A constructive presentation of rigged Hilbert spaces

TL;DR

This paper constructs a mathematically rigorous rigged Hilbert space incorporating discrete and continuous operators, providing a toy model that extends the algebraic framework of quantum mechanics beyond traditional Hilbert spaces.

Contribution

It introduces a new construction of rigged Hilbert spaces with a discrete operator, extending the algebraic structure and providing a well-defined toy model for operators with different cardinalities.

Findings

Rigged Hilbert space constructed with discrete and continuous operators.

Representation of the algebra io(2) on L^2(R) and R is irreducible.

Extension to higher-dimensional orthogonal spaces via tensorialization.

Abstract

We construct a rigged Hilbert space for the square integrable functions on the line L^2(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.