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Journal of Mathematical Physics

Path integrals and supercoherent states

Масуд Чайчиан, Demosthenes Ellinas, Peter Prešnajder

Published Dec 1, 1991• 21 citations
Publisher

Abstract

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.

DOI
10.1063/1.529451

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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.
A general approach is presented for constructing coherent states for supersymmetric systems. It uses Rogers's supermanifold formulation of supergroups to extend the group-theoretic method. Supercoherent states are explicitly obtained for the supersymmetric harmonic oscillator. They are shown to be eigenstates of the supersymmetric annihilation operator and to be minimum-uncertainty states. Two more-complex situations with extended physical supersymmetries are also considered: an electron moving in a constant magnetic field, and the electron-monopole system. The supercoherent states for these systems are found using super Baker-Campbell-Hausdorff relations and their interpretation is elucidated.