A constructive presentation of rigged Hilbert spaces. (June 2015)

• Record Type: Journal Article
• Authors: Celeghini, Enrico
• Is Part Of: Journal of physics. Number 626(2015)
• Publication Date: 2015-06
• Subjects: Physics -- Congresses; 530.5
• DOI: 10.1088/1742-6596/626/1/012047
• Languages: English
• ISSNs: 1742-6588

Abstract

Abstract: We construct a rigged Hilbert space for the square integrable functions on the line L2(R) adding to the generators of the Weyl-Heisenberg algebra a new discrete operator, related to the degree of the Hermite polynomials. All together, continuous and discrete operators, constitute the generators of the projective algebra io (2). L 2 (R) and the vector space of the line R are shown to be isomorphic representations of such an algebra and, as both these representations are irreducible, all operators defined on the rigged Hilbert spaces L 2 (R) or R are shown to belong to the universal enveloping algebra of io (2). The procedure can be extended to orthogonal and pseudo-orthogonal spaces of arbitrary dimension by tensorialization. Circumventing all formal problems the paper proposes a kind of toy model, well defined from a mathematical point of view, of rigged Hilbert spaces where, in contrast with the Hilbert spaces, operators with different cardinality are allowed.