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Journal of Mathematical Physics

A Unified Variational Formulation of Classical and Quantum Dynamics. I

Edward M. Corson

Published Jan 1, 1963• 2 citations
Publisher

Abstract

We have critically re-examined the role and status of Lagrangian concepts in both classical and quantal contexts. We find a new, unified variational principle which, subject to the superadded postulates of determinism or indeterminism, respectively, leads uniquely to the classical concept of state and Lagrange equations of motion or to the quantal concept of state and Schrödinger equations of motion.

DOI
10.1063/1.1703886

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We have critically re-examined the role and status of Lagrangian concepts in both classical and quantal contexts. We find a new, unified variational principle which, subject to the superadded postulates of determinism or indeterminism, respectively, leads uniquely to the classical concept of state and Lagrange equations of motion or to the quantal concept of state and Schrödinger equations of motion.
In a previous article in this series [E. M. Corson, J. Math. Phys. 4, 42(1963); herein referred to as I], we showed that a new variational principle, of Lagrangian structure, includes Hamilton's principle as a unique derived consequent and encompasses both the classical and quantal definitions of ``state'' and the corresponding Lagrangian and Schrödinger equations of motion, respectively. In this theory, the essential difference between classical and quantal domains was shown to arise solely from the superadded postulates of determinism vs indeterminism; other than this, the two basic postulates may be reduced to (a) Newton's equations and (b) the new variation principle. It was shown that in the quantal domain, Schrödinger's equations obtain for the transition amplitudes (propagators) between pure eigenstates of homologous variables in configuration space, at different times, thus clearly implying all the results of conventional Hamiltonian-operator quantum formalism. In the present article we show that our variational formulation, under the familiar classical canonical transformation to momentum space, also automatically leads to the corresponding Schrödinger equations for the transition amplitudes between pure eigenstates of homologous variables in momentum space, at different times, the unitarity of the formalism and the reciprocal Fourier integral relations between coordinate and momentum representations. Thus we bridge a long-standing gap in theory and derive, uniquely and apparently for the first time, the basic quantum commutators (P.B.'s) which are postulated in the more familiar Hamiltonian theory.
The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle employing differential two-form ? is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail.
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.
A theoretical scheme, based on a probabilistic generalization of Hamilton's principle, is elaborated to obtain a unified description of more general dynamical behaviors determined both from a Lagrangian function and by mechanisms not contemplated by this function. Within this scheme, quantum mechanics, classical field theory and a quantum theory for scalar fields are discussed. As a by-product of the probabilistic scheme for classical field theory, the equations of the De Donder–Weyl theory for multi-dimensional variational problems are recovered.
The symmetry properties of quantum variational principles are considered. Euler-Poincaré reduction theory is applied to the Dirac-Frenkel variational principle for Schrödinger's dynamics producing new variational principles for the different pictures of quantum mechanics (Schrödinger's, Heisenberg's and Dirac's) as well as for the Wigner-Moyal formulation. In addition, new variational principles for mixed states dynamics have been formulated. The already known geometric characterization of quantum mechanics on the complex projective space is shown to emerge naturally from the Euler-Poincaré variational principle. Semidirect-product structures are seen to produce new variational principles for Dirac's picture. \n \nOn the other hand, the variational and Hamiltonian approach to Ehrenfest expectation values dynamics is proposed. In the Lagrangian framework, the quantum variational principle for Schrödinger's dynamics is extended to account for both classical and quantum degrees of freedom. First, it is shown that the mean field model of any quantum mechanical system can be derived from a classical-quantum Euler-Poincaré Lagrangian on the direct sum Lie algebra of the Heisenberg and unitary groups. Then, the semidirect-product structure (named Ehrenfest group), is constructed using the displacement operator from the theory of coherent quantum states ( the unitary action of the Heisenberg group on the space of wavefunctions). New classical-quantum equations for Ehrenfest's expectation values dynamics are derived redefining the mean-field model Euler-Poincaré Lagrangian on the Lie algebra of the Ehrenfest group. In the Hamiltonian framework, first expectation values of the canonical observables are shown to be equivariant momentum maps for the unitary action of the Heisenberg group on quantum states. Then, the Hamiltonian structure for Ehrenfest's dynamics is shown to be Lie-Poisson for the Ehrenfest group. The variational formulation is then given a corresponding Hamiltonian structure. The classical-quantum Ehrenfest dynamics equations produce classical and quantum dynamics as special limit cases. \n \nIn the particular case of Gaussian states, expectation values couple to second order moments, so that GS are completely characterized by first and second moments. When the total energy is computed with respect to a Gaussian state, higher moments can be expressed in terms of the first two, so that the moment hierarchy closes for Gaussian states. Second moments are shown to be equivariant momentum maps for the action of the symplectic group on the space of Wigner functions. Eventually, Gaussian states are shown to possess a Lie-Poisson structure on the Jacobi group. This structure produces an energy-conserving variant of a class of Gaussian moment models that have appeared in the chemical physics literature.