SU(2), Associated Laguerre Polynomials and Rigged Hilbert Spaces
aa r X i v : . [ m a t h - ph ] A p r SU (2), Associated Laguerre Polynomialsand Rigged Hilbert Spaces Enrico Celeghini, Manuel Gadella and Mariano A del Olmo
Abstract
We present a family of unitary irreducible representations of SU (2)realized in the plane, in terms of the Laguerre polynomials. These functionsare similar to the spherical harmonics defined on the sphere. Relations withan space of square integrable functions defined on the plane, L ( R ), areanalyzed. We have also enlarged this study using rigged Hilbert spaces thatallow to work with iscrete and continuous bases like is the case here. The representations of a Lie algebra are usually considered as ancillary tothe algebra and developed starting from the algebra, i.e. from the generatorsand their commutation relations. The universal enveloping algebra (UEA) isconstructed and a complete set of commuting observables selected, choosingbetween the invariant operators of the algebra and of a chain of its subalge-bras. The common eigenvectors of this complete set of operators are a basisof a vector space where the Lie algebra generators are realized as operators.We propose here an alternative construction that allows to add to therepresentations obtained following the reported recipe, new ones not achiev-able following the previous approach. Starting from a concrete vector space
E. CeleghiniDpto di Fisica, Universit`a di Firenze and INFN–Sezione di Firenze, Firenze, Italy,Dpto. de F´ısica Te´orica, Universidad de Valladolid, E-47005, Valladolid, Spaine-mail: celeghini@fi.infn.itM. Gadella and M.A. del OlmoDpto de F´ısica Te´orica and IMUVA, Univ. de Valladolid, E-47005, Valladolid, Spaine-mail: [email protected] , e-mail: [email protected] of the 10-th International Symposium ”Quantum Theory and Symme-tries” (QTS10), 19-25 June 2017, Varna, Bulgaria (to be published by Springer). 1 Enrico Celeghini, Manuel Gadella and Mariano A del Olmo of functions with discrete labels and continuous variables, we consider therecurrence relations that allow to connect functions with different values ofthe labels. These recurrence relations are not operators but allow us to intro-duce, for each label and for each continuous variable, an operator that readsits value. In this way, recurrence relations are rewritten in terms of risingand lowering operators built by means of the above defined operators.Theserising and lowering operators are often genuine generators of the Lie algebrasconsidered by Miller [1] and the procedure gives simply the representationsof the algebras in a well defined function space [2, 3]. However it can happenthat the commutators, besides the values required by the algebra, have addi-tional contributions. The essential point of this paper is that these additionalcontributions (as exhibited here) can be proportional to the null identity thatdefines the starting vector space. As this identity is zero on the whole repre-sentation, the Lie algebra is well defined and a new representation in a spaceof functions has been found.We do not discuss here the general approach, but we limit ourselves to asimple example where all aspects are better understandable. We start thusfrom the associated Laguerre functions (ALF) and, following the proposedconstruction, we realize the algebra su (2) in terms of the appropriate risingand lowering operators. The ALF support in reality a larger algebra [4] but weprefer to consider here only the subalgebra su (2). The reasons for this choiceare twofold: first in this way the technicalities are reduced at the minimumand second it has been very nice for us to discover that not all representationsof a so elementary group like SU (2) where known.As discussed in [5, 6, 7] the presence of operators with spectrum of differentcardinality implies that, as considered for the first time in Lie algebras in [8],the space of the group representation is not a Hilbert space but a riggedHilbert space (RHS) [9]. Thus, we introduce the above setting within thecontext of RHS since the RHS is the perfect framework where discrete andcontinuous bases coexist. In addition, the same RHS serves as a support for arepresentation on it of a Lie algebra as continuous operators as well as for itsUEA. Therefore, the connection between discrete and continuous bases andLie algebras with RHS is well established. The ALP [10], L ( α ) n ( x ), depend from a real continuous variable x ∈ [0 , ∞ )and from two other real labels ( n, α ) : n = 0 , , , . . . and α (usually assumedas a fixed parameter) continuous and > − α = 0 and are defined by the second order differential equa-tion (cid:20) x d dx + (1 + α − x ) ddx + n (cid:21) L ( α ) n ( x ) = 0 . (1) U (2), Associated Laguerre Polynomials and Rigged Hilbert Spaces 3 From the many recurrence relations that can be found in literature [10, 11],we consider the following ones, all first order differential recurrence relations: (cid:20) x ddx + ( n + 1 + a − x ) (cid:21) L ( α ) n ( x ) = ( n + 1) L ( α ) n +1 ( x ) , (cid:20) − x ddx + n (cid:21) L ( α ) n ( x ) = ( n + α ) L ( α ) n − ( x ) , (cid:20) − ddx + 1 (cid:21) L ( α ) n ( x ) = L ( α +1) n ( x ) , (cid:20) x ddx + α (cid:21) L ( α ) n ( x ) = ( n + α ) L ( α − n ( x ) . (2)Starting from L ( α ) n ( x ), by means of repeated applications of eqs. (2), L ( α + h ) n + k ( x )–with h and k arbitrary integers– can be obtained through a differential re-lation of higher order. But, by means of eq. (1), every differential relation oforder two or higher can be rewritten as a differential relation of order one.In particular we can obtain (cid:20) ddx + nα + 1 (cid:21) L ( α ) n ( x ) = − αα + 1 L ( α +2) n − ( x ) , (cid:20) x ( α − ddx − x (cid:16) n + 3 α (cid:17) + α ( α − (cid:21) L ( α ) n ( x )= ( j + α )( α + 1) L ( α − n +1 ( x ) , (3)that are the recurrence relations we employ in this paper.The ALP L ( α ) n ( x ) are –for α > − n with respectthe weight measure dµ ( x ) = x α e − x dx [10]: Z ∞ dx x α e − x L ( α ) n ( x ) L ( α ) n ′ ( x ) = Γ ( n + α + 1) n ! δ nn ′ , ∞ X n =0 x α e − x L ( α ) n ( x ) L ( α ) n ( x ′ ) = δ ( x − x ′ ) . (4)The parameter α can be extended to arbitrary complex values [10] and,in particular, for α integer and such that 0 ≤ | α | ≤ n , we have the relation L ( − α ) n ( x ) = ( − x ) α ( n − α )! n ! L ( α ) n − α ( x ) . (5)Here we assume consistently that n ∈ N , α ∈ Z and n − α ∈ N , and wealso consider α as a label, like n , and not a parameter fixed at the beginning.Following the approach of [2], we introduce now a set of alternative variablesand include the weight measure inside the functions, in such a way to obtainthe bases we are used in quantum mechanics. We define indeed j := n + α/ Enrico Celeghini, Manuel Gadella and Mariano A del Olmo and m := − α/ j ∈ N / , j − m ∈ N and | m | ≤ j . Notethat they look like the parameters j and m used in SU (2). Now we write L mj ( x ) := s ( j + m )!( j − m )! x − m e − x/ L ( − m ) j + m ( x )so that, from eq. (5), L mj ( x ) is symmetric/antisymmetric in the exchange m ↔ − m since L mj ( x ) = ( − j L − mj ( x ). From eqs. (4), we see that the L mj ( x ) verify, for m fixed, the following orthonormality and completenessrelations Z ∞ L mj ( x ) L mj ′ ( x ) dx = δ jj ′ , ∞ X j = | m | L mj ( x ) L mj ( x ′ ) = δ ( x − x ′ ) , (6)and are thus, for any fixed value of m , an orthonormal basis of L ( R + ).Note that, in the algebraic description of the spherical harmonics, thefunctions T mj ( x ) = q ( j − m )!( j + m )! P mj ( x ), related to the associated Legendre func-tions P ml ( x ) and introduced in [2], satisfy T mj ( x ) = ( − m T − mj ( x ) whichis a relation similar to those verified by the L mj ( x ) . Moreover the T mj ( x ),like the L mj ( x ) on the half-line, are orthogonal –for fixed m – in the interval( − , +1) ⊂ R and a basis for L [ − , SU (2) representations in the plane Following now Ref. [2], we define four operators X , D x , J and M such that X L mj ( x ) = x L mj ( x ) , D x L mj ( x ) = L mj ( x ) ′ ,J L mj ( x ) = j L mj ( x ) , M L mj ( x ) = m L mj ( x ) . (7)and we can rewrite eq. (1) in terms of the L mj ( x ) and in operatorial form as E L mj ( x ) ≡ (cid:20) X D x + D x − X M − X J + 12 (cid:21) L mj ( x ) = 0 . (8)Thus, the identity E ≡ L ( R + ).The relations (3) can now be rewritten on terms of the L mj ( x ) as K + L mj ( x ) = p ( j − m )( j + m + 1) L m +1 j ( x ) ,K − L mj ( x ) = p ( j + m )( j − m + 1) L m − j ( x ) , (9)where U (2), Associated Laguerre Polynomials and Rigged Hilbert Spaces 5 K + = − D x (cid:18) M + 12 (cid:19) + 2 X M (cid:18) M + 12 (cid:19) − (cid:18) J + 12 (cid:19) ,K − = 2 D x (cid:18) M − (cid:19) + 2 X M (cid:18) M − (cid:19) − (cid:18) J + 12 (cid:19) . (10)Since, from eqs. (9), we have [ K + , K − ] L mj ( x ) = 2 m L mj ( x ) and assuming K := M (i.e. K L mj ( x ) = m L mj ( x )) we get the relations[ K + , K − ] L mj ( x ) = 2 K L mj ( x ) , [ K , K ± ] L mj ( x ) = ± K ± L mj ( x ) , (11)that display the fact that, for fixed j , under the action of K ± and K , the L mj ( x ) supports the irreducible representation of dimension 2 j + 1 of su (2).However, while as exhibited by (6) the space {L mj ( x ) } has an inner prod-uct for m fixed and j ≥ | m | (thus supporting a set of UIR of SU (1 ,
1) [4]), therepresentation (11) of SU (2) is not faithful, since L mj ( x ) = ( − j L − mj ( x ),and not unitary. The definition of a scalar product is indeed one of the prob-lems we have in the connection of hypergeometric functions and Lie algebras.Hence, we have two problems: the L mj ( x ) are not orthonormal for j fixed andfunctions with opposite m are not independent (as it happens also with the P mj ( x )). Following the same approach of the spherical harmonics to constructthe inner product space for j fixed and | m | ≤ j we, thus, introduce a newreal variable φ ( − π < φ ≤ π ) and the new objects Z mj ( r, φ ) := e i m φ L mj ( r ) , that verify Z mj ( r, φ + 2 π ) = ( − j Z mj ( r, φ ) . Under the change of variable x → r equation (8) becomes for Z mj ( r, (cid:20) d dr + 1 r ddr − m r − r + 4( j + 12 ) (cid:21) Z mj ( r,
0) = 0 . (12)The functions Z mj ( r, φ ) are the analogous on the plane of the spherical har-monics Y lm ( θ, φ ) on the sphere. The orthonormality and completeness of the Z mj ( r, φ ) is similar to that of Y mj ( θ, φ )1 π Z π − π dφ Z ∞ r dr Z mj ( r, φ ) ∗ Z m ′ j ′ ( r, φ ) = δ j,j ′ δ m,m ′ , X j,m Z mj ( r, φ ) ∗ Z mj ( r ′ , φ ′ ) = πr δ ( r − r ′ ) δ ( φ − φ ′ ) . (13)This means that {Z mj ( r, φ ) } is a basis of the Hilbert space L ( R ) with mea-sure dµ ( r, φ ) = r dr dφ/π like { Y mj ( θ, φ ) } is a basis of L ( S ) with dΩ .Now we consider an abstract Hilbert space H supporting the 2 j + 1 di-mensional IR of su (2) spanned by the eigenvectors of J and M (see eq. (7)) Enrico Celeghini, Manuel Gadella and Mariano A del Olmo J | j, m i = j | j, m i , M | j, m i = m | j, m i , j ∈ N , | m | ≤ j . These vectors | j, m i constitute a basis of H verifying the properties of or-thogonality and completeness h j, m | j ′ , m ′ i = δ j,j ′ δ m,m ′ , ∞ X j =0 j X m = − j | j, m ih j, m | = I Any | f i ∈ H may be written as | f i = P ∞ j =0 P jm = − j f j,m | j, m i if and only if ∞ X j =0 j X m = − j | f j,m | < ∞ , f l,m = h l, m | f i . (14)A canonical injection S : H → L ( R ) can be defined by | j, m i → Z mj ( r, φ )and extended by linearity and continuity to the whole H . One can easily checkthat S is unitary. For any | f i ∈ H we have the following expression S | f i = ∞ X j =0 j X m = − j f j,m S | j, m i = ∞ X j =0 j X m = − j f j,m Z mj ( r, φ ) . We now introduce a continuous basis, {| r, φ i} , depending on the values ofthe variables r and φ with the help of the discrete basis {| j, m i} by h r, φ | j, m i := Z mj ( r, φ ) . (15)In reality, because of the different cardinality of r and j , we are dealing witha RHS (see next Section). The Z mj ( r, φ ) can be seen as the transformationmatrices from the irreducible representation states {| j, m i} to the localizedstates in the plane {| r, φ i} , like Y mj ( θ, φ ) = h j, m | θ, φ i are the correspondingones to the localized states {| θ, φ i} in the sphere [7, 12]. Indeed | j, m i = 1 π Z R | r, φ iZ mj ( r, φ ) rdrdφ, | j, m i = Z S | θ, φ i p j + 1 / Y mj ( θ, φ ) dΩ. We continue with the analogy and, from K ± and K (10), we define J ± := e ± iφ K ± , J := K , (16)with act on the Z mj ( r, φ ) as J + Z mj ( r, φ ) = p ( j − m )( j + m + 1) Z m +1 j ( r, φ ) ,J − Z mj ( r, φ ) = p ( j + m )( j − m + 1) Z m − j ( r, φ ) ,J Z mj ( r, φ ) = m Z mj ( r, φ ) . (17) U (2), Associated Laguerre Polynomials and Rigged Hilbert Spaces 7 The functions Z mj ( r, φ ) with j fixed and | m | ≤ j , are orthonormal and de-termine the representation of dimension 2 j + 1 of su (2) as it happens forthe Y mj ( θ, φ ). However there is a essential difference between the operators { J ± , J } that act on the sphere S that are true generators of su (2) and the { J ± , J } of (16), defined in R , that do not close a Lie algebra. Indeed, whenwe calculate the commutator [ J + , J − ] in terms of the differential operatorsdefined in the eqs. (10) and (16), we obtain [ J + , J − ] = 2 J + 8 R J E , andonly when E ≡ L ( R ) , the su (2) algebrais recovered. On the other hand, E is related to the su (2) Casimir C E = − R J + 1 [ C − J ( J + 1)] ≡ − R J + 1 (cid:20) J + 12 { J + , J − } − J ( J + 1) (cid:21) , so equation E = 0 is equivalent to the su (2) Casimir condition C − J ( J + 1) =0, that entails the usual Lie algebra in each su (2) representation space. A RHS (or Gelf’and triplet) is a triplet of spaces Φ ⊂ H ⊂ Φ × , where H isan infinite dimensional separable Hilbert space, Φ is a dense subspace of H endowed with its own topology, and Φ × is the dual (or the antidual) spaceof Φ [9, 13, 14]. The topology considered on Φ is finer (contains more opensets) than the topology that Φ has as subspace of H , and Φ × is equippedwith a topology compatible with the dual pair ( Φ, Φ × ) [15], usually the weaktopology. The topology of Φ [16, 17] allows that all sequences which convergeon Φ , also converge on H but the converse is not true. The difference betweentopologies gives rise that Φ × is bigger than H , which is self-dual.Here, any F ∈ Φ × is a continuous linear mapping from Φ into C .An essential property is that if A is a densely defined operator on H , suchthat Φ be a subspace of its domain and that Aϕ ∈ Φ for all ϕ ∈ Φ , we saythat Φ reduces A or that Φ is invariant under the action of A , (i.e., AΦ ⊂ Φ ).Then A may be extended unambiguously to Φ × by the duality formula h A × F | ϕ i := h F | Aϕ i , ∀ ϕ ∈ Φ , ∀ F ∈ Φ × . (18)Moreover if A is continuous on Φ , then A × is continuous on Φ × .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 > || − || p , || − || p , . . . , || − || p r such that for any ϕ ∈ Φ , one has [18] || Aϕ || n ≤ K n ( || ϕ || p + || ϕ || p + · · · + || ϕ || p r ) . (19) Enrico Celeghini, Manuel Gadella and Mariano A del Olmo
Now let us go to define and use the RHS G ⊂ H ⊂ G × where discrete andcontinuous bases coexist and the meaningful operators are well defined andcontinuous. Since we have a representation in terms of the Z mj ( r, φ ), it wouldbe more convenient to start with an equivalent RHS D ⊂ L ( R ) ⊂ D × , such as D is a test functions space with f ( r, φ ) ∈ L ( R ), which thereforeadmit the span f ( r, φ ) = ∞ X j =0 j X m = − j f j,m Z mj ( r, φ ) , (20)where the series converges in the sense of the norm in L ( R ). A necessaryand sufficient condition for it is P ∞ j =0 P jm = − j | f j,m | < ∞ . Thus, from (20),we define D as the space of functions f ( r, φ ) in L ( R ) such that || f ( r, φ ) || n := ∞ X j =0 j X m = − j ( j + | m | +1) n | f j,m | < ∞ , n = 0 , , , . . . . (21)Obviously, all the finite linear combinations of the Z mj ( r, φ ) are in D , hence D is dense in L ( R ). Thus, the family of norms || − || n on D (21) gives atopology such that D is a Fr`echet space (metrizable and complete). Since for n = 0 we have the Hilbert space norm, the canonical injection from D into L ( R ) is continuous.Because j goes from 0 to ∞ , the operators J ± , J are all unbounded and,therefore, their respective domains are densely defined on L ( R ), but noton the whole L ( R ). We can prove that all these operators are defined onthe whole D and are continuous with the topology on D . The proof is simpleand it is essentially the same for all operators. As an example, let us give theproof for J + . For any function f in D , we have J + f , i.e., J + ∞ X j =0 j X m = − j f j,m Z mj ( r, φ ) = ∞ X j =0 j X m = − j f j,m p ( j − m )( j + m + 1) Z m +1 j ( r, φ ) . To show that J + f ∈ D we have to prove that for any n ∈ N , it satisfies (21).So taking into account the shift on the index m (17) we have ∞ X j =0 j X m = − j | f j,m | ( j − m )( j + m + 1) ( j + 1 + | m | + 1) n . (22)The following two inequalities are straightforward:( j − m )( j + m +1) ≤ ( j + | m | +1) , ( j +1+ | m | +1) n ≤ n ( j + | m | +1) n . Using these inequalities we see that (22) is bounded by U (2), Associated Laguerre Polynomials and Rigged Hilbert Spaces 9 n ∞ X j =0 j X m = − j | f j,m | ( j + 1 + | m | + 1) n +2 , (23)which converges after (21). Hence, J + f ∈ D . In order to show the continuityof J + on D , we use (19). Thus, applying J + to any f ( r, φ ) ∈ D we get || J + f ( r, φ ) || n ≤ n || f ( r, φ ) || n +1 = ⇒ || J + f ( r, φ ) || n ≤ n || f ( r, φ ) || n +1 , which satisfies (19) for all n = 0 , , , . . . . Hence, the continuity of J + on D has been proved. By means of the duality formula, we extend J + to a weaklycontinuous operator on D × . Same properties can be proved for J − and J .Now we are able to define the abstract RHS G ⊂ H ⊂ G × using theunitary mapping S : H → L ( R ) introduced in the previous section. Thus,we define G := S − D . Hence the topology on G is the transported topologyfrom D by S , so that if f ∈ G , the semi-norms are || f || n = ∞ X j =0 j X m = − j ( j + | m | + 1) n | f j,m | < ∞ , n = 0 , , , . . . . The topology on G uniquely defines G × . Moreover there exists a one-to-onecontinuous mapping from G onto D with continuous inverse. It is given byan extension, e S , of S defined via the duality formula h e Sf | e SF i = h f | F i , with f ∈ G and F ∈ G × .On the other hand, if an operator O satisfies O D ⊂ D with continuity, thesame property works for b O = S − OS on G . Starting from the recurrence relations (3) we obtained the operators { J ± , J } (16). Their general linear algebra is not a Lie algebra. However its represen-tation on L ( R ), characterized by the eigenvalue zero of the operator E , isisomorphic to the regular representation {| j, m i} of su (2) and it has thereforea stronger symmetry than the general linear operator structure itself.We are used in Lie algebra theory to representations that preserve thesymmetry of the algebra and to algebras that have the same symmetry ofthe space where the representation is defined. This is exactly what happenswith the spherical harmonics, that are solution of Laplace equation and, thus,have the same intrinsic symmetry of the group SU (2) of which they are rep-resentation bases. However, here the situation is different since we represent SU (2) in the plane R which geometry preserves only the subgroup SO (2) of SU (2). Indeed { J ± , J } (16) are defined for arbitrary E , but they generate su (2) only under the assumption E ≡
0, i.e. when we restrict ourselves to functions f verifying the Casimir condition C f = J ( J + 1) f, , i.e. that belongto L ( R ).Reversing the connection, the representations of a Lie algebra have beenrelated not only to the Lie algebra itself but also to a set of operators thatdo not close a Lie algebra in an universal way but reduce to a Lie algebraonly when applied to well defined vector spaces.This paper offers a method to introduce representations of Lie groups inspaces that are not symmetric under the group action and in situations wherethe general linear group of operators is not a Lie group in a universal way.We have also constructed two RHS ( G ⊂ H ⊂ G × and D ⊂ L ( R ) ⊂ D × )supporting two UIR of SU (2), the first one is related with the discrete basis {| j, m i} and the other RHS with the continuous one {| r, φ } . Both are relatedby the unitary map S : | j, m i → Z mj ( r, φ ) that also transports the topologiesof the first RHS and other properties to the second RHS. Acknowledgements
Partial financial support is acknowledged to the Junta deCastilla y Le´on and FEDER (Project VA057U16) and MINECO of Spain (ProjectMTM2014-57129-C2-1-P).
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