Non-commutative Solitons in Finite Quantum Mechanics

TL;DR

This paper develops a framework for quantizing linear maps in finite quantum mechanics, revealing how the arithmetic properties of discretization influence the evolution, uncertainty, and soliton-like states.

Contribution

It introduces a method to construct unitary evolution operators for finite quantum systems and explores their dependence on the discretization parameter N.

Findings

Unitary operators depend non-trivially on the arithmetic nature of N.

Constructed coherent states can be interpreted as non-commutative solitons.

Discussed the implications for the uncertainty principle in finite quantum systems.

Abstract

We construct the unitary evolution operators that realize the quantization of linear maps of SL(2,R) over phase spaces of arbitrary integer discretization N and show the non-trivial dependence on the arithmetic nature of N. We discuss the corresponding uncertainty principle and construct the corresponding coherent states, that may be interpreted as non-commutative solitons.