On the geometry of the energy operator in quantum mechanics

TL;DR

This paper examines how the energy operator in geometric quantum mechanics can include or exclude scalar curvature terms, showing that such modifications are mathematically flexible across different formulations.

Contribution

It demonstrates the arbitrary addition or cancellation of scalar curvature terms in the energy operator within geometric quantum mechanics frameworks.

Findings

Scalar curvature terms can be added or canceled in the energy operator.

The flexibility applies to both Geometric Quantization and Covariant Quantum Mechanics.

The modifications are consistent across multiple formulations.

Abstract

We analyze the different ways to define the energy operator in geometric theories of quantum mechanics. In some formulations the operator contains the scalar curvature as a multiplicative term. We show that such term can be canceled or added with an arbitrary constant factor, both in the mainstream Geometric Quantization and in the Covariant Quantum Mechanics, developed by Jadczyk and Modugno with several contributions from many authors.