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Journal of Physics A Mathematical and General

Path integral formalism for Osp(1/2, R) coherent states

H. A. Schmitt, A. Mufti

Published Aug 7, 1991• 7 citations
Publisher

Abstract

A path integral formulation in the representation of coherent states for the non-compact supergroup Osp(1/2, R) is introduced. An expression for the transition amplitude connecting two Osp(1/2, R) coherent states is constructed, and the corresponding canonical equations of motion derived. A set of generalized Poisson brackets is introduced and interpreted.

DOI
10.1088/0305-4470/24/15/005

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A path integral formulation in the representation of coherent states for the non-compact supergroup Osp(1/2, R) is introduced. An expression for the transition amplitude connecting two Osp(1/2, R) coherent states is constructed, and the corresponding canonical equations of motion derived. A set of generalized Poisson brackets is introduced and interpreted.
For the real supergroup Osp(1‖2;R), with both its compact and noncompact versions, supercoherent states are introduced with a method close to the one by Perelomov for the even subgroups SU(2) or SU(1,1). These states labeled by a complex c number and Grassmann variable minimize the uncertainty of the quadratic Casimir operator of the group. A path integral formalism is developed for the transition amplitude between supercoherent states for a Hamiltonian linear in the generators of the superalgebra, which leads to a super-Riccati equation. Finally, in the classical limit the canonical equations of motion are derived which involve a generalized super Poisson bracket.
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.