Deformation Quantization of Geometric Quantum Mechanics

TL;DR

This paper explores deformation quantization methods for geometric quantum mechanics, comparing Weyl-Wigner-Moyal and Berezin approaches, and introduces a novel second quantization framework with unique eigenstate properties.

Contribution

It develops a new second quantization scheme for geometric quantum mechanics using Berezin quantization on $CP^{ obreakinite}$, revealing distinct eigenstate characteristics.

Findings

Wigner function and evolution equation derived for particle states.

Comparison of Weyl-Wigner-Moyal and Berezin quantizations.

New second quantization yields eigenstates with eigenvalue 1/ħ.

Abstract

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always ${1 \over \hbar}$.