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The London Edinburgh and Dublin Philosophical Magazine and Journal of Science

CXXV. Schwinger's dynamical principle

W. K. Burton, B. F. Touschek

Published Oct 1, 1953• 2 citations
Publisher

Abstract

A comparison is made of the methods applied by Schwinger and by the authors to the derivation of commutation relations for systems whose equations of motion are of the first order in the time derivatives.
DOI
10.1080/14786441008520378

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Progress of Theoretical Physics•Dec 1, 1972•6 citations
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.
Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian Lc(\dotx, x) = (M/2) f-1(x)\dotx2 - v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lagrangian operator is essential in our formulation.