Variations on a theme of Heisenberg, Pauli and Weyl
arXiv:0807.2837 · doi:10.1088/1751-8113/41/37/375302
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.
Dedicated to the memory of Moshé Flato on the occasion of the tenth anniversary of his death
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Cited by in corpus (13)
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- Symmetries of the finite Heisenberg group for composite systems
- SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms
- Symmetries of finite Heisenberg groups for multipartite systems
- Topological decompositions of the Pauli group and their influence on dynamical systems
- Complementarity and phases in SU(3)
- On the Pauli group on 2-qubits in dynamical systems with pseudofermions
- Matrix reduction and Lagrangian submodules
- Bases for spin systems and qudits from angular momentum theory
- A classification of finite quantum kinematics
- Special bases for the vector space of square matrices