Dirac's monopole, quaternions, and the Zassenhaus formula
Michael A. Soloviev (Lebedev Inst.)
TL;DR
This paper develops a method to compute the star product in quaternionic quantum mechanics with magnetic monopoles, using the Zassenhaus formula and twisted convolution, and demonstrates it with explicit calculations.
Contribution
It introduces an algorithm for deriving the differential representation of the star product in quaternionic phase space, expanding the mathematical tools for monopole quantum theories.
Findings
Explicit second-order star product calculation
Application of Zassenhaus formula to quaternionic operators
Development of an algorithm for star product representation
Abstract
Starting from the quaternionic quantization scheme proposed by Emch and Jadczyk for describing the motion of a quantum particle in the magnetic monopole field, we derive an algorithm for finding the differential representation of the star product generated by the quaternionic Weyl correspondence on phase-space functions. This procedure is illustrated by the explicit calculation of the star product up to the second order in the Planck constant. Our main tools are an operator analog of the twisted convolution and the Zassenhaus formula for the products of exponentials of noncommuting operators.
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