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arXiv:1905.07597 · Journal of Physics A Mathematical and Theoretical

Correspondence rules for Wigner functions over S U ( 3 ) / U ( 2 )

Alex Clésio Nunes Martins, Andrei B Klimov, Hubert de Guise

Published May 17, 2019• 4 citations
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Abstract

Abstract We present results on the product for Wigner functions over . In particular, we present a form of the so-called correspondence rules, which provide a differential form of the product and when is an generator. For the Wigner map, these rules must contain second order derivatives and thus substantially differ from the rules of other known cases.

DOI
10.1088/1751-8121/ab226c

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Abstract We present results on the product for Wigner functions over . In particular, we present a form of the so-called correspondence rules, which provide a differential form of the product and when is an generator. For the Wigner map, these rules must contain second order derivatives and thus substantially differ from the rules of other known cases.
We point out that there is a close connection between the Weyl correspondence and coherent state correspondence. By virtue of differential and integral method within normal ordered products we demonstrate that the essence of the Weyl correspondence is just coherent state correspondence in both boson and fermion cases. Furthermore, coherent state form and explicit operator form for the Wigner function are given. With these forms we can derive some important operator formulas from the Weyl correspondence. The Wigner function's zero-point value is also discussed.
Differential quotient rule of composite function operator and its applications in quantum physics, quantum statistics, operator ordering theory, matrix theory and control theory are given. The integration problem of Wigner operator and Weyl corresponding rules are studied. Two kinds of typical operator identity formulas are proved. The differential form of Wigner operator in ordered product of operators and new differential form of important functions are obtained. Finally, a Wigner operator with parameter for unifying regular order, Weyl sequencing and abnormal order is introduced.
It is shown that the number-phase (or rotational) Wigner function is obtained from the Weyl symmetrization rule for the correspondence between classical functions and quantum operators. In spite of the complicated form of the commutator for the number and phase operators, the Weyl symmetrization rule is similar to that in the case of the position and momentum operators. In addition, it is found that the ordering of the number and phase operators also has the same structure as that for the position-momentum pair.