n-symplectic quantization on LRn

TL;DR

This paper develops a novel quantization scheme based on n-symplectic geometry on LRn, extending standard quantum mechanics principles to a new geometric framework for particles in Rn.

Contribution

It introduces a full polynomial quantization method on LRn using n-symplectic geometry, aligning with the Schrödinger representation.

Findings

Constructed a consistent polynomial quantization on LRn.

Extended standard quantization axioms to the n-symplectic setting.

Achieved compatibility with the Schrödinger representation.

Abstract

n-symplectic geometry, a generalization of symplectic geometry on the cotangent bundle of a manifold M, is formulated on the bundle of linear frames LM using the Rn-valued soldering 1-form as the generalized n-symplectic potential. In this paper we use n-symplectic geometry on LRn to formulate a quantization scheme for a single particle moving in Rn. By retaining the essence of the standard axioms for quantization on T*Rn, but adapting them to LR^n, we show it is possible to construct a full polynomial quantization that is consistent with the Schr\"odinger representation on a 2n-dimensional subbundle of LRn.