Lie algebra representations and rigged Hilbert spaces: the SO(2) case

It is well known that related with the irreducible representations of the Lie group $SO(2)$ we find a discrete basis as well a continuous one. In this paper we revisited this situation under the light of Rigged Hilbert spaces, which are the suitable framework to deal with both discrete and bases in the same context and in relation with physical applications.


Introduction
In the last years we have been involved in a program of revision of the connection between special functions (in particular, Classical Orthogonal Polynomials), Lie groups, differential equations and physical spaces. We have obtained the ladder algebraic structure for different orthogonal polynomials, like Hermite, Legendre, Laguerre [1], Associated Laguerre Polynomials, Spherical Harmonics, etc. [2,3]. In all cases, we have obtained a symmetry group. The corresponding orthogonal polynomial is associated to a particular representation of its Lie group. For instance, for the Associated Laguerre Polynomials and the Spherical Harmonics, we obtain the symmetry group SO (3,2) and in both cases they support a unitary irreducible representation (UIR) with quadratic Casimir −5/4. Both are bases of square integrable functions defined on (−1, 1) × Z and on the sphere S 2 , respectively. In any case we get discrete basis and continuous basis.
On the other hand, the Rigged Hilbert Space (RHS) is a suitable framework for a description of quantum states, when the use of both discrete basis, i.e., complete orthonormal sets, and generalized continuous basis like those used in the Dirac formalism are necessary [4]- [15]. As mentioned above, this is a typical situation arisen when we deal with special functions, which hold discrete labels and depend on continuous variables. We have analysed this situation for the Hermite and Laguerre functions motivated also by possible applications on signal theory in recent papers [11,12]. Moreover, the RHS fit very well with Lie groups [16] and also with semigroups, see [17] and references therein.
In this paper, we continue the study of the relation between Lie algebras, special functions and RHS. Here we propose a revision of the simplest case provided by the Lie algebra so(2) related to the Fourier series, both for its practical interest and as an introduction to our ideas. Since continuous and discrete bases are involved [13], the framework of the RHS is the appropriate one for this case too. Thus, we use the so(2) Lie algebra and its UIR as to construct the RHS associated to these representations.

Rigged Hilbert Spaces
There are several reasons to assert that Hilbert spaces are not sufficient for a thoroughly formulation of QM even within the non-relativistic context. We can mention, for instance, the Dirac formulation [18] where operators with continuous spectrum play a crucial role (see also [10] and references therein) and their eigenvectors are not in the Hilbert space of square integrable wave functions. Another example is related with the proper definition of Gamow vectors [9], which are widely used in calculations including unstable quantum systems and are non-normalizable. We can also refer to formulations of time asymmetry in QM that may require the use of tools more general than Hilbert spaces [19].
The proper framework that includes naturally the Hilbert space and its features, which are widely used in QM, is the RHS.
The Rigged Hilbert Spaces were introduced by Gel'fand and collaborators [4] in connection with the spectral theory of self-adjoint operators. They also proved, together with Maurin [14], the nuclear spectral theorem [10,15]. The RHS formulation of Quantum Mechanics (QM) was introduced by Bohm and Roberts around 1965 [5,8].
A Rigged Hilbert Space (also called Gelf'and triplet) is a triplet of spaces with H an infinite dimensional separable Hilbert space, Φ (test vectors space) a dense subspace of H endowed with its own topology, and Φ × is the dual /antidual space of Φ.
The topology considered on Φ is finer (contains more open sets) than the topology that Φ has as subspace of H, and Φ × is equipped with a topology compatible with the dual pair (Φ, Φ × ) [20], usually the weak topology. One consequence of the topology of Φ [21,10] is that all sequences which converge on Φ, also converge on H, the converse being not true. The difference between topologies gives rise that the dual space of Φ, Φ × , is bigger than H, which is self-dual.
Here, we shall consider the dual Φ × of Φ, i.e., any F ∈ Φ × is a continuous linear mapping from Φ into C. Linearity and antilinearity mean, respectively, that for any pair of vectors ψ, ϕ ∈ Φ, any pair F, G ∈ Φ × and any pair α, β ∈ C we have where we have followed the Dirac bra-ket notation and the star denotes complex conjugation.
A crucial property to be taken under consideration is that if A is a densely defined operator on H, such that Φ be a subspace of its domain and that Aϕ ∈ Φ for all ϕ ∈ Φ, we say that Φ reduces A or that Φ is invariant under the action of A, (i.e., AΦ ⊂ Φ). In this case, A may be extended unambiguously to the dual Φ × by making use of the duality formula The topology on Φ is given by an infinite countable set of norms {|| − || ∞ n=1 }. A linear operator A on Φ is continuous if and only if for each norm || − || n there is a K n > 0 and a finite sequence of norms || − || p 1 , || − || p 2 , . . . , || − || pr such that for any ϕ ∈ Φ, one has [22] ||Aϕ|| n ≤ K n (||ϕ|| p 1 + ||ϕ|| p 2 + · · · + ||ϕ|| pr ) .
The same result applies to check the continuity of any linear or antilinear mapping F : Φ −→ C. In this case, the norm ||Aϕ|| p should be replaced by the modulus |F (ϕ)|.

A paradigmatic case: RHS for SO(2)
As mentioned before, we have considered the most elementary situation provided by SO(2), where we have two RHS serving as support of unitary equivalent representations of SO(2). One of these RHS is a concrete RHS constructed with functions or generalised functions and the other one is an abstract RHS. A mapping of the test vectors of the abstract RHS gives the test functions of the concrete one. Also we have to adjust the topologies so that the elements of the Lie algebra be continuous operators on both test spaces and their corresponding duals.
Let us remember that SO (2) is the group of rotations on the Euclidean plane. It is a one-dimensional (1D) abelian Lie group, parametrized by φ ∈ [0, 2π).
The elements R(φ) of SO(2) satisfy the product law Here, we are considering two equivalent families of UIR of SO (2): one of them supported by the Hilbert space L 2 [0, 2π] (via the regular representation, that contains once all the UIR, each one related to an integer number) and another set of UIR (also labelled by Z) supported by an abstract infinite dimensional separable Hilbert space H.

UIR supported by the HS L 2 [0, 2π]
We consider the UIR characterised by the unitary operator on L 2 [0, 2π] An orthonormal basis for L 2 [0, 2π] is given by the sequence of functions φ m labelled by Thus, any Lebesgue square integrable function f (φ) of L 2 [0, 2π] can be written as with under the condition that Note that the complex numbers f m are the Fourier coefficients of f (φ).
The functions U m = e −imφ satisfy the following orthogonality and completeness relations: 1 2π

UIR on an infinite-D separable HS
Equivalently, we may construct another set of UIR's of SO(2) labelled by Z and supported on an abstract infinite dimensional separable Hilbert space H.
Let {|m } m∈Z be an orthonormal basis of H. There is a unique natural unitary mapping S such that Let us consider the subspace Φ of H of vectors such that for any p = 0, 1, 2, . . . Since Φ contains all finite linear combinations of the basis vectors |m is dense on H. We endow Φ with the metrizable topology generated by the norms ||f || p , (p = 0, 1, 2, . . . ). In this way we have constructed a RHS: Φ ⊂ H ⊂ Φ × .
Considering that the unitary mapping S transports the topologies, we get two RHS such that Φ and H have the discrete basis {|m } m∈Z and (SΦ) and L 2 [0, 2π] have its equivalent discrete basis {φ m } m∈Z . Now, we may define continuous basis in both RHS as follows. Since these two RHS are unitarily equivalent, it is enough to construct the continuous basis on the abstract RHS and to induce the equivalent one in the other RHS.
Let us consider the abstract RHS Φ ⊂ H ⊂ Φ × . Since |m ∈ H we can consider m| ∈ H × = H. Then, for any φ ∈ [0, 2π), we can define a ket |φ such that From the duality relation φ|m = m|φ * and for any |f = ∞ m=−∞ a m |m ∈ Φ we get where a m = f m as in (4). The action of φ| on Φ, φ|f , is well defined since the following series is absolutely convergent Note that both series in the second row of this inequality converge: the first one because it verifies eq. (7) for p = 1, and it is obvious for the second series.
On the other hand, {|φ } φ∈[0,2π) , is a continuous basis. In fact, if we apply the map S to an arbitrary |f ∈ Φ as in (6), we obtain that S|f ∈ SΦ ⊂ L 2 [0, 2π] and If |f , |g ∈ Φ, then f (φ) = S|f and g(φ) = S|g belong to (SΦ) ⊂ L 2 [0, 2π]. Thus, and due to the unitarity of S, we get and thus Applying this identity to |f ∈ Φ, we have This gives a span of |f in terms of |φ with coefficients f (φ) for all φ ∈ [0, 2π), which shows that {|φ } is a continuous basis on Φ, although its elements are not in Φ but instead in Φ × . Since φ| acts on Φ only (not on all H), then for an arbitrary |g ∈ Φ we have Because of the definition of RHS to any |f ∈ Φ corresponds a f | ∈ Φ × and the action of f | on any |g ∈ Φ is given by the scalar product f |g (9) from L 2 [0, 2π]. Thus, I is the canonical injection I : Φ −→ Φ × , and it is continuous. Moreover, |φ is a continuous linear functional on Φ then φ| is a continuous antilinear functional on Φ. Consequently, and then, Therefore, the set {|φ } satisfies the relations of orthogonality (12) and completeness (10) that allow us to write showing, once more, that {|φ } is a continuous basis. Note that {|φ } spans Φ and not H.
There are some formal relations between both basis, {|m } and {|φ }. For instance, replacing f by |m in (13) we have that Since {|m } is a basis in H, the following completeness relation holds where I is the identity on H and also on Φ. Do not confuse this identity with I previously defined (10).
Because |m ∈ Φ, we may apply to it any element of Φ × so that I becomes a well defined identity on the dual Φ × which gives the second formal identity (14). Nevertheless and due to the absolute convergence of the series it is easy to prove that φ| I converges in the weak topology on Φ × .
The unitary map S : H −→ L 2 [0, 2π] also allows us to transport R to an equivalent representation R supported on H by for any value of θ then R(θ) is also unitary on H due to the unitarity of S.
Unitary operators leaving Φ invariant can be extended to Φ × by the duality formula (1), i.e., Therefore, Combining both expressions and dropping the arbitrary |f ∈ Φ we arrive to which is a rigorous expression in Φ × . In fact, let |f ∈ Φ as in (6). Then, we have Hence, we see that R(θ)Φ ⊂ Φ and since The UIR' s, U m , on H are given in terms of the U m , defined in (3), by Let J be the infinitesimal generator associated to this representation, i.e. U m (φ) = e −iJφ .
Since U m is unitary then J is self-adjoint. Its action on the vectors |m is Hence for any |f ∈ Φ as in (6), we have that From the set of norms ||Jf || 2 p , p = 0, 1, 2, . . . , that we have defined in (7), we obtain the following inequality which shows that JΦ ⊂ Φ. Also, this inequality may be also read as ||Jf || 2 p ≤ ||f || p+1 , ∀|f ∈ Φ , ∀p . Thus, we have proved that J is continuous on Φ. Moreover, since the self-adjoint operator J verifies JΦ ⊂ Φ, it can be extended to Φ × using the duality form, i.e., Furthermore, since J is continuous on Φ, this extension is weakly (with the weak topology) continuous on Φ × . In fact, since the series is weakly convergent, then where the operator D φ is defined as follows: for any |f in Φ we know that S|f = f (φ) ∈ (SΦ) as in (8). Then, We easily conclude that the operator i d/dφ is continuous on SΦ with the topology transported by S from Φ (norms on SΦ look like exactly as the norms on Φ). Hence, This definition implies that iD φ is continuous on Φ. Moreover, it is self-adjoint on H, so that it can be extended to a weakly continuous operator on Φ × as the last identity in (15) shows. Therefore J ≡ iD φ on Φ × .

Conclusions
We have construct two RHS that support the UIR of the Lie group SO (2), Φ ⊂ H ⊂ Φ × and (SΦ) ⊂ L 2 [0, 2π] ⊂ (SΦ) × . The first one is related with the discrete basis {|m } and in some sense is an abstract RHS, but the second one, related with the continuous basis {|φ }, is obtained by means of a unitary map S : |m → e −imφ / √ 2π that allows to translate the topologies of the first RHS to the second one as well as all the relevant properties.
Another interesting point to stress is the fact that RHS, from one side, and Lie algebras and Universal Enveloping Algebras, from the other, are closely related. This means that starting from a Lie algebra we can construct RHS that support its UIR and also that generators and universal enveloping elements can be represented by continuous operators avoiding domain difficulties [5].