Quantum Physics and Signal Processing in Rigged Hilbert Spaces by means of Special Functions, Lie Algebras and Fourier and Fourier-like Transforms

Quantum Mechanics and Signal Processing in the line R, are strictly related to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A Rigged Hilbert space is found and a new discrete basis in R obtained. The operators {O[R]} defined on R are shown to belong to the Universal Enveloping Algebra UEA[io(2)] allowing, in this way, their algebraic discussion. Introducing in the half-line a Fourier-like Transform, the procedure is extended to R^+ and can be easily generalized to R^n and to spherical reference systems.


INTRODUCTION
The starting point of Quantum Mechanics in the line R (and its wave functions counterpart L 2 (R)) are the Weyl-Heisenberg algebra (WHA), X, P, I [1] that defines position and momentum (or time and frequency) and the Fourier Transform [FT] [2] that relates them each other. This approach is easily extended to Signal Processing both in Informatics and Optics also if, there, the operatorial structure is scarcely taken in consideration.
We consider here a more predictive approach based on the Hermite functions (HF) [3], introduced as transition matrices between discrete and continuous bases in a quantum space [4].
The central point is to add to the WHA a new operator N that reads the degree of the HF. A Rigged Hilbert Space (RHS) [5] is found and a projective algebra io(2) [6,7]  Three bases are indeed obtained, two continuous, {|x } and {|p } related to the position X and momentum P of WHA and a new one, discrete, {|n } connected to the HF: as in a Hilbert space (HS) all bases have the same cardinality, the mathematical structure is a RHS, the true space of Quantum Mechanics. A state in Quantum Mechanics was indeed originally defined as a ray in a RHS, an intricate concept involving a Gelfand triple φ ⊆ H ⊆ φ ′ : H is a HS, φ (dense subset of H) is the "ket" space and φ ′ (dual of φ) is the "bra" space. However quickly RHS has been considered an unnecessary complication as all results can be found (after elaborated limits on functions with compact support) in the HS obtained representing, by means of the axiom of choice, each entire ray by a vector of norm one and phase zero.
We attempt to show here that RHS are not so complicated to justify its elimination from all recent textbook and can be more powerful that the standard HS because they allow an operatorial description and an inclusion inside the same algebra of variables with different cardinality. Hoping that it could convince more than a general discussion, that can be found in [5], we discuss here the 1-dimensional case.
The projective algebra io(2) ≡ X, P, N, I above introduced is isomorphic to the so called algebra of harmonic oscillator H, a, a † , I [6,7], usually described inside the UEA[WHA]. As the ∞-dimensional UEA[io (2)] is a complex object, a set of relevant substructures emerges in R and {O[R]} as well as in L 2 (R) and {O[L 2 (R)]}. In particular, as ∞-many copies of io (2) are contained into UEA[io (2)], ∞-many operators that close io(2) are found.
It should be noted that in L 2 (R) (but not in R, {O[R]} and {O[L 2 (R)]}) the subspaces structure derived from the algebra could be obtained also from the Fractional Fourier Transform ([FrFT]) [8]. Indeed the eigenvectors of the [FrFT] are, as the ones of [FT], the Hermite functions and for each k ∈ N a [FrFT] can be constructed such that its eigenvalues are the kth roots of unity, each one corresponding to one subspace.
To exhibit that the approach in not peculiar of the line R, the discussion is repeated on the half line R + . For this purpose, starting from the HF and the relations between HF and generalized Laguerre functions, two Fourier-like Transforms [T ± ] are constructed with appropriate eigenvectors on the half-line.
The extension to R n and to spherical coordinates will be discussed elsewhere.

THE LINE R
To describe the line we start from the unitary irreducible representations of the translation group The regular representation {|p } (−∞ < p < ∞) is such that |p dp p| = I, and the conjugate basis {|x }, defined by the operator X, is obtained using the [FT] The operators X and P together with I close the WHA [1]. At this point, we move to a non standard approach related to the ray representations [6,7,9] of the inhomogeneous orthogonal algebra io (2). In addition to X and P (which spectra have the cardinality of the continuum ℵ 1 ) an operator N (with spectrum of cardinality ℵ 0 ), related to the index n of the Hermite functions (HF) (where H n (x) are the Hermite polynomials) is introduced. As the Hermite functions are a basis of the space of square integrable functions on the line L 2 ((−∞, ∞)) ≡ L 2 (R) [2]: the basic idea is now to introduce the vectors {|n } that, by inspection, are an orthonormal and complete set in R The set {|n } is thus a discrete basis in the real line R.
Relations among the three bases are easily established, as {ψ n (x)} are eigenvectors of [FT], For an arbitrary vector |f ∈ R we thus have , and the wave functions f (x) , f (p) and the sequence {f n } describe |f in the three bases.
All seems trivial, but {|n } has the cardinality of the natural numbers ℵ 0 and, as all bases in a Hilbert space have the same cardinality, the structure we have constructed (the quantum space on the line R) is not an Hilbert space but a Rigged Hilbert space.
In this way, by means of [FT] and HF, we went back to the foundations of Quantum Mechanics, where a physical state was defined as a ray in a Hilbert space and not as a vector in HS as we are used, considering the RHS as a unnecessary complication. However the problem is apparent also in the simple case of the harmonic oscillator: the algebraic description consider H (≡ N + 1/2) , a, a † and I. But X = 1 √ 2 (a + a † ) and P ≡ −i D x = −i √ 2 (a − a † ) do not belong to the algebra because they are not operators in a HS. In a RHS, instead, X and P are standard operators also if their spectra are continuous and we have no problems to consider them as generators of a Lie algebra together with the number operator N [4,5]. This allows us to extend the Lie algebra to the differential structure and, in general, to include all operators inside the involved Universal Enveloping Algebra.
To describe the structure of the operators on the line R we thus begin introducing in the The recurrence relations of Hermite polynomials can be rewritten as where the operators a and a † are defined in terms of the operators X and P a : The algebra contains indeed the rising and lowering operators on the HF [10,11] [N, and is isomorphic to the projective algebra io(2) [6,7]: Because of eq. (1) the RHS R and L 2 (R) are isomorphic and support a representation with zero value of the Casimir operator of iso(2) (3) (or equivalently (2)), as discussed in [9], So, in the RHS of the line R, we have that in L 2 (R) (the RHS of square integrable functions defined on the line) can also be written as (4) is indeed the operatorial identity that defines the RHS R and L 2 (R) by inspection equivalent to the Hermite differential equation The representation is irreducible, so that, on both spaces L 2 (R) and R, all operators of the UEA[io (2) From the analytical point of view, an ordered monomial X α D x β N γ ∈ UEA[io (2)] is an order β differential operator but, because of the operatorial identity (5), we have that on the basis vector ψ n (x) becomes The UEA[io (2)] is a rich structure. In particular it contains ∞-many io(2) algebras that allow to construct an intricate structure of subspaces. In particular for any k ∈ N we have where k and r are parameters that define each copy io k,r (2) of the algebra io(2). Indeed, starting from n and k , we can define two other integers q = Quotient[n, k] and r = Mod[n, k] so that n = k q + r and the operators Q and R are, of course, diagonal on {|k q + r } Q |kq + r = q |kq + r R |kq + r = r |kq + r .

By inspection, the operators
are defined on the whole set {|k q + r } and give A † k,r |k q + r = q + 1 |k(q + 1) + r , A k,r |k q + r = √ q |k(q − 1) + r .
Each couple, k and 0 ≤ r < k, gives us a irreducible representation with C = 0 of io(2) that we denote io k,r (2) In particular, for k = 4 and 0 ≤ r < 4, we can define and L 2 (R) and R can be split in four subspaces each one representation of one of io 4,r (2) These results are general since eqs. (6) allow to write for all k ∈ N, 0 ≤ r < k and q = 0, 1, 2, . . . L 2 k,r (R) := {ψ k q+r (x)}, R k,r := {|k q + r }, The eq. (7) is well-known from the Fourier Transform (2) that splits L 2 (R) in four subspaces each one corresponding to one eigenvalue of [FT] (but, of course, does not consider the operators). Its generalization to all k ∈ N, that allows to divide L 2 (R) in k subspaces (again disregarding the operators), can be obtained from the Fractional Fourier Transform ([FrFT]) [8]: [FrFT] α ψ n (x) = e iαn ψ n (x ′ ), α ∈ C, by specializing α to α = 2π/k where N -diagonal on each ψ n (x)-on the whole space is rewritten as N ≡ (X 2 −D 2 x −1)/2. As an example, for k = 3 , eq. (8) gives [FrFT] 2π/3 ψ n (x) = e i 2πn 3 ψ n (x ′ ), that exhibits three subspaces of L 2 (R)

THE HALF-LINE R +
The results obtained on the line R can be rewritten in R + , the vector space defined by the operator Y , with basis {|y } with y into the open set (0, +∞): Hence, like the case of {ψ n (x)} in L 2 (R) , for any fixed α the set {M α n (x)} is a basis in L 2 (R + ) [2]. Again like in (1) we define, for n ∈ N and fixed α, the vector |n |n := Since a discrete basis {|n } and a continuous one {|y } have been found also the half-line is a RHS.
We have thus on the half-line the operatorial identity Also in the half line, the representation is irreducible so that the operators acting on the L 2 (R + ) and R + belong to the UEA[su(1, 1)] i.e. they can be written as The monomials J + α J 3 β J − γ ∈ UEA[su(1, 1)] look to be differential operators of the order α + γ but the identity (12) can also be read so that all the off-diagonal operators are at most first order differential operators and all the diagonal ones are equivalent to eq. (11).
Also the UEA[su(1, 1)] contains, in analogy with the UEA[io (2)] , ∞-many su(1, 1), denoted by su k,r (1, 1) . We can define L 2 k,r (R + ) = {M k q+r (R + )} , R + k,r = {|k q + r } and for each k ∈ N the spaces L 2 (R + ) and R + are direct sums of r depending subspaces  In conclusion, two alternative conjugate basis of {|y } have been found where N is diagonal on M ±1/2 n (y) and, in general, can be obtained from the identity (12).
The hope that, for a generic α, an Integral Transform [T α ], with eigenvectors {M α n } could be found (allowing ∞-many equivalent bases α-dependent on the same footing) seems reasonable. We are working on it.

CONCLUSIONS
Rigged Hilbert spaces are shown to be more predictive than Hilbert spaces in Quantum Physics and Signal Processing in Optics and Informatics since operators of different cardinality can be considered together.
In Rigged Hilbert spaces continuous and discrete bases exist with special functions as transformation matrices between them.
In a RHS operators of different cardinality can be together generators of a Lie algebra and/or elements of its Universal Enveloping Algebra.
An elaborated algebraic structure inside both, the Quantum space and the space of operators defined on it, is found. In particular, an infinite set of substructures emerges both in the space of the states and in the space of operators acting on them.
An alternative definition of an Integral Transform, as an operator that has special functions as eigenvectors, has been introduced.