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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. III. A Generalized Wick Theorem and Multitime Mapping

G. S. Agarwal, E. Wolf

Published Nov 15, 1970• 173 citations
Publisher

Abstract

The new $c$-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 $c$-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 $c$-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.

DOI
10.1103/physrevd.2.2206

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Physical Review•Jan 5, 1969•16 citations
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.
A new calculus for functions of noncommuting operators is developed, based on the notion of mapping of functions of operators onto ccc-number functions. The class of linear mappings, each member of which is characterized by an entire analytic function of two complex variables, is studied in detail. Closed-form solutions for such mappings and for the inverse mappings are obtained and various properties of these mappings are studied. It is shown that the most commonly occurring rules of association between operators and ccc-numbers (the Weyl, the normal, the antinormal, the standard, and the antistandard rules) belong to this class and are, in fact, the simplest ones in a clearly defined sense. It is shown further that the problem of expressing an operator in an ordered form according to some prescribed rule is equivalent to an appropriate mapping of the operator on a ccc-number space. The theory provides a systematic technique for the solution of numerous quantum-mechanical problems that were treated in the past by ad hoc methods, and it furnishes a new approach to many others. This is illustrated by a number of examples relating to mappings and ordering of operators.
In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto ccc-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.
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.
After a brief summary of recently derived general results relating to the mapping of functions of noncommuting operators on functions of ccc numbers, equations are given which describe the time evolution of the ccc-number equivalents (phase-space representations) of the density operator and of a Heisenberg operator. The evaluation of time-ordered functions of operators by ccc-number techniques is also briefly discussed.
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.
Using the recently developed phase-space techniques for the treatment of quantum-mechanical problems, we set up a procedure for calculating multitime-correlation functions in terms of the joint distribution functions. The correlation functions are then expressed simply as integrals over the associated phase space. Explicit expressions are given for these joint distribution functions, in terms of Green's functions of the ccc-number equations of motion for the phase-space equivalent of the density operator. Using these joint distribution functions, an exact regression theorem is rederived, and the connection with the multitime correspondence between classical and quantum stochastic processes is discussed.