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Journal of Beijing University of Technology

Quantum Poincaré-Cartan Integral Invariant for Singular System

Ziping Li

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

In order to study the symmetry of a singular system, based on phase-space generation of Green function of a singular system with finite degrees of freedom, an integral invariant of quantum Poincare-Cartan (excluding ground state sign |0) is established. The relationship between the Poincare-Cartan integral invariant and the Hamilton-Jacobi equation is discussed at the quantum level. It is pointed that the Hamilton-Jacobi equation can be deduced from integral invariant of the quantum Poincare-Cartan.

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In order to study the symmetry of a singular system, based on phase-space generation of Green function of a singular system with finite degrees of freedom, an integral invariant of quantum Poincare-Cartan (excluding ground state sign |0) is established. The relationship between the Poincare-Cartan integral invariant and the Hamilton-Jacobi equation is discussed at the quantum level. It is pointed that the Hamilton-Jacobi equation can be deduced from integral invariant of the quantum Poincare-Cartan.
Universal quantum integrals of motion are introduced, and their relation with the universal quantum invariants is established. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, which are preserved in time independently of the concrete form of the coefficients of the Schrödinger equation, provided the Hamiltonian is either a generic quadratic form of the coordinate and momenta operators, or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics in terms of the Wigner function, the relations between the quantum invariants and the classical universal integral invariants by Poincaré and Cartan are elucidated. Examples of the `universal invariant solutions' of the Schrödinger equation, i.e. self-consistent eigenstates of the universal integrals of motion, are given. Applications to the physics of optical and particle beams are discussed.
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.
We give a brief review of the theory of quantum and optical universal invariants, i.e., certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators or the transverse phase-space coordinates of optical paraxial beams that are preserved in time (or along the axis of the beam) independently of the concrete form of the coefficients of the Hamiltonian or the parameters of the optical system, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of certain finite-dimensional algebras. Using the phase-space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation between the quantum invariants and the classical universal integral invariants of Poincaré and Cartan. The specific features of the Gaussian beams are discussed as examples.
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.