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Journal of Mathematical Physics

Field quantization for pure and mixed states and its classical limit

Pimon Ajanapon

Published Aug 1, 1988• 1 citation
Publisher

Abstract

The path-integral formulation of quantum mechanics in terms of the density matrix proposed in a previous paper is here extended to quantum field theory. The corresponding classical field equation is derived as the deterministic classical limit by imposing an appropriate mixed-state initial condition. Relativistic invariance is insured by construction of ten generators of the inhomogeneous Lorentz group.

DOI
10.1063/1.527890

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The path-integral formulation of quantum mechanics in terms of the density matrix proposed in a previous paper is here extended to quantum field theory. The corresponding classical field equation is derived as the deterministic classical limit by imposing an appropriate mixed-state initial condition. Relativistic invariance is insured by construction of ten generators of the inhomogeneous Lorentz group.
Instead of regarding the classical limit as the limit ℏ→0, an alternative view based on the physical interpretation of the elements of the density matrix is proposed. The derivation of the classical limit of quantum mechanics is carried out in two stages. First, the statistical classical limit is derived, and then, with an appropriate initial condition, the deterministic classical limit is obtained. The derivation hinges on the use of the Feynman path-integral formulation of quantum mechanics in terms of the density matrix. It is shown that deterministic (Hamilton’s or Newton’s) classical mehanics is not the classical limit of Schrödinger’s wave mechanics. The classical limit of Schrödinger’s wave mechanics is only statistical and corresponds to the classical Liouville equation. In order to obtain the deterministic classical limit, it is necessary to start out initially with a quantum mechanical mixture.
Instead of regarding the classical limit as the limit ℏ →\hbar\ \toℏ → 0, an alternative view based on the physical interpretation of the elements of the density matrix is proposed. According to this alternative view, taking the classical limit corresponds to taking the diagonal elements and ignoring the off-diagonal elements of the density matrix. As illustrations of this alternative approach, the classical limits of quantum mechanics and quantum electrodynamics are derived. The derivation is carried out in two stages. First, the statistical classical limit is derived. Then with an appropriate initial condition, the deterministic classical limit is obtained. In the case of quantum mechanics, it is found that the classical limit of Schrodinger's wave mechanics is at best statistical, i.e., Schrodinger's wave mechanics does not reduce to deterministic (Hamilton's or Newton's) classical mechanics. In order to obtain the latter, it is necessary to start out initially with a mixture at the level of statistical quantum mechanics. The derivation hinges on the use of the Feynman path integral rigorously defined with the aid of nonstandard analysis. Nonstandard analysis is also applied to extend the method to the case of quantum electrodynamics. The fundamental decoupling problem arising from the use of Grassmann variables is circumvented by the use of c-number electron fields, but antisymmetrically tagged. The basic classical (deterministic) field equations are obtained in the classical limit with appropriate initial conditions. The result raises the question as to what the corresponding classical field equations obtained in the classical limit from the renormalized Lagrangian containing infinite counterterms really mean.
The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.
This book is a self-contained and concise introduction to the techniques and applications of path integral quantization and functional techniques, aimed at students and practitioners. The first half of the text focuses on quantum mechanics, including a review of the action formulation of classical mechanics and quantum mechanics in the Dirac operator and state formalism, and further examination of the path integral. The second part examines relativistic field theories, reviewing special relativity, as well as derivation of the path integral representation of the vacuum transition element for quantized scalar, spinor, and vector fields from the coherent state representation of the respective field theories
The path integral formulation in quantum mechanics corresponds to the first quantization since it is just to rewrite the quantum mechanical amplitude into many dimensional integrations over discretized coordinates xnx_nxn​. However, the path integral expression cannot be connected to the dynamics of classical mechanics, even though, superficially, there is some similarity between them. Further, the field theory path integral in terms of many dimensional integrations over fields does not correspond to the field quantization. We clarify the essential difference between Feynman's original formulation of path integral in QED and the modern version of the path integral method prevailing in lattice field theory calculations, and show that the former can make a correct second quantization while the latter cannot quantize fields at all and its physical meaning is unknown.
In the standard formulation of quantum mechanics, observables are represented by operators in a Hilbert space and states by vectors in it. Quantum field theory too is formulated in terms of field operators. States in field theory include the vacuum state, one particle states, two particle states and so on. One does not measure operators or state vectors, however; measurements involve eigenvalues of operators or transition probabilities between states. It turns out that one can formulate quantum mechanics and field theory without using operators. Indeed, the probability amplitude for a transition from a position q1 at time t1 to a position q2 at time t2 can be expressed as a path integral ∫ dt L(q(t),q˙(t)), (5.1) where q(t) runs over all possible functions satisfying the conditions q(t1) = q1, q(t2) = q2 (5.2) and the integral over q(t) is an integral over all such paths of the exponential evaluated for each path. The measure of integration is not well defined a priori, but it is possible to give prescriptions that make sense. In classical mechanics, positions at two different times determine a path from the second order differential equation of motion, which yields a definite solution if two conditions are imposed. In contrast, quantum mechanics does not allow us to talk about trajectories. The path integral formulation manifestly refers to the myriad paths that may be taken by a particle in going from one point to another. It goes so far as to assign a probability amplitude to each path, and it is related to the classical action corresponding to that path. However, it is complex and it also involves a superposition, characteristic of quantum mechanics.