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arXiv:quant-ph/9707025

Quasiclassical Path-Integral Approach to Quantum Mechanics Associated with a Semisimple Lee Algebra

E. A. Kochetov

Published Jul 10, 1997• 0 citations
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Abstract

A closed (in terms of classical data) expression for a transition amplitude between two generalized coherent states associated with a semisimple Lee algebra underlying the system is derived for large values of the representation highest weight, which corresponds to the quasiclssical approximation. Consideration is based upon a path-integral formalism adjusted to quantization of symplectic coherent-state manifolds that appear as one-rank coadjoint orbits.

DOI
10.48550/arxiv.quant-ph/9707025

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A closed (in terms of classical data) expression for a transition amplitude between two generalized coherent states associated with a semisimple Lee algebra underlying the system is derived for large values of the representation highest weight, which corresponds to the quasiclssical approximation. Consideration is based upon a path-integral formalism adjusted to quantization of symplectic coherent-state manifolds that appear as one-rank coadjoint orbits.
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The general formalism for path integrals expressed in terms of an arbitrary continuous representation (generalized coherent states) is applied to give ccc-number formulations for the canonical algebra and for the spin algebra, and is used to derive meaningful stationary-phase approximations for these two cases. Some clarification of a recent discussion by Jevicki and Papanicolaou for the spin case is given.
Path integral in the representation of coherent state for the simplest semisimple Lie group SU(2) and its classical consequences are investigated. Using the completeness relation of the coherent state, we derive a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states. In the classical limit we arrive at a canonical equation of motion in a ’’curved phase space’’ (two-dimensional sphere) which reproduces the ordinary Euler’s equation of a rigid body when applied to a rotator.