Localization and the canonical commutation relations

Let W_n (R) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of W_n (R) onto its image in a localization, or a quotient thereof, of U(so(2, q)), the universal enveloping algebra of so(2, q), forn depending upon q [1]. Here we treat the so(2, 1) case in complete detail. We establish an isomorphism of skew fields, specifically, D(so(2, 1)) ~ D_(1,1)(R) where D_1,1(R) is the fraction field of W_1.1(R) ~ W₁(R) ⊗ R(y) with R(y) being the ring of polynomials in the indeterminate y and D(so(2, 1)) is a certain extension of the skew field of fractions of U(so(2, 1)), which is described below. We give applications of this result to representations. In particular we are able to construct representations of W₁ (R) out of representations of so (2, 1). Thus, we are able, for this lowest dimensional case, to obtain the canonical commutation relations and representations of them out of so(2, 1) symmetry. Using similar results in higher dimensions [1] we are able to construct representations of W_n(R) out of representations of so (2, q).