Discrete Heisenberg-Weyl Group and Modular Group

It is shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers θ and −1/θ generate the whole algebra B of bounded operators on L2(R). The natural action of the modular group in B is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed. Writing a contribution to a memorial volume one always feels the mixture of the admiration for passed away and sorrow of a loss. Because of a difference in age and geographical (and/or political) obstructions I did not have any significant personal scientific encounters with J.Schwinger. But our brief exchanges during several short meetings and most of all reading his papers influenced my way of thinking and writing to a great extent. Together with his colossal work on QED and general quantum field theory, J.Schwinger did pay attention to technical problem, pertaining to the ordinary nonrelativistic quantum mechanics [1]. So I hope, that the comments in this paper, which are confined to the objects in a simplest Hilbert space of quantum theory, namely L2(R), still would amuse him. My excuse is that these comments were generated by a dynamical problem in quantum field theory. What I want to discuss is to my belief the base of several ”duality principles”, which appear nowadays, i.e., in connection with string theory and conformal field theory. However I shall not go into technicalities of these subjects. More on this wll be mentioned in the conclusion of this paper. I consider a Hilbert spase H, where the usual coordinate and momentum operators Q and P , satisfying the Heisenberg commutation rule [P,Q] = −ih̄I are irreducibly represented. For example, we can take H = L2(R) with elements ψ(x) and usual action of Q and P Pψ(x) = h̄ i d dx ψ(x); Qψ(x) = xψ(x) (coordinate representation). Together with Q and P consider their exponentials u = e , v = e with the Weyl commutation relations uv = evu, where θ = pq 2πh̄ . Here p and q are fixed real numbers with dimensions of momentum and coordinate, respectively, so that θ is dimensionless. I could of course fix the units in such a way, that the dimensions disappear. Only the parameter θ is essential in what follows. It is clear that (for fixed p and q) the map