Wick’s theorem which lies at the bottom of any diagrammatic expansion. It states that the expectation value, with respect to exp[−βH0], of a product of 2n annihilation and creation operators is given by a sum of terms where each term is a product of expectation values of only two annihilation and creation operators. Depending on whether the model is fermionic or bosonic or complex or scalar, this sum is given by a determinant, a pfaffian, a permanent or simply the sum over all pairings which is the bosonic analog of a pfaffian. For the many-electron system, one obtains an n × n determinant. The final result is summarized in Theorem 3.2.4 below.