TY - GEN

T1 - Localization and the canonical commutation relations

AU - Moylan, Patrick

N1 - Publisher Copyright:
© Springer Nature Singapore Pte Ltd. 2016.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2016

Y1 - 2016

N2 - Let Wn (R) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of Wn (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 D1,1(R) is the fraction field of W1.1(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 Wn(R) out of representations of so (2, q).

AB - Let Wn (R) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of Wn (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 D1,1(R) is the fraction field of W1.1(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 Wn(R) out of representations of so (2, q).

UR - http://www.scopus.com/inward/record.url?scp=85009730035&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009730035&partnerID=8YFLogxK

U2 - 10.1007/978-981-10-2636-2_30

DO - 10.1007/978-981-10-2636-2_30

M3 - Conference contribution

AN - SCOPUS:85009730035

SN - 9789811026355

T3 - Springer Proceedings in Mathematics and Statistics

SP - 423

EP - 430

BT - Lie Theory and Its Applications in Physics

A2 - Dobrev, Vladimir

PB - Springer New York LLC

T2 - Proceedings of the 11th International Workshop on Lie Theory and Its Applications in Physics, 2015

Y2 - 15 June 2015 through 21 June 2015

ER -