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

TL;DR

This paper extends Schwinger's quantization to supermanifolds derived from the Poincare group, constructing quantum mechanics for superparticles with non-commuting spatial coordinates, highlighting the role of Killing vectors.

Contribution

It introduces a novel approach to quantum mechanics on supermanifolds using Schwinger's method, emphasizing non-commutative coordinates and the geometric role of Killing vectors.

Findings

Quantum mechanics on supermanifolds involves non-commuting spatial coordinates.

Killing vectors are central to the analysis of supermanifold quantum systems.

Wave functions are represented by integral operators due to non-commutativity.

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.