Change of representation and the rigged Hilbert space formalism in quantum mechanics
Introduction
The rigged Hilbert space (RHS) is a generalization of the following construction from distribution theory where is the Schwarz space and is the space of tempered distributions. The RHS theory was developed in the 1950s, mainly by Gel'fand, Kostutchenko, Vilenkin, Shilov and Maurin [1, 2, 3]. A cornerstone of the theory is the nuclear spectral theorem [2, 4, 5], which refines von Neumann's spectral theorem [6]. The use of the RHS for a rigorous mathematical formulation of Dirac's formalism was suggested and investigated by several authors, in many papers. Among them we mention the works of Foias [7], Roberts [8, 9], Böhm [10] and Antoine [11]. The RHS theory allows us to associate to every quantum system a Hilbert space , a construction (or more [12, 13]) of the form (Φ ⊂ ⊂ Φ×, A) where Φ is a dense linear subspace of endowed with a locally convex topology τ finer than the topology induced by the norm of Φ× is its topological antidual space and A is an algebra of continuous operators on Φ. For more details on the rigged Hilbert space formalism see for instance [10, 12, 13, 14, 15, 16, 17, 18] and references therein. The basic tools of the theory are the generalized eigenvectors of self-adjoint or unitary operators which leave the space Φ invariant. Recall that if A is a linear operator in that leaves Φ invariant, a generalized eigenvector of A is an eigenvector of the algebraic dual of the operator A | Φ, which is continuous with respect to the topology τ. We refer the reader to [19, 20] for conditions that ensure tight rigging of the operator A, where the spectrum of the operator A coincides with an adequate set built from generalized eigenvalues of A. For the study of other constructions that generalise the RHS, see e.g. [14, 21, 22]. Finally, let us mention the work of Bergeron [23], where another explanation of Dirac's formalism is proposed.
In practice, in the study of a quantum system, we may consider observables and other fundamental operators. To every fundamental operator (or a commuting family of fundamental operators) we may associate a Hilbert space and a representation. The goal of any change of representation is to facilitate calculations and understanding of the system. The following two problems arise naturally: Given an initial representation of the system and an associated algebra of operators containing fundamental observables of the system, how to construct new useful representations? How to identify all the fundamental operators of the system? Important examples of representations are the so-called “orthonormal representations” which are associated to self-adjoint operators, like the well-known discrete basis {|n〉}n, or continuous bases {|x〉}x and {|p〉}p [24, 25, 26] or the representations associated to coherent states, these are representations associated somehow to the eigenvectors of some lowering operator [25, 27].
In this paper we are concerned with the existence of representations realized with a family of generalized eigenvectors of an operator. Given a Hilbert space , we mainly deal with constructions of the form Λ ⊂ ⊂ Λ¯*, where Λ is a dense linear subspace of and Λ¯* is its algebraic antidual space, and we consider linear operators that leave Λ invariant. Observe that this may allow a greater generality since Λ can be constructed using some physical states, and we can define a topology on Λ that makes some generalized eigenvectors continuous. At this stage, let us mention that the topology cannot be obtained from physical data, see for instance [10, p. 4]. We first show some simple and basic results on algebraic generalized eigenvectors of certain unbounded linear operators in . Then we discuss conditions that ensure the existence of associated representations. This may be considered as a practical method for finding new representations, and more fundamental operators. We may classify elements of the algebra A of operators on Λ by the behaviour of their generalized eigenvectors. If Λ is endowed with a topology τ, we may define coherence-like properties in the sense of Klauder–Skagerstam [27] with respect to τ. In this paper we touch only a few aspects of the subject, the idea we wish to convey is that generalized eigenvectors arise naturally in many situations. On the other hand, we believe that the use of generalized eigenvectors may enrich our understanding of the expansion of a state into eigenfunctions of hermitian or nonhermitian operators, see for instance [25, 28, 29].
As an application, we review and refine Böhm's algebraic study of the quantum harmonic oscillator [10] (see also [13, 30]). This is a basic, theoretical and elementary quantum system, without boundary conditions. We focus on the study of the change of representations. Hence we use the number representation, which can be constructed using the algebraic span of the eigenstates of the Hamiltonian in any representation and construct other representations, by using the generalized eigenvectors of suitable operators. As it was shown by Böhm, the rigged Hilbert space structure Φ ⊂ ⊂ Φ× arise naturally. in particular, we recover well known results by using elementary methods, without using the nuclear spectral theorem. indeed, the concept of generalized eigenvectors allows a rigorous and elementary mathematical study of this basic example. The position, the momentum and the Fock–Bargmann representation are respectively associated to generalized eigenvectors of the position, momentum and creation operators. The |x〉 and the |p〉 vectors do not lie in the Hilbert space. However, with respect to the topology of Φ, we see that they exhibit coherence like properties in the sense of Klauder–Skagerstam.
Section snippets
Preliminaries
Let Λ be a complex vector space and let T: Λ → Λ be a linear map. The spectrum of T is
We denote by Λ* the dual space of Λ, elements of Λ* are the linear functionals F: Λ → ℂ. Denote by T°: Λ* → Λ* the dual of T, which is defined by T°F = FT. It is easy to see that if the range of T has finite codimension n, then the dimension of Ker T° is equal to n. If F is an eigenvector of T° with respect to the eigenvalue λ, we say that (F, λ) (or F) is a generalized
Generalized eigenvectors and associated representations
Throughout this section, X is a subset of ℂ and is a Hilbert space. To deal freely with evaluation maps, we first consider spaces of functions in a general sense. We shall say that a complex linear space Ω is a space of functions on X if all elements of Ω are functions f: X → ℂ; in particular, f = 0 if and only if f (x) = 0 for every x ∈ X.
Our discussion is based on the following elementary lemma.
The algebra of the harmonic oscillator
We describe in this section the number representation and the associated RHS. Let Ψ be a complex linear space and let P, Q :Φ → Φ be linear operators satisfying the canonical commutation rule (CCR), that is . Put
The following properties, which are well known, are easy to prove
Eigenstates of the Hamiltonian in function spaces
Next we pursue our study of the algebra of the quantum harmonic oscillator, and we see how the three other fundamental representations (position, momentum and Bargmann–Fock) arise naturally, using fundamental operators of the system and their generalized eigenvectors. The three spaces are spaces of functions, associated to pairs (A, B) of operators in the first Weyl algebra that satisfy [A, B] = I, Aψ(x) = xψ(x) and Bψ(x) = – dψ(x)/dx. At this stage, a natural question presents itself: Are
Acknowledgements
We thank the referee for various useful comments and for drawing our attention to papers [17, 18].
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Supported by the National Center for Scientific and Technical Research (CNRST, Morocco) as part of the research excellence scholarship program.