Algebraic determination of scattering matrices

Revista Mexicana de Fúica 39, Suplemento 2 (1993) 64-75 AIgebraic determination of scattering matrices A. FRANK Instituto de Ciencias Nucleares and Instituto de Física-Laborotorio de Cuernavaca, UNAM Apartado postal 70-543, 04510 Meneo, D.F., Mexico C.E. ALONSO AND J. GÓMEZ-CAMACHO Departamento de Física Atómica, Molecular y Nuclear Facultad de Física, Universidad de Sevilla Apartado 1065, 41080 Sevilla, Spain ABSTRACT.The a1gebraic approach to scattering allows the determination of S-matrices associated to a potential gronp describing the interaction region, through contraetion and expansion mechanisms connecting the potential and asymptotic Lie a1gebras. We show that this procedure can be generalized to the SO,(2, 1) algebra and extraet the corresponding S-matrix. Possible applications of a three-dimensional generalization of our results are a1sodiscussed. RESUMEN.El método algebraico permite determinar las matrices S asociadas a grupos de potencial que describen la región de interacción, mediante mecanismos de contracción y expansión que conectan las a1gebras de Lie del potencial con un a1gebra asintótica. Mostramos que este procedimiento puede ser generalizado a un a1gebra SO,(2, 1) y determinamos la matriz S correspondiente. Posibles aplicaciones de una generalización tridimensional de nuestros resultados son también discutidas. PACS: 03.80.+r; 11.20.Dj; 11.30.-j 1. INTRODUCTION The use of group contractions has a long, albeit not widely known history, starting from the work of Inonu and Wigner, who used them as a means to determine representations of non-semisimple groups [1]. The opposite operation, that of group expansion [1], is even less familiar and has had few applications. Although most physicists have sorne degree of familiarity with the concepts and techniques of group theory and Lie algebras in connection with the use of symmetries and conserved quantities in physical systems, the contraction and expansion of these mathematical structures is by no means common knowledge. In the last years, however, these concepts were realized to be of central importance for the algebraic description of scattering processes 12,3], and the reason for it may be schematically understood from Fig. 1. The interaction regio n in the algebraic approach is described by means of a group (or, more precisely, by its associated Lie algebra) which is refered to as the "potential group" and denoted by g in the figure. The asymptotic regio n is likewise described by the group g'. These correspond in the algebraic language to the potentials associated to these regions in the usual integro-differential framework. The algebras are related to each other through ALGEBRAICDETERMINATIONOFSCATTERINGMATRICES 65 --+ g' Asymptotic Region g Interaction Region --+ g' Asymptotic Region FIGURE1. Schematicrepresentation of the algebraicapproacb to scattering. a contraction-expansion procedure contraction 9 ~ g' expansion as will be explained below. In contrast to the case of bound systems, where the role of group theory is well understood, symmetry methods were very seldom useful for the description of continuous spectra, other than in the trivial form of angular momentum and energy conservation. The relevant information is in these cases contained in the S-matrix and the question is whether group theory can provide sorne information about it. In a series of papers [2-4] in the last years it was shown that the S-matrix of a scattering system may be evaluated by establishing a connection between the generators of the potential group, describing the interaction potential, and appropriate asymptotic generators, describing the long range behavior of the system. This connection formula was refered to as the "Euclidean connection" in Re£. [31. It was shown in Re£. [4]that the connection formula is equivalent to the expansion [11 of the potential group generators in terms of asymptotic ones. This interpretation permits a fully algebraic determination of S-matrices for abstract potential groups of the general form SO(n, m). The algebraic approach was later applied to heavy ion collisions [5]' nuclear reactions [6,7], and to relativistic systems [8]. Prom a different perspective, quantum algebras have become a subject of great current interest [9-16]. Although they have up to now no direct physical interpretation, they have been shown to be a powerful tool to solve Yang-Baxter equations. The solutions are of interest both to integrable lattice models in statistical mechanics and to link and knot theories. Many properties of Lie algebras and groups and their representations have been extended to their quantum analogs. While Jimbo [10]has supplied the relations that define this extension for any classical Lie algebra, Celeghini et al. [11]have devised a contraction procedure to establish the representations of non-semisimple quantum groups, in the same spirit as in the work of Inonu and Wigner for Lie algebras. Many other q-generalizations have been carried out [13-16] and there is much interest in finding physical applications for these mathematical structures. The purpose of the present paper is two-fold. On the one hand, since the algebraic approach to scattering can be viewed as an abstract procedure, the question arises as to whether it can be extended to the case where the potential region is described by a q-algebra. If the contraction-expansion procedure can indeed be generalized to cover these cases, we will then be able to extract the corresponding S-matrices and obtain, as a bonus, a new realization for the q-algebra generators in terms of asymptotic ones. We shall analyze the case of the contraction-expansion procedure for the SO.(2, 1) •..•E(2) algebras 66 A. FRANKET AL. and carry out the above-mentioned steps. On the other hand, we study a three-dimensional version of this procedure, which corresponds to a q-deformation of Coulomb scattering and interpret the results in terms of a screened Coulomb potential, which may be useful for the study of electron.atom scattering. 2. SCATTERING FROM AN SOq(2,1) POTENTIAL We first define the SOq(2, 1) representations, following a recent paper by Maekawa [17]. The quantum algebra SUq(1,I) (isomorphic to SOq(2,1)) is defined by the operators i+,L and io, satisfying qio _ q-io ql/2 _ q-l/2' (1) Introducing w == In q, we find a different form for the second commutator sinh(wio) sinh(w/2)' (2) These relations reduce to the SU(I, 1) ones for q --+ l(w --+ O), and, except for the minus sign on the r.h.s. of (2), they coincide with the SUq(2) commutation relations. The Casimir invariant of SUq(l, 1) takes the form [17] • 2 - C- ( ) - h( /2) smh (wJ o /2) 1 (JJJJ- ) 2W-COSW 2 --2+-+-+ sinh (w/2) = cosh(w/2)[io]2 - ~(i+L + Li+), where we have introduced the notation [XI = sinh(wX/2), sinh(w/2) (3) (4) which reduces to X for w --+ O. The SUq(l, 1) unitary irreps are discussed in Re£. [17]. We shall be interested only in the continuous (principal) series, de/ined by C2(w)lj, m) = [jl[j +Il1j,m), iolj,m) = mli,m), irlj, m) = ([m:¡: jJ[m:l: j :1: lj)1/2Ij, m:l: 1), (5) where j = -1/2:1: ia, with real a, and m takes al! integral or half-integral values [17]. The quantities in square brackets are defined in (4). ALGEBRAIC DETERMINATION OF SCATTERING MATRICES 61 We now carry out the contraction of (1) and (2) to the Euclidean group E(2), by first considering the change of scale transformation and then taking the limit when f -> O. We find [j8,p~J = ::f:P~; (6) (7) which are the E(2) commutation relations [IJ. From now on we omit the superindex "O" in these operators. Can we now expand E(2) back to 8U q (l, 1)? In the usual expansion to 8U(I, 1), we use the formula [3,5) - 12QJr. = 2ik[Jo ,Poi J + ¡¿Poi, (8) where Q is a constant which depends on the representation label j and k = J P+ .P_. Using (7) we can verify that the operators in (8) satisfy the 8U(I, 1) commutation relations. The expansion formula (8) can be understood in the following way [3,4]: the Casimir invariant JJ of the compact subalgebra 80(2) (which is not modified by the contraction process) is not present in the contracted E(2) invariant P2. We should then use it to reconstruct the original scattering algebra. Because jJ is an 80(2) scalar and Poi transform as an 80(2) vector, a new vector in 80(2) which is non linear in the E(2) generators can be constructed in the form (8), which therefore automatically satisfies the correct commutation relations with jo. The particular combination In (8) further guarantees that [j+, LJ = -2jo. We now attempt to repeat this argument for 8U q (l, 1). The form of the Casimir invariant (3) suggests that instead of jJ, we may try the 80(2) invariant • 2 - <;}w = cosh(w/2)smh ( wJ o/2) sinh2(w/2) = cosh(w/2)[jof To test this idea we need the basic commutators [Poi, sinh(wjo)] = Poi (sinh(wjo) - sinh(wjo::f: w)), [Poi, cosh(wjo)J = Poi (COSh(wjo)- cosh(wjo ::f: w)), (9) (10) (11) which can be derived by using (7). We find that it is not precisely <;}w of (9) which leads to the 8U q (l, 1) commutators (1), (2), but the slightly different form j = 4cosh(w/4) [sinh2(wjo/4) p] '2p oi 2ik sinh2(w/2) , oi + k oi, 68 A. FRANK ET AL. In the notation (4), formula (11) can be written in the alternative form j = 4 cosh(w/4) [[j /2]2 p] '!.P " 2ik o, " +k '" (12) which clearly reduces to (8) for w --+ o. The j" generators may be calculated from (11) and (10) to give the new SU q (l, 1) realization j = _l_p(COSh(wjO/2):l: sinh(wjo/2) 2.) " 2ik" cosh(w/4) sinh(w/4) + w , jo = jo (13) (14) in terms of the E(2) generators. To define the constant 0<, we compute the Casimir operator (3) using (13). After sorne algebra, we find the simple relation C2(W) = _0<2 _ 1 . 4cosh2(w/4) Returning to equation (5), C2(w)lj, m) = lilli + l]lj, m), we find, for the principal series j = -1/2:l: ia, that lilli +1] = sinh2(iaw/2) sinh2(w/2) ia 2 _ 1 [] 4cosh2(w/4). (15) By comparing with (14), we identify O< as O< = ili +1/2] = :l:i[iaj. (16) Equation (16)fixes O< in terms of the SUq(l,l) representation label j and gives its particular value for the principal series. We further define the appropriate sign in (16) below. Inserting (16) into (13) and rearranging terms, we arrive at the final expression for the SUq(l, 1) generators in the continuous series representation je; = ~; ([jo +1/2] :l: [ial) , "p(" ) J~ = ik [Jo - 1/2] :l: tia) , (17) ALGEBRAIC DETERMINATION OF SCATTERING MATRICES 69 where we attach the superindex "00" to indicate that they arise from their expansion from the asymptotic E(2) generators. Again, these formulas reduce to the usual ones for w -> O[2-4]. Both signs in (17) are permissible in principie. We now calculate the S-matrix for scattering from an SU q (I,I) potential group by following the usual procedure [2-4] and the explicit form of j'f in (17). In the algebraic approach to scattering, the representation index u becomes a real but otherwise arbitrary function of the momentum k, i.e., u(k). The asymptotic (contracted) form for the SUq(l, 1) wave functions is then given by Ij, m)~ = A~Ik, m) + B~lk, m), (18) where [-k,m) and ¡k,m) are identified with E(2) (i.e. free) incoming and outgoing waves, respectively. We now impose the equality (19) which implies that the SU(I, l)q relations are valid asymptotically [2-4] and use (5), (18) and the E(2) defining equations ?+I:I: k, m) = :l:kl:l: k, m+1), jol :1: k, m) = mi :1: k, m), to find recurrence relations for the S-matrix S:;'. '= B'::./ A';,.: 1m +1/2] + [iu(k)] ~+l = [m +1/2] - [ia(k)J~' (20) (21) where the sign in (17) is fixed by whether the j+ generator acts on the +k or -k E(2) representation [2-4]. For half-integer m-values, we find m-l/2 sw = TI ([n] + [iU(k)]) i",_(k) m [n]- [ia(k)J e , n=l (22) where 'Pw(k) is an arbitrary function. Equation (22) describes SU q (l, 1)- like S-matrices and their m-dependence in terms of u(k) and the para meter w = In q. The form of a(k) is determined by the specific function of the Casimir invariant (14) which is taken as the Hamiltonian of the scattering system [3,5]. For example, for SU(I, 1) scattering (w = O) off a Poschl-Teller potential, the scattering Hamiltonian turns out to be given by [2] fH!(x) = (-6 2(0) - 1/4) w(x) = ew(x), (23) for which (14) and (16) imply u(k) = :l:k. The S-matrix (22) reduces to the ratio of two gamma functions for w -> O.Note that a minar modification of Eq. (13) leads to a realization for the quantum group SU q (2). 70 A. FRANK ET AL. We have thus shown that the algebraic approach to scattering can be extended to q-algebras by means of an appropriate modification of the usual expansions formulas. In the next section we study a three-dimensional generalization of the procedure considered in this section and analyze its possible physical interpretation. 3. Q-COULOMB SCATTERING AND SCREENED POTENTIALS While the discussion of the previous section demonstrates that the contraction-expansion procedures can be successfuly applied to more complex potential-group structures, it does not shed much light into the nature of the new physical features which are in this way incorporated to the scattering processes. The usual approach, which deals with SO(n, m) Lie algebras and their contraction, leads to S-matrices which are ratios of r functions [2-7]. In particular, for heavy ion scattering and reactions, both SO(3,1) and SO(3,2) [5-71have been proposed as potential groups which incorporate both the Coulomb and short-range interactions, although the 1-dependen ce is uSllally modified through an additional parametrization of the group labels [5-71. In this section we shall show that a simple generalization of formula (21) for three-dimensional systems leads to an interesting modification of Coulomb scattering. Before considering this generalization, however, we briefly indicate the results of the algebraic approach to pure Coulomb scattering [5). The relevant contraction-expansion procedure is applied in this case to the algebras 50(3,1) contraction ---+ ~ expansion E(3), as SO(3, 1) is well known to constitute a symmetry gronp for (positive energy) hydrogenic systems [18]. The analysis is fully analogous to the 5U(1,1) "" 50(2,1) - E(2) one discussed in the last section. For Coulomb potentials, the Hamiltonian may be written as -2 f3 f¡ = ~+-, 2/1 r (24) (25) where C 2 is the SO(3, 1) second order Casimir invariant [5,181. The corresponding recurrence relation for the 5-matrix turns out to be given by [5] 1+1+if(k) S'+l(k) = 1+1 _ if(k) S,(k), where f(k) is fixed by (25) and given by f(k) = /1f3/k. (26) (27) ALGEBRAIC DETERMINATION OF SCATTERING MATRICES 71 Note the similarity with (21) (for the case w --+ O).The 1plays the role ofm and f(k) that of u(k), while the 1/2 is substituted by 1 due to the higher dimensionality of 80(3.1). Relation (26) leads to the well known result for Coulomb scattering S (k) = f(1 + 1 + i/l(3/k) i",(k) 1 f(l + 1_ i/l(3/k) e , (28) where <p(k) is fixed by the s-waveamplitude (and cannot be determined by the algebraic procedure). We may now consider the q-deformation of the 80(3,1) algebra [18]which is defined by the commutators [ti, t)] = ifi)ktk [ti, K)] = ifi}kKk •• . sinh(wLk) . 1 [Ki,K)] = -Ui)k2sinh(w/2) - -ui)k2'12Lk], (29.a) (29.b) (29.c) where the square bracket on the r.h.s. of (29.c) wasdefinedin (4). Note that these relations constitute a natural generalization of the commutators (1) corresponding to 80.(2,1). In faet, this kind of algebraic structures can be simply defined for 80(n, 1) with arbitrary n. The 80(n, 1) generators can be divided into compact and non-compaet ones. The compaet generators are those corresponding to the 80(n) subalgebra (n(n - 1)/2 of them) while the non-compaet ones are the n remaining operators. Thus, in 80(2,1) there is a single compact generator (jo) and two non-compact ones (ix), while in 80(3,1) there are three compaet (t¡) and three non-compact (K¡) generators. The q-deformation we are defining corresponds to leaving invariant all commutators except the ones among the non-compaet generators, which are deformed as in (29.c). We now return to the 8-matrix associated to the contraction-expansion of the algebras contraction 80.(3, 1) ;:::: E(3), expansion where we denote by 80.(3,1) the mathematical structure defined by Eqs. (29). By following a procedure closely analogous to the one carried out in the last section, we find the recurrence relations for the 80.(3,1) 8-matrix w 1I + 11 + [if(k)) S'+l(k) = 1I + 1] _ [if(k)1 S'¡', (30) where f(k) should be determined by the relation between the Hamiltonian and the 80.(3,1) Casimir invaríant. We define the q-Coulomb Hamiltonian by generalizing (25) to (31) 72 A. FRANK ET AL. where C 2(W) is the Casimir invariant of the algebra (29) and w is related to q through w = In q, as before. This generalization follows from an analysis of the 80 q (3, 1) representations. In the continuous series, n = -1 :l:: if(k), the eigenvalue equation for C 2 (w) gives C2(w)lnlm)", = [n)[n +2J1nlm)", = ([if(k W- / ) Inlm)",. (32) cosh (w/4) Comparing with (31) and using H",lnlm)", = ~:Inlm)"" leads to the identification of f(k): [if(k)] =/':, which reduces to (27) for w --+ O, as it should. 8ubstitution into (30) gives S'" (k) = [1 + 11 + il-'f3/k S"'(k) ¡+l [1 + 11 - il-'f3/k ¡ (33) (34) for the 8-matrix recurrence relation associated to q-Coulomb scattering. This form differs markedly in its I-dependence from relation (26). Writing SI = exp(2io¡(w)), we find (71sinh(W/2) ) <5¡(w) = 0l-1(w) +arctan sinh(lw/2) , (35) where 71 = 1-'f3/k is the 80mmerfeld parameter. The classical deflection function, that is, the angle of deviation as a function of I [19]' is then given by Bo¡ (71sinh(W/2)) 8",(1) = 28/ '" 2arctan sinh(lw/2) , to be compared with the Coulomb result (w --+ O) 8 e (l) = 2 arctan(71/I). (36) (37) Thus 8",(1) tends to zero exponentially with increasing 1, which is typical of interaction potentials which fall exponentially. The determination of this potential involves either solving the inverse scattering problem or finding appropriate coordinate realizations for the algebraic relations (29) [20]. We shall follow a simpler route by carrying out a semiclassical analysis in search of an l-independent (but otherwise energy dependent) potential reproducing (34). We use the expression [20] l OO ñl dr 8(1)=,,-2 ro r 2 P(r); p2(r) ñ 212 --=E-V---, 21-' 2p.r2 (38)