# No anomalous canonical commutators induced by Berry's phase

## TL;DR

This paper demonstrates that Berry's phase does not cause anomalous canonical commutators in quantum mechanics, challenging previous claims of such deformations, especially in the context of monopole-like singularities.

## Contribution

Using an exactly solvable Berry's model, the authors show that Berry's phase does not induce anomalous canonical commutators, clarifying misconceptions about quantum mechanical deformations.

## Key findings

- Berry's phase does not deform quantum mechanics in the form of anomalous commutators.
- Genuine Dirac monopole-type Berry's phase is not supported by the models analyzed.
- The monopole-like Berry's phase has a magnetic charge leading to negligible anomalous terms of order O(ħ²).

## Abstract

The monopole-like singularity of Berry's adiabatic phase in momentum space and associated anomalous Poisson brackets have been recently discussed in various fields. With the help of the results of an exactly solvable version of Berry's model, we show that Berry's phase does not lead to the deformation of the principle of quantum mechanics in the sense of anomalous canonical commutators. If one should assume Berry's phase of genuine Dirac monopole-type, which is assumed to hold not only in the adiabatic limit but also in the non-adiabatic limit, the deformation of the principle of quantum mechanics could take place. But Berry's phase of the genuine Dirac monopole-type is not supported by the exactly solvable version of Berry's model nor by a generic model of Berry's phase. Besides, the monopole-like Berry's phase in momentum space has a magnetic charge $e_{M}=2\pi\hbar$, for which the possible anomalous term in the canonical commutator $[x_{k},x_{l}]=i\hbar\Omega_{kl}$ would become of the order $O(\hbar^{2})$.