Algebraic determination of scattering matrices

Re is a Mexicana de Fúica 39, Suplemen o 2 (1993) 64-75 AIgeb aic de e mina ion o sca e ing ma ices A. FRANK Ins i u o de Ciencias Nuclea es and Ins i u o de Física-Labo o o io de Cue na aca, UNAM Apa ado pos al 70-543, 04510 Meneo, D.F., Mexico C.E. ALONSO AND J. GÓMEZ-CAMACHO Depa amen o de Física A ómica, Molecula y Nuclea Facul ad de Física, Uni e sidad de Se illa Apa ado 1065, 41080 Se illa, Spain ABSTRACT.The a1geb aic app oach o sca e ing allows he de e mina ion o S-ma ices associ- a ed o a po en ial g onp desc ibing he in e ac ion egion, h ough con ae ion and expansion mechanisms connec ing he po en ial and asymp o ic Lie a1geb as. We show ha his p ocedu e can be gene alized o he SO,(2, 1) algeb a and ex ae he co esponding S-ma ix. Possible ap- plica ions o a h ee-dimensional gene aliza ion o ou esul s a e a1sodiscussed. RESUMEN.El mé odo algeb aico pe mi e de e mina las ma ices S asociadas a g upos de po encial que desc iben la egión de in e acción, median e mecanismos de con acción y expansión que conec an las a1geb as de Lie del po encial con un a1geb a asin ó ica. Mos amos que es e p ocedimien o puede se gene alizado a un a1geb a SO,(2, 1) y de e minamos la ma iz S co es- pondien e. Posibles aplicaciones de una gene alización idimensional de nues os esul ados son ambién discu idas. PACS: 03.80.+ ; 11.20.Dj; 11.30.-j 1. INTRODUCTION The use o g oup con ac ions has a long, albei no widely known his o y, s a ing om he wo k o Inonu and Wigne , who used hem as a means o de e mine ep esen a ions o non-semisimple g oups [1]. The opposi e ope a ion, ha o g oup expansion [1], is e en less amilia and has had ew applica ions. Al hough mos physicis s ha e so ne deg ee o amilia i y wi h he concep s and echniques o g oup heo y and Lie algeb as in connec ion wi h he use o symme ies and conse ed quan i ies in physical sys ems, he con ac ion and expansion o hese ma hema ical s uc u es is by no means common knowledge. In he las yea s, howe e , hese concep s we e ealized o be o cen al im- po ance o he algeb aic desc ip ion o sca e ing p ocesses 12,3], and he eason o i may be schema ically unde s ood om Fig. 1. The in e ac ion egio n in he algeb aic app oach is desc ibed by means o a g oup (o , mo e p ecisely, by i s associa ed Lie algeb a) which is e e ed o as he "po en ial g oup" and deno ed by g in he igu e. The asymp o ic egio n is likewise desc ibed by he g oup g'. These co espond in he algeb aic language o he po en ials associa ed o hese egions in he usual in eg o-di e en ial amewo k. The algeb as a e ela ed o each o he h ough ALGEBRAICDETERMINATIONOFSCATTERINGMATRICES 65 --+ g' Asymp o ic Region g In e ac ion Region --+ g' Asymp o ic Region FIGURE1. Schema ic ep esen a ion o he algeb aicapp oacb o sca e ing. a con ac ion-expansion p ocedu e con ac ion 9 ~ g' expansion as will be explained below. In con as o he case o bound sys ems, whe e he ole o g oup heo y is well unde s ood, symme y me hods we e e y seldom use ul o he desc ip ion o con inuous spec a, o he han in he i ial o m o angula momen um and ene gy conse a ion. The ele an in o ma ion is in hese cases con ained in he S-ma ix and he ques ion is whe he g oup heo y can p o ide so ne in o ma ion abou i . In a se ies o pape s [2-4] in he las yea s i was shown ha he S-ma ix o a sca e ing sys em may be e alua ed by es ablishing a connec ion be ween he gene a o s o he po en ial g oup, desc ibing he in e ac ion po en ial, and app op ia e asymp o ic gene a o s, desc ibing he long ange beha io o he sys em. This connec ion o mula was e e ed o as he "Euclidean connec ion" in Re£. [31. I was shown in Re£. [4] ha he connec ion o mula is equi alen o he expansion [11 o he po en ial g oup gene a o s in e ms o asymp o ic ones. This in e p e a ion pe mi s a ully algeb aic de e mina ion o S-ma ices o abs ac po en ial g oups o he gene al o m SO(n, m). The algeb aic app oach was la e applied o hea y ion collisions [5]' nuclea eac ions [6,7], and o ela i is ic sys ems [8]. P om a di e en pe spec i e, quan um algeb as ha e become a subjec o g ea cu en in e es [9-16]. Al hough hey ha e up o now no di ec physical in e p e a ion, hey ha e been shown o be a powe ul ool o sol e Yang-Bax e equa ions. The solu ions a e o in e es bo h o in eg able la ice models in s a is ical mechanics and o link and kno heo ies. Many p ope ies o Lie algeb as and g oups and hei ep esen a ions ha e been ex ended o hei quan um analogs. While Jimbo [10]has supplied he ela ions ha de ine his ex ension o any classical Lie algeb a, Celeghini e al. [11]ha e de ised a con ac ion p ocedu e o es ablish he ep esen a ions o non-semisimple quan um g oups, in he same spi i as in he wo k o Inonu and Wigne o Lie algeb as. Many o he q-gene aliza ions ha e been ca ied ou [13-16] and he e is much in e es in inding physical applica ions o hese ma hema ical s uc u es. The pu pose o he p esen pape is wo- old. On he one hand, since he algeb aic app oach o sca e ing can be iewed as an abs ac p ocedu e, he ques ion a ises as o whe he i can be ex ended o he case whe e he po en ial egion is desc ibed by a q-algeb a. I he con ac ion-expansion p ocedu e can indeed be gene alized o co e hese cases, we will hen be able o ex ac he co esponding S-ma ices and ob ain, as a bonus, a new ealiza ion o he q-algeb a gene a o s in e ms o asymp o ic ones. We shall analyze he case o he con ac ion-expansion p ocedu e o he SO.(2, 1) •..•E(2) algeb as 66 A. FRANKET AL. and ca y ou he abo e-men ioned s eps. On he o he hand, we s udy a h ee-dimensional e sion o his p ocedu e, which co esponds o a q-de o ma ion o Coulomb sca e ing and in e p e he esul s in e ms o a sc eened Coulomb po en ial, which may be use ul o he s udy o elec on.a om sca e ing. 2. SCATTERING FROM AN SOq(2,1) POTENTIAL We i s de ine he SOq(2, 1) ep esen a ions, ollowing a ecen pape by Maekawa [17]. The quan um algeb a SUq(1,I) (isomo phic o SOq(2,1)) is de ined by he ope a o s i+,L and io, sa is ying qio _ q-io ql/2 _ q-l/2' (1) In oducing w == In q, we ind a di e en o m o he second commu a o sinh(wio) sinh(w/2)' (2) These ela ions educe o he SU(I, 1) ones o q --+ l(w --+ O), and, excep o he minus sign on he .h.s. o (2), hey coincide wi h he SUq(2) commu a ion ela ions. The Casimi in a ian o SUq(l, 1) akes he o m [17] • 2 - C- ( ) - h( /2) smh (wJ o /2) 1 (J- J- J- J- ) 2W-COSW 2 --2+-+-+ sinh (w/2) = cosh(w/2)[io]2 - ~(i+L + Li+), whe e we ha e in oduced he no a ion [XI = sinh(wX/2), sinh(w/2) (3) (4) which educes o X o w --+ O. The SUq(l, 1) uni a y i eps a e discussed in Re£. [17]. We shall be in e es ed only in he con inuous (p incipal) se ies, de/ined by C2(w)lj, m) = [jl[j +Il1j,m), iolj,m) = mli,m), i lj, m) = ([m:¡: jJ[m:l: j :1: lj)1/2Ij, m:l: 1), (5) whe e j = -1/2:1: ia, wi h eal a, and m akes al! in eg al o hal -in eg al alues [17]. The quan i ies in squa e b acke s a e de ined in (4). ALGEBRAIC DETERMINATION OF SCATTERING MATRICES 61 We now ca y ou he con ac ion o (1) and (2) o he Euclidean g oup E(2), by i s conside ing he change o scale ans o ma ion and hen aking he limi when -> O. We ind [j8,p~J = :: :P~; (6) (7) which a e he E(2) commu a ion ela ions [IJ. F om now on we omi he supe index "O" in hese ope a o s. Can we now expand E(2) back o 8U q (l, 1)? In he usual expansion o 8U(I, 1), we use he o mula [3,5) - 1- 2- Q- J . = 2ik[Jo ,Poi J + ¡¿Poi, (8) whe e Q is a cons an which depends on he ep esen a ion label j and k = J P+ .P_. Using (7) we can e i y ha he ope a o s in (8) sa is y he 8U(I, 1) commu a ion e- la ions. The expansion o mula (8) can be unde s ood in he ollowing way [3,4]: he Casimi in a ian JJ o he compac subalgeb a 80(2) (which is no modi ied by he con ac ion p ocess) is no p esen in he con ac ed E(2) in a ian P2. We should hen use i o econs uc he o iginal sca e ing algeb a. Because jJ is an 80(2) scala and Poi ans o m as an 80(2) ec o , a new ec o in 80(2) which is non linea in he E(2) gene a o s can be cons uc ed in he o m (8), which he e o e au oma ically sa is ies he co ec commu a ion ela ions wi h jo. The pa icula combina ion In (8) u he gua an ees ha [j+, LJ = -2jo. We now a emp o epea his a gumen o 8U q (l, 1). The o m o he Casimi in a ian (3) sugges s ha ins ead o jJ, we may y he 80(2) in a ian • 2 - <;}w = cosh(w/2)smh ( wJ o/2) sinh2(w/2) = cosh(w/2)[jo To es his idea we need he basic commu a o s [Poi, sinh(wjo)] = Poi (sinh(wjo) - sinh(wjo:: : w)), [Poi, cosh(wjo)J = Poi (COSh(wjo)- cosh(wjo :: : w)), (9) (10) (11) which can be de i ed by using (7). We ind ha i is no p ecisely <;}w o (9) which leads o he 8U q (l, 1) commu a o s (1), (2), bu he sligh ly di e en o m j = 4cosh(w/4) [sinh2(wjo/4) p] '2p oi 2ik sinh2(w/2) , oi + k oi, 68 A. FRANK ET AL. In he no a ion (4), o mula (11) can be w i en in he al e na i e o m j = 4 cosh(w/4) [[j /2]2 p] '!.P " 2ik o, " +k '" (12) which clea ly educes o (8) o w --+ o. The j" gene a o s may be calcula ed om (11) and (10) o gi e he new SU q (l, 1) ealiza ion j = _l_p(COSh(wjO/2):l: sinh(wjo/2) 2.) " 2ik" cosh(w/4) sinh(w/4) + w , jo = jo (13) (14) in e ms o he E(2) gene a o s. To de ine he cons an 0<, we compu e he Casimi ope a o (3) using (13). A e so ne algeb a, we ind he simple ela ion C2(W) = _0<2 _ 1 . 4cosh2(w/4) Re u ning o equa ion (5), C2(w)lj, m) = lilli + l]lj, m), we ind, o he p incipal se ies j = -1/2:l: ia, ha lilli +1] = sinh2(iaw/2) sinh2(w/2) ia 2 _ 1 [] 4cosh2(w/4). (15) By compa ing wi h (14), we iden i y O< as O< = ili +1/2] = :l:i[iaj. (16) Equa ion (16) ixes O< in e ms o he SUq(l,l) ep esen a ion label j and gi es i s pa icula alue o he p incipal se ies. We u he de ine he app op ia e sign in (16) below. Inse ing (16) in o (13) and ea anging e ms, we a i e a he inal exp ession o he SUq(l, 1) gene a o s in he con inuous se ies ep esen a ion je; = ~; ([jo +1/2] :l: [ial) , "p(" ) J~ = ik [Jo - 1/2] :l: ia) , (17) ALGEBRAIC DETERMINATION OF SCATTERING MATRICES 69 whe e we a ach he supe index "00" o indica e ha hey a ise om hei expansion om he asymp o ic E(2) gene a o s. Again, hese o mulas educe o he usual ones o w -> O[2-4]. Bo h signs in (17) a e pe missible in p incipie. We now calcula e he S-ma ix o sca e ing om an SU q (I,I) po en ial g oup by ollowing he usual p ocedu e [2-4] and he explici o m o j' in (17). In he algeb aic app oach o sca e ing, he ep esen a ion index u becomes a eal bu o he wise a bi a y unc ion o he momen um k, i.e., u(k). The asymp o ic (con ac ed) o m o he SUq(l, 1) wa e unc ions is hen gi en by Ij, m)~ = A~I- k, m) + B~lk, m), (18) whe e [-k,m) and ¡k,m) a e iden i ied wi h E(2) (i.e. ee) incoming and ou going wa es, espec i ely. We now impose he equali y (19) which implies ha he SU(I, l)q ela ions a e alid asymp o ically [2-4] and use (5), (18) and he E(2) de ining equa ions ?+I:I: k, m) = :l:kl:l: k, m+1), jol :1: k, m) = mi :1: k, m), o ind ecu ence ela ions o he S-ma ix S:;'. '= B'::./ A';,.: 1m +1/2] + [iu(k)] ~+l = [m +1/2] - [ia(k)J~' (20) (21) whe e he sign in (17) is ixed by whe he he j+ gene a o ac s on he +k o -k E(2) ep esen a ion [2-4]. Fo hal -in ege m- alues, we ind m-l/2 sw = TI ([n] + [iU(k)]) i",_(k) m [n]- [ia(k)J e , n=l (22) whe e 'Pw(k) is an a bi a y unc ion. Equa ion (22) desc ibes SU q (l, 1)- like S-ma ices and hei m-dependence in e ms o u(k) and he pa a me e w = In q. The o m o a(k) is de e mined by he speci ic unc ion o he Casimi in a ian (14) which is aken as he Hamil onian o he sca e ing sys em [3,5]. Fo example, o SU(I, 1) sca e ing (w = O) o a Poschl-Telle po en ial, he sca e ing Hamil onian u ns ou o be gi en by [2] H!(x) = (-6 2(0) - 1/4) w(x) = ew(x), (23) o which (14) and (16) imply u(k) = :l:k. The S-ma ix (22) educes o he a io o wo gamma unc ions o w -> O.No e ha a mina modi ica ion o Eq. (13) leads o a ealiza ion o he quan um g oup SU q (2). 70 A. FRANK ET AL. We ha e hus shown ha he algeb aic app oach o sca e ing can be ex ended o q-algeb as by means o an app op ia e modi ica ion o he usual expansions o mulas. In he nex sec ion we s udy a h ee-dimensional gene aliza ion o he p ocedu e conside ed in his sec ion and analyze i s possible physical in e p e a ion. 3. Q-COULOMB SCATTERING AND SCREENED POTENTIALS While he discussion o he p e ious sec ion demons a es ha he con ac ion-expansion p ocedu es can be success uly applied o mo e complex po en ial-g oup s uc u es, i does no shed much ligh in o he na u e o he new physical ea u es which a e in his way inco po a ed o he sca e ing p ocesses. The usual app oach, which deals wi h SO(n, m) Lie algeb as and hei con ac ion, leads o S-ma ices which a e a ios o unc ions [2-7]. In pa icula , o hea y ion sca e ing and eac ions, bo h SO(3,1) and SO(3,2) [5-71ha e been p oposed as po en ial g oups which inco po a e bo h he Coulomb and sho - ange in e ac ions, al hough he 1-dependen ce is uSllally modi ied h ough an addi ional pa ame iza ion o he g oup labels [5-71. In his sec ion we shall show ha a simple gene aliza ion o o mula (21) o h ee-dimensional sys ems leads o an in e es ing modi ica ion o Coulomb sca e ing. Be o e conside ing his gene aliza ion, howe e , we b ie ly indica e he esul s o he algeb aic app oach o pu e Coulomb sca e ing [5). The ele an con ac ion-expansion p ocedu e is applied in his case o he algeb as 50(3,1) con ac ion ---+ ~ expansion E(3), as SO(3, 1) is well known o cons i u e a symme y g onp o (posi i e ene gy) hyd ogenic sys ems [18]. The analysis is ully analogous o he 5U(1,1) "" 50(2,1) - E(2) one discussed in he las sec ion. Fo Coulomb po en ials, he Hamil onian may be w i en as -2 3 ¡ = ~+-, 2/1 (24) (25) whe e C 2 is he SO(3, 1) second o de Casimi in a ian [5,181. The co esponding ecu - ence ela ion o he 5-ma ix u ns ou o be gi en by [5] 1+1+i (k) S'+l(k) = 1+1 _ i (k) S,(k), whe e (k) is ixed by (25) and gi en by (k) = /1 3/k. (26) (27) ALGEBRAIC DETERMINATION OF SCATTERING MATRICES 71 No e he simila i y wi h (21) ( o he case w --+ O).The 1plays he ole o m and (k) ha o u(k), while he 1/2 is subs i u ed by 1 due o he highe dimensionali y o 80(3.1). Rela ion (26) leads o he well known esul o Coulomb sca e ing S (k) = (1 + 1 + i/l(3/k) i",(k) 1 (l + 1_ i/l(3/k) e , (28) whe e <p(k) is ixed by he s-wa eampli ude (and canno be de e mined by he algeb aic p ocedu e). We may now conside he q-de o ma ion o he 80(3,1) algeb a [18]which is de ined by he commu a o s [ i, )] = i i)k k [ i, K)] = i i}kKk •• . sinh(wLk) . 1 [Ki,K)] = -Ui)k2sinh(w/2) - -ui)k2'12Lk], (29.a) (29.b) (29.c) whe e he squa e b acke on he .h.s. o (29.c) wasde inedin (4). No e ha hese ela ions cons i u e a na u al gene aliza ion o he commu a o s (1) co esponding o 80.(2,1). In ae , his kind o algeb aic s uc u es can be simply de ined o 80(n, 1) wi h a bi a y n. The 80(n, 1) gene a o s can be di ided in o compac and non-compae ones. The compae gene a o s a e hose co esponding o he 80(n) subalgeb a (n(n - 1)/2 o hem) while he non-compae ones a e he n emaining ope a o s. Thus, in 80(2,1) he e is a single compac gene a o (jo) and wo non-compac ones (ix), while in 80(3,1) he e a e h ee compae ( ¡) and h ee non-compac (K¡) gene a o s. The q-de o ma ion we a e de ining co esponds o lea ing in a ian all commu a o s excep he ones among he non-compae gene a o s, which a e de o med as in (29.c). We now e u n o he 8-ma ix associa ed o he con ac ion-expansion o he algeb as con ac ion 80.(3, 1) ;:::: E(3), expansion whe e we deno e by 80.(3,1) he ma hema ical s uc u e de ined by Eqs. (29). By ol- lowing a p ocedu e closely analogous o he one ca ied ou in he las sec ion, we ind he ecu ence ela ions o he 80.(3,1) 8-ma ix w 1I + 11 + [i (k)) S'+l(k) = 1I + 1] _ [i (k)1 S'¡', (30) whe e (k) should be de e mined by he ela ion be ween he Hamil onian and he 80.(3,1) Casimi in a ían . We de ine he q-Coulomb Hamil onian by gene alizing (25) o (31) 72 A. FRANK ET AL. whe e C 2(W) is he Casimi in a ian o he algeb a (29) and w is ela ed o q h ough w = In q, as be o e. This gene aliza ion ollows om an analysis o he 80 q (3, 1) ep e- sen a ions. In he con inuous se ies, n = -1 :l:: i (k), he eigen alue equa ion o C 2 (w) gi es C2(w)lnlm)", = [n)[n +2J1nlm)", = ([i (k W- / ) Inlm)",. (32) cosh (w/4) Compa ing wi h (31) and using H",lnlm)", = ~:Inlm)"" leads o he iden i ica ion o (k): [i (k)] =/':, which educes o (27) o w --+ O, as i should. 8ubs i u ion in o (30) gi es S'" (k) = [1 + 11 + il-' 3/k S"'(k) ¡+l [1 + 11 - il-' 3/k ¡ (33) (34) o he 8-ma ix ecu ence ela ion associa ed o q-Coulomb sca e ing. This o m di e s ma kedly in i s I-dependence om ela ion (26). W i ing SI = exp(2io¡(w)), we ind (71sinh(W/2) ) <5¡(w) = 0l-1(w) +a c an sinh(lw/2) , (35) whe e 71 = 1-' 3/k is he 80mme eld pa ame e . The classical de lec ion unc ion, ha is, he angle o de ia ion as a unc ion o I [19]' is hen gi en by Bo¡ (71sinh(W/2)) 8",(1) = 28/ '" 2a c an sinh(lw/2) , o be compa ed wi h he Coulomb esul (w --+ O) 8 e (l) = 2 a c an(71/I). (36) (37) Thus 8",(1) ends o ze o exponen ially wi h inc easing 1, which is ypical o in e ac ion po en ials which all exponen ially. The de e mina ion o his po en ial in ol es ei he sol ing he in e se sca e ing p oblem o inding app op ia e coo dina e ealiza ions o he algeb aic ela ions (29) [20]. We shall ollow a simple ou e by ca ying ou a semi- classical analysis in sea ch o an l-independen (bu o he wise ene gy dependen ) po en ial ep oducing (34). We use he exp ession [20] l OO ñl d 8(1)=,,-2 o 2 P( ); p2( ) ñ 212 --=E-V---, 21-' 2p. 2 (38)