Variations on a theme of Heisenberg, Pauli and Weyl1

نویسنده

Maurice R Kibler

چکیده

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 Zd. The main characteristics of the latter group, an abstract group of order d noted Pd, are given (conjugacy classes and irreducible representation classes or equivalently Lie algebra of dimension d associated with Pd). 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 Pd 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 Pd. This leads to a development of the Lie algebra of U(d) in a basis consisting of d 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. 1Dedicated to the memory of my teacher and friend Moshé Flato on the occasion of the tenth anniversary of his death. 1 in 2p 300 30 01 84 , v er si on 1 17 J ul 2 00 8 Author manuscript, published in "Journal of Physics A Mathematical and Theoretical 41 (2008) 375302" DOI : 10.1088/1751-8113/41/37/375302

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