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

From classical mechanics to Feynman graphs with *-products

B. Lesche

Published May 15, 1984• 6 citations
Publisher

Abstract

Quantum mechanics can be formulated on a phase space using so-called *-products. This formulation illustrates the relations to classical mechanics particularly clearly. We show that *-products, furthermore, are closely related to Feynman graphs.
DOI
10.1103/physrevd.29.2270

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Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
The Feynman's path integral in space-time is closely related, via classical Lagrangian theory, to Dirac's formulation of quantum mechanics.The essential point of this approach is in making use of the Lagrangian function in space-time and Hamilton's first action principle to construct the probability amplitude in conventional quantum mechanics.This formulation has, as mentioned by Feynman, mathematical and physical incompleteness.We have investigated a part of the physical incompleteness, which can be treated by using the classical technique of eliminating ignorable.coordinates.The corresponding conjugate momenta will then provide constraint relations, which can be used to reduce the Lagrangian function and the path integral to a modified form.The paths are now restricted, via the constraint relations, to limited configuration spaces.A general modified formulation is constructed for paths in a limited space-time region.An important special case is demonstrated.In the special case, Jacobi's action principle is employed to replace Hamilton's action principle, therefore, the paths thus formulated are in space-energy rather than in space-time.
Basically, there are two approaches to the formulation of quantum mechanics, i.e., the operator and the Feynman path integral approach. The main object in quantum mechanics is referred to as the transition amplitude between those states, matrix elements of physical operators and, in particular, the S- matrix in the problem of quantum scattering. On the basis of quantum mechanics, the operator method is closed, well formulated mathematically, and so is an appropriate universal language describing quantum field theory, the quantum physics of condensed states and quantum relativistic statistical physics. The differential equation defining the path of the system in the configuration space is called the equation of motion of the system. The aim of classical as well as quantum mechanics is to predict the time evolution of the system, starting from knowledge of the state of the system at a given point in time.