Operators in Phase Space
3.1 Introduction
Ordinarily, the formulation of quantum mechanics is a blend of Heisenberg matrix mechanics and Schrodinger wave mechanics. This, of course, is the most fundamental, having its own strength that is, broadly speaking, non-classical. The Feynman path integral formulation of quantum mechanics reveals new aspects of physical and mathematical construction that might suggest new hints for development of the quantum theory in addition to its added practical applications.
This heading introduces the concept of the Feynman path integral by developing the general construction scheme with particular emphasis on establishing the interconnections between quantum mechanical path integral, Hamiltonian mechanics and statistical mechanics. Basically, there are two approaches to the formulation of quantum mechanics, i.e., the operator (based on the canonical quantization of physical observables together with the associated operator algebra) and the Feynmanl path integral approach. The path integral formulation has the following merits:
- • It makes explicit use of classical mechanics as a basic tool on which to construct the theory of quantum fluctuations.
- • It is an efficient formulation of non-perturbative approaches to the solution of quantum mechanical problems.
- • It is a prototype of higher dimensional functional field integrals.
- • It is of relevance to a wide variety of applications in many-body physics.
The path-integral formalism was originally formulated by Richard P. Feynman in his 1942 PhD Thesis
[1,9] in the mid-1940s, following a hint from an earlier paper by Dirac [6,7] and became the Feynman new formulation of quantum mechanics [3,9,10] as a new space-time approach to non-relativistic quantum mechanics. This serves as an answer to the question of how to extend the classical principle of stationary action, the so-called principle of least action to quantum mechanics. It is for this reason that the principle of stationary action is first introduced [1,2] The principle of least action in quantum physics, introduced path integrals tailored by the classical action functional, the integral of the Lagrangian over time.
The essence of Feynman functional integration is to transform the Schrodinger differential equation into an integral equation. So, this Feynman functional integration should be the main object of quantum mechanics describing the real world via the classical Hamiltonian operator H or Lagrangian L. This may be done without the use of operators or states in the Hilbert space. 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. In more complex problems, it is appropriate to introduce the method of functional integration in the language of quantum mechanics principally based on the variational method. The transition amplitude is derived based on minimization of the action of a classical system and so the initial discussion is the classical notion of action [1,2].
