SU(2) coherent state path integrals labeled by a full set of Euler angles: basic formulation

TL;DR

This paper develops a formulation for SU(2) coherent state path integrals using arbitrary fiducial vectors, expanding the theoretical framework and clarifying the mathematical structure of spin systems with potential applications in quantum physics.

Contribution

It introduces a new basic formulation of SU(2) coherent state path integrals based on arbitrary fiducial vectors, generalizing previous approaches.

Findings

Overcompleteness relation established for the states

Path integral representation derived for general systems

Complex variable forms of states and path integrals obtained

Abstract

We develop a basic formulation of the spin (SU(2)) coherent state path integrals based not on the conventional highest or lowest weight vectors but on arbitrary fiducial vectors. The coherent states, being defined on a 3-sphere, are specified by a full set of Euler angles. They are generally considered as states without classical analogues. The overcompleteness relation holds for the states, by which we obtain the time evolution of general systems in terms of the path integral representation; the resultant Lagrangian in the action has a monopole-type term a la Balachandran etal. as well as some additional terms, both of which depend on fiuducial vectors in a simple way. The process of the discrete path integrals to the continuous ones is clarified. Complex variable forms of the states and path integrals are also obtained. During the course of all steps, we emphasize the analogies and correspondences to the general canonical coherent states and path integrals that we proposed some time ago. In this paper we concentrate on the basic formulation. The physical applications as well as criteria in choosing fiducial vectors for real Lagrangians, in relation to fictitious monopoles and geometric phases, will be treated in subsequent papers separately.