Abelian groups and the Weyl approach to kinematics nonlocal function-algebras

This paper is motivated by the Weyl-Wigner-Moyal approach to quantum mechanics [8], [12]. The nonlocal multiplication of phase space functions is generalized to functions over locally compact Abelian groups or over homogeneous spaces of such groups. It is shown that such nonlocal algebras of functions are closely related to the ray-representations of the corresponding dual groups. An arbitrary continuous solution of the factor- equation on ? gives rise to some translationally invariant nonlocal algebra over G or over its homogeneous space. This enables us to find the group-theoretical interpretation of the well-known symmetrization rules (e.g. the standard, antistandard and Weyl rules [2]). Some properties of the nonlocal products are studied (e.g. the existence of the identity). It should be noted that some of our results (e.g. the theory of weighted convolutions of measures) are valid in the non-Abelian cases, as well, and they generalize some ideas of Edwards and Lewis [14]. When working with the Fourier analysis on the locally compact Abelian groups, we mostly use the language of W. Rudin's book [10].