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The Moyal representation for spin. (English) Zbl 0652.46028

The phase-space approach to spin is developed from two basic principles, SU(2)-covariance and traciality, as a theory of Wigner functions on the sphere. The twisted product of phase-space functions is related to group convolution on SU(2) by means of a Fourier transform theory on the coadjoint orbits, which yields the Plancherel-Parseval formula. Coherent spin states provide an alternative route to the same phase-space description of spin. The Wigner functions for spin states and transitions are exhibited up to \(j=2\). It is shown that for Hamiltonians which arise from time-dependent magnetic fields, the quantum spin dynamics is given entirely by the classical motion on the sphere. The Majorana formula becomes transparent in the Moyal representation.
Reviewer: J.C.Várilly

MSC:

46F05 Topological linear spaces of test functions, distributions and ultradistributions
46N99 Miscellaneous applications of functional analysis
81S10 Geometry and quantization, symplectic methods
81Q99 General mathematical topics and methods in quantum theory
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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