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arXiv:hep-th/9709200 · Physical Review Letters

Quantum Hamilton-Jacobi Equation

Vipul Periwal

Published May 18, 1998• 16 citations
arXivHTMLPDF

Abstract

The nontrivial transformation of the phase space path integral measure under certain discretized analogs of canonical transformations is computed. This Jacobian is used to derive a quantum analog of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A perturbative solution of the quantum Hamilton-Jacobi equation is given. This solution gives a new way to compute quantum corrections for any soliton equation for which action-angle variables are known.

DOI
10.1103/physrevlett.80.4366

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The nontrivial transformation of the phase space path integral measure under certain discretized analogs of canonical transformations is computed. This Jacobian is used to derive a quantum analog of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A perturbative solution of the quantum Hamilton-Jacobi equation is given. This solution gives a new way to compute quantum corrections for any soliton equation for which action-angle variables are known.
Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.
Anomalies associated with canonical transformations in quantum-mechanical systems are examined in the context of the path-integral formalism, where they appear as nontrivial Jacobians. The quantum-mechanical anomaly, for the case of a transformation to cyclic coordinates, is shown to be equivalent to a term in the action, similar to a gauge field configuration, whose presence maintains self-adjointness of the Hamiltonian or corrects the phase-space measure of the path integral to reflect the bound-state spectrum of the original problem. Various nonperturbative aspects of quantum-mechanical transition amplitudes are calculated by using classical solutions to Hamilton's equations of motion combined with the anomaly generated by the canonical transformation.
An ordinary unambiguous integral representation for the finite propagator of a quantum system is found by starting of a privileged skeletonization of the functional action in phase space, provided by the complete solution of the Hamilton-Jacobi equation. This representation allows to regard the propagator as the sum of the contributions coming from paths where the momenta generated by the complete solution of the Hamilton-Jacobi equation are conserved -as it does happen on the classical trajectory-, but are not restricted to having the classical values associated with the boundary conditions for the original coordinates.
We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi (HJ) theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton’s characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ formalism is energy, not time. Thus, we are led to consider the Fourier transform of the path integral, the spectral path integral Z ˜ ( E ) . The evaluation of path integrals reduces to determining the quantum Hamilton characteristic functions (which can be achieved via an asymptotic analysis) and a discrete sum over the quantum period lattice, generalizing Gutzwiller’s sum.
The canonical functional action in the path integral in phase space is discretized by linking each pair of consecutive vertebral points - and or and - through the invariant complete solution of the Hamilton-Jacobi equation associated with the classical path defined by these extremes. When the measure is chosen to reflect the geometrical character of the propagator (it must behave as a density of weight in both of its arguments), the resulting infinitesimal propagator is cast in the form of an expansion in a basis of short-time solutions of the wave equation, associated with the eigenfunctions of the initial momenta canonically conjugated to a set of normal coordinates. The operator ordering induced by this prescription is a combination of a symmetrization rule coming from the phase, and a derivative term coming from the measure.