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Journal of Mathematical Physics

Do trilinear commutation relations in quantum mechanics admit coordinate space realization in three dimensions?

Ranjan Bhattacharya, Siddhartha Bhowmick

Published Jun 1, 1987• 1 citation
Publisher

Abstract

The trilinear commutation relations involving coordinates and momenta introduced by Wigner [E. P. Wigner, Phys. Rev. 77, 711 (1950)] are generalized to three dimensions. It is shown that the only realizable coordinate space representation of the momenta implies the usual bilinear commutation relations.
DOI
10.1063/1.527532

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This paper contributes to a recent series discussing quantum mechanics defined on a finite-dimensional Hilbert space in which Weyl's commutation relation for unitary operators holds. In an earlier paper, Santhanam and Tekumalla (1976) showed that the commutation relation for hermitian operators with a bounded spectrum tends to Hdsenberg's standard canonical commutation relation as the spectrum becomes continuous and the dimension n -> 00. The present paper offers a formulation which is coordinate free in the limit n -> 00 and makes the limiting procedure especially ransparent.