Quantum mechanics on Riemannian manifold in Schwinger's quantization approach IV

Abstract

Abstract. In this paper we extend Schwinger's quantization approach to the case of a supermanifold considered as a coset space of the Poincare group by the Lorentz group. In terms of coordinates parametrizing a supermanifold, quantum mechanics for a superparticle is constructed. As in models related to the usual Riemannian manifold, the key role in analyzes is played by Killing vectors. The main feature of quantum theory on the supermanifold consists of the fact that the spatial coordinates are not commute with each other and therefore are represented on wave functions by integral operators.

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach III

“…The main idea on which the supersymmetry is based consists of the usage in (4) the extension of the group ISO(1, 3) instead of ISO (1,3). Such an extension can be developed by adding to the Lie algebra new generators that must correspond to some representation of SO (1,3) . The simplest way of doing it is to include spinor generators Q, Q with anticommutative components with the following properties…”

“…where s(a, b) is the signature factor for the operators a and b defined in (1). The velocity operator can be defined as…”

“…The results obtained at the classical level are briefly discussed in sections 3 and 4. As it has appeared in [1]- [3], the permissible variations δz A are closely related to Killing vector fields defined, however, on s M n instead of M n .…”

“…The present paper accomplishes our series of works (see [1]- [3]) devoted to extending Schwinger's quantization procedure to the case of quantum theory defined on manifolds with a group structure. The aim of the present paper is to develop quantum mechanics for a particle moving on a superspace treated as a coset space s M n ≃ s ISO(p, q)/SO(p, q), where p + q = n , where SO(p, q) denotes a special orthogonal rotation group and s ISO(p, q) is a supersymmetrical extension of the Poincare group ISO(p, q).…”

“…The quantum mechanics of a superparticle is constructed in terms of a logical scheme presented in [1]- [3]. To obtain the explicit form of permissible variations of coordinates we have analyzed the symmetries of the Lagrangian which determines the dynamical behavior of a superparticle.…”