Dynamical equivalence, commutation relations and noncommutative geometry

TL;DR

This paper explores how different Hamiltonian structures with the same dynamics can be quantized, revealing new phenomena such as noncommutative geometry in multi-degree systems and their relation to symmetries and quantum corrections.

Contribution

It provides a framework for understanding the quantization of dynamically equivalent Hamiltonian structures, especially in systems with symmetries and multiple degrees of freedom.

Findings

Dynamically equivalent Hamiltonian structures can lead to different quantum commutation relations.

Noncommutative geometry emerges in multi-degree systems like the 2D oscillator in a magnetic field.

Quantum corrections are necessary for nonlinear equations of motion.

Abstract

We revisit Wigner's question about the admissible commutation relations for coordinate and velocity operators given their equations of motion (EOM). In more general terms we want to consider the question of how to quantize dynamically equivalent Hamiltonian structures. A unique answer can presumably be given in those cases, where we have a dynamical symmetry. In this case arbitrary deformations of the symmetry algebra should be dynamically equivalent. We illustrate this for the linear as well as the singular 1d-oscillator. In the case of nonlinear EOM quantum corrections have to be taken into account. We present some examples thereof New phenomena arise in case of more then one degree of freedom, where sometimes the interaction can be described either by the Hamiltonian or by nonstandard commutation relations. This may induce a noncommutative geometry (for example the 2d-oscillator in a constant magnetic field). Also some related results from nonrelativistic quantum field theory applied to solid state physics are briefly discussed within this framework