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Physical Review Letters

Conditions for Existence of a Diagonal Representation for Quantum Mechanical Operators

R. Bonifacio, Lorenzo M. Narducci, E. Montaldi

Published Jun 13, 1966• 14 citations
Publisher
DOI
10.1103/physrevlett.16.1125

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"The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".
We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with a unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch–Gordan theory and the reciprocity theorem associated with induced unitary group representations. Applications to several examples involving SU(2), SU(3), and the Heisenberg–Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase–space description of quantum mechanics and quantum state reconstruction problems.
Necessary and sufficient conditions are given for a quantummechanical system to possess a coordinate system with respect to which its behaviour at discrete times is that of a universal digital computer. The form of the diagonal representation for the unitary time evolution operator for quantum universal computers is derived; aspects of the transformation between the diagonal representation and the computational representation are shown to be uncomputable. A quantum-mechanical treatment of macroscopic, dissipative computers is given.