Alternative linear structures for classical and quantum systems

TL;DR

This paper explores how changing the linear structure on the tangent bundle of a classical system can lead to new quantization methods, potentially bypassing the von Neumann uniqueness theorem.

Contribution

It introduces a novel approach of modifying linear structures on tangent bundles to obtain alternative quantum descriptions of classical systems.

Findings

Constructed alternative linear structures on tangent bundles.

Demonstrated potential to use Weyl quantization differently.

Showed possibility to evade von Neumann's theorem.

Abstract

The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by changing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical configuration space Q that can be considered as "adapted" to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in different non equivalent ways, "evading", so to speak, the von Neumann uniqueness theorem.