No anomalous canonical commutators induced by Berry's phase

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 $e_M=2\pi \hslash$, for which the possible anomalous term in the canonical commutator $[x_k,x_l]=i\hslash {\Omega }_{kl}$ would become of the order $O({\hslash }^2)$.

Comments: 29 pages and 3 figures. Substantially expanded and a detailed account of the absence of no anomalous cononical commutators induced by Berry's phase is given. The title has also been changed accordingly. This version is to be published in Ann. of Phys