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Physical Review A

Functional integration and the Onsager-Machlup Lagrangian for continuous Markov processes in Riemannian geometries

H. Dekker

Published May 1, 1979• 67 citations

Abstract

The Onsager-Machlup Lagrangian defined by the functional-integral representation of general continuous Markov processes is derived explicitly and concisely. Attention is given both to the mathematics and its physical interpretation. The method is based on (i) the consideration of continuous and differentiable paths connecting fixed end points in the convariant propagator, (ii) the application of a recently developed Fourier-series analysis of these trajectories in locally flat spaces, and (iii) the use of the proper transformations between locally Euclidean and globally Riemannian geometries. The spectral analysis allows for arbitrary paths rather than an a priori straight line even in the short-time propagator and avoids any ad hoc discretization rule. Specialized to globally flat spaces the result agrees with formulas given by Stratonovich, Horsthemke and Bach, Graham, and Dekker. It is demonstrated that a unique covariant path integral is equivalent to a whole class of stochastically equivalent lattice expressions.

DOI: 10.1103/physreva.19.2102

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The Onsager-Machlup Lagrangian defined by the functional-integral representation of general continuous Markov processes is derived explicitly and concisely. Attention is given both to the mathematics and its physical interpretation. The method is based on (i) the consideration of continuous and differentiable paths connecting fixed end points in the convariant propagator, (ii) the application of a recently developed Fourier-series analysis of these trajectories in locally flat spaces, and (iii) the use of the proper transformations between locally Euclidean and globally Riemannian geometries. The spectral analysis allows for arbitrary paths rather than an a priori straight line even in the short-time propagator and avoids any ad hoc discretization rule. Specialized to globally flat spaces the result agrees with formulas given by Stratonovich, Horsthemke and Bach, Graham, and Dekker. It is demonstrated that a unique covariant path integral is equivalent to a whole class of stochastically equivalent lattice expressions.

We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a

The direct transformation from the polygonal to the Fourier series representation of Feynman path integrals, via a change of integration variables, is effected explicitly for cases where the Lagrangian is of the form L(x,ẋ,t)= 1/2 ẋ2−V(x,t). This transformation involves a ‘‘functional Jacobian’’ stemming solely from the velocity term in the Lagrangian; this is because N-segment polygonal paths of the type contributing significantly to the integral, and their N-term Fourier series approximants, coalesce together as N→∞, but not their derivatives. We also consider integrals over paths with fixed means T−1∫T0 dt x(t) =const. The usefulness of the Fourier representation is illustrated with the harmonic oscillator case V(x)= 1/2 ω2x2, in both the free and fixed means situations; in particular, the Fourier evaluation of the path integrals trivially determines the large time phases (Maslov indices), and the ranges of ω2 values for which the integrals are finite or infinite in the imaginary time cases.

The quantization of physical systems moving on group and symmetric spaces has been an area of active research over the past three decades. This book shows that it is possible to introduce a representation-independent propagator for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irreducible representations of the Lie group generated by these kinematical variables, which nonetheless correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables.Furthermore, the book shows that it is possible to construct regularized lattice phase space path integrals for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations, and although the configuration space is in general a multidimensional curved manifold, it is shown that the resulting lattice phase space path integral has the form of a lattice phase space path integral on a multidimensional flat manifold. Hence, a novel and extremely natural phase space path integral quantization is obtained for general physical systems whose kinematical variables are the generators of a connected and simply connected Lie group. This novel phase space path integral quantization is (a) exact, (b) more general than, and (c) free from the limitations of the previously considered path integral quantizations of free physical systems moving on group manifolds.To illustrate the general theory, a representation-independent propagator is explicitly constructed for SU(2) and the affine group