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Discrete Moyal-type Representations for a Spin

Stephan Heiss·Stefan Weigert
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Abstract

In Moyal's formulation of quantum mechanics, a quantum spin s is described in terms of continuous symbols, i.e. by smooth functions on a two-dimensional sphere. Such prescriptions to associate operators with Wigner functions, P- or Q-symbols, are conveniently expressed in terms of operator kernels satisfying the Stratonovich-Weyl postulates. In analogy to this approach, a discrete Moyal formalism is defined on the basis of a modified set of postulates. It is shown that appropriately modified postulates single out a well-defined set of kernels which give rise to discrete symbols. Now operators are represented by functions taking values on (2s+1)(2s+1) points of the sphere. The discrete symbols contain no redundant information, contrary to the continuous ones. The properties of the resulting discrete Moyal formalism for a quantum spin are worked out in detail and compared to the continuous formalism, and it is illustrated by the example of a spin 1/2.

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Stephan Heiss, Stefan Weigert. 2000-04-05. Discrete Moyal-type Representations for a Spin. https://doi.org/10.1103/physreva.63.012105

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