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

Uniqueness of the functional approach

H. O. Girotti, T. J. M. Simões

Published Sep 15, 1980• 1 citation
Publisher

Abstract

It is shown that the phase-space Schwinger equations can only be appropriately formulated in terms of a specific discrete procedure. Nevertheless, the solutions to Schwinger equations turn out to be independent of the discrete procedure. We stress the point that a multiplicity of representations for the quantum-mechanical amplitudes is not a peculiar effect of the time-slicing definition of the path integral, but rather a necessity of the functional approach.

DOI
10.1103/physrevd.22.1385

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It is shown that the phase-space Schwinger equations can only be appropriately formulated in terms of a specific discrete procedure. Nevertheless, the solutions to Schwinger equations turn out to be independent of the discrete procedure. We stress the point that a multiplicity of representations for the quantum-mechanical amplitudes is not a peculiar effect of the time-slicing definition of the path integral, but rather a necessity of the functional approach.
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We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
The generalized Weyl transform of index $\ensuremath{\alpha}$ is used to implement the time-slice definition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter $\ensuremath{\alpha}$. We succeed in proving that the $\ensuremath{\alpha}$-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian involves products of noncommuting operators originating from the noncommutativity. The antisymmetry of the matrix parametrizing the noncommutativity plays a key role in the cancellation mechanism of the $\ensuremath{\alpha}$-dependent terms.