Variations on a theme of Heisenberg, Pauli and Weyl

TL;DR

This paper explores the mathematical relationships between Heisenberg-Weyl groups, generalized Pauli groups, and unitary groups, providing new insights into their structures and representations in finite dimensions.

Contribution

It offers a detailed analysis of the structure of generalized Pauli groups and their connection to Weyl pairs and Lie algebras, including explicit constructions and decompositions.

Findings

Characterization of the generalized Pauli group P_d and its conjugacy classes.

Construction of Weyl pairs from angular momentum theory.

Development of the Lie algebra of U(d) using generalized Pauli matrices.

Abstract

The parentage between Weyl pairs, generalized Pauli group and unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field R and then switch to the discrete Heisenberg-Weyl group or generalized Pauli group on a finite ring Z_d. The main characteristics of the latter group, an abstract group of order d**3 noted P_d, are given (conjugacy classes and irreducible representation classes or equivalently Lie algebra of dimension d**3 associated with P_d). Leaving the abstract sector, a set of Weyl pairs in dimension d is derived from a polar decomposition of SU(2) closely connected to angular momentum theory. Then, a realization of the generalized Pauli group P_d and the construction of generalized Pauli matrices in dimension d are revisited in terms of Weyl pairs. Finally, the Lie algebra of the unitary group U(d) is obtained as a subalgebra of the Lie algebra associated with P_d. This leads to a development of the Lie algebra of U(d) in a basis consisting of d**2 generalized Pauli matrices. In the case where d is a power of a prime integer, the Lie algebra of SU(d) can be decomposed into d-1 Cartan subalgebras.