Quantum mechanics on Riemannian Manifold in Schwinger's Quantization Approach I

TL;DR

This paper extends Schwinger's quantization to formulate quantum mechanics on Riemannian manifolds with group structures, emphasizing Killing vectors and invariance under coordinate transformations.

Contribution

It develops a unified procedure for quantum theory on manifolds with group actions, incorporating geometric structures into the quantization scheme.

Findings

Quantum Lagrangian for a free particle is defined.

Operators' algebraic properties are consistent with manifold geometry.

Quantum mechanics on manifolds with transitive group actions is analyzed.

Abstract

Schwinger's quantization scheme is extended in order to solve the problem of the formulation of quantum mechanics on a space with a group structure. The importance of Killing vectors in a quantization scheme is showed. Usage of these vectors provides algebraic properties of operators to be consistent with the geometrical structure of a manifold. The procedure of the definition of the quantum Lagrangian of a free particle and the norm of velocity (momentum) operators is given. These constructions are invariant under a general coordinate transformation. The unified procedure for constructing the quantum theory on a space with a group structure is developed. Using it quantum mechanics on a Riemannian manifold with a simply transitive group acting on it is investigated.