LYCEN 7112 Février 1972 notice 92ft On the Generalized Exchange Operators for SU(n) A. Partensky Institut de Physique Nucléaire, Université Claude Bernard de Lyon Institut National de Physique Nucléaire et de Physique des Particules 43, Bd du 11 novembre 1918, 69-Villeurbanne, France Abstract - By following the work of Biedenharn we have redefined the k-particles generalized exchange operators (g.e.o) and studied their properties. By a straightforward but cur-jersome calculation we have derived the expression, in terms of the SU(n) Casimir operators, of the ?., 3 or 4 particles g.e.o. acting on the A-particles states which span an irreducible representation of the grown SU(n). A striking and interesting result is that the eigenvalues of each of these g.e.o. do not depend on the n in SU(n) but only on the Young pattern associated with the irreducible representation considered. For a given g.e.o. , the eigenvalues corresponding to two conjugate Young patterns are the same except for the sign which depends on the parity of the g.e.o. considered. Three Appendices deal with some related problems and, more specifically, Appendix C 2 contains a method of obtaining the eigenvalues of Gel'fand invariants in a new and simple way. 1 . Introduction The interest attributed by physicists to the Lie groups has 3-4 been continuously growing since the pionner works of Weyl , Wigner and Racah • In particular, unitary groups have been used successfully in various domains of physics, although the physical reasons for their introduction have not always been completely explained. The use of unitary groups in physicsl problems, has in turn given rise to a lot of works devoted to the study of these groups considered as entities . In this paper, we shall use the group U(n) to define, to investigate the properties of and to calculate the eigen­ values of the so-called generalized exchange operators (g.e.o.). This research opening was suggested by Biedenharn , who defined a g.e.o. P^ ' 2" '* k £Qr k particles and relative to U(n) as an extension of the two particles exchange operator. The two particles exchange operator has proved to be very useful in Nuclear Physics. 8 (cf. the spin exchange operator or Bartlett forces , the space 9 exchange operator or Majorana forces , and the charge exchange operator or Heisenberg potential ). These two particles exchange operators are a simple transcription of two-body forces. In a • similar way, the k-particles g.e.o. (k > 2) derive from k-body forces. For a long time, the existence of many-body forces among : 11 12 nucléons has been postulated . Recently, some papers have been devoted to 3-body forces in an attempt to define their importance 13 relative to 2-body forces and William G. Harter using Young tableau, derived sum rules for k-body operator spectra. I The material will be organized as follows. In Sec. 2, we • study the properties of the g.e.o.1 P. * Z'" k defined in a 1 I somewhat different manner to the one of Biedenharn . In Sec. 3, ] A "' we extend this definition to produce an operator P for A-particles J A A i • (A >k). Sec. 4 is devoted to the three particular cases P, , P '|j A AAA I and P. By a direct derivation,we get P? , P and P. as !j functions of the invariant operators for SU(h). Foi: each of the :* AAA 4 operators P , P and P. , we then obtain the eigenvalues | relative to an A-particles state which carries an irreducible repre- I sentation of SU(n). Making use of the Young diagram corresponding || yp •:%:,, 3. A A to the representation, we show that the eigenvalues of P , P A and P do not depend on n in SU(n). In Sec. 5, we write the g.e.o.'s in Weyl basis. As is well-known, the Weyl basis makes the study of U(n) easier. The study of our g.e.o.'s is simplified with this basis. The Weyl basis is however less convenient for 14-15 physical problems such as the one suggested by Biedenharn and used in the first three parts of this paper. In particular Biedenharn's basis enables us to render apparent, through the weight operators H , quantities of physical interest, while Weyl's basis does not. Finally, three Appendices deal with problems connected with the main body of the paper, although they are more or less self-contained* Appendix A discusses the algebra whose elements are the structure constants and the coupling coefficients for U(n). Appendix B is devoted with sums of type Last of all, Appendix C gives a general expression for the eigen- 2 values of Gel'fand's invariants For the purpose of unifying the notation, we follow the same definitions and notations as those used in a previous paper , with one exception : summation of the generators X. (or x ) always ir.cludes the number operator H (or h )• As a consequence the coupling coefficient [AB ] is now defined by C [ xA, xB ]+ = [ AB ] xc (I.!) In other words, we extend the metric for SU(n) , to the one of U(n), by letting gAB = &A with gAQ = 6A . 2. Definition and significance of the SU(n) k-particle g.e.o. acting on the k-particle states We shall define such an operator by the following formula : ?k « if £ Wc -EJ x? aw x<*>\ .. X«A) (ni) which differs from that of Biedenharn by the norm and the coefficient [ABC ...E]. The interests of our definition will appear below. InEq.(II.l) a., a. ..., a, stand for the k-particles, whereas x , x , ... x are the generators of the fundamental ABE representation [1,0] of SU(n) and, finally [ABC ... E] is the completely symmetric coefficient in all indices A, B, ..., E obtained from the coefficient [ A B-M] [M ...]...[. .E] . More precisely, [AB] (or gAR) and [ABC], which correspond to k = 2 and k = 3 respectively, are completely symmetric by defini- tion . For k $• 4, the coefficient [ABC.. E] can be defined via the following recurrency formula : It should be emphazised that the summation in Eq. (II. 1) has to be extended over all the values of the indiceB A B. E including the value 0 which corresponds to the unit generator. Let us denote, for the particle a., by i (a.) the m state (1 < m < n) of the fundamental representation of SU(n).. The generator x. 1 (x. = h, or e ) acting on * giveB s : A Ai a' • n The index n is superfluous and we shall bmit it in what follows . 5. Let us now examine the significance of P aj ' a2" " ,ak operating on a tensorial product of such one-particle functions. For k = 2, Eq. (II. 1) gives : and by making use of Eq. (II.3) we get : . <b («,)éil) = (b il) ai id.) as is well-known, i.e., P^l a2 simply exchanges the particles a. and a,. For k = 3, firstly using the definition relation of the coupling coeffi­ cient [AB C] written here in the following form : tX-ï .X^CVBCl^ ' (II.6) and secondly, by taking advantage of the useful relation.: where em. is the nxn matrix consisting of unity in the (m.i) position and zeros elsewhere. We should obtain : P3 ^T«,)^^)^7l,)^x^T«()4>T^^7«3^^f;^^*;J (n.s) As a matter of fact, this operator induces on the three particles all the permutations which do not leave any of them unchanged. To be more precise, those permutations are, in cyclic notation, (123) and (13 2), i.e., form the 3-cycle class of the symmetric group S3. For k = 4, we shall use the following relation definition for the coupling coefficient. (a. 9) and with Eq. (II.7) we thus get : 6. «r.r . -,, (»J ("'1 '">' PJjv*" A1"1» A.*** J.""* J f SrteP C " *B *C D ^tfl,)^ }^^ ^ ~«A* ? (11.10) Here we do not get all the permutations leading none of the particle unchanged, because the coefficient [AB-M] [MCD] is not comple­ tely symmetric in all indices. This point leads us to define our g.e.o. by introducing a coefficient [ABCD] completely symmetric in all indices. Thus, for P,al 2 3 4 obtain : 4 we T 4«( mz **£ rtH. «% l»tt< ,*Hj »Wt rt n 1 w (11.11) Jt+s, ,'* 3 ,'" i «*tt »*"15 j.^* ^ , 0*ï» j_' '( 1 The significance of the P. 2 3 4 operator is now clear. It induces on the 4-particles the six permutations (1234).(1243) , (1 324), (1342), (1423), (1432) which belong to the 4-cycle class of S4- By an inductive process, we might generalize the results obtained for k = 2, 3 and 4. The significance of a k-particle g.e.o. P. * 2'" k for SU(n) (n ^. k) acting on a k-particle state function corresponds to the permutations on the k-objects belonging to the class defined by the cycles of length k. 7. 3. a) Definition of the SU(n) k-particle g.e.o. acting on the A-particle states ( A > k ) We shall define it through : which is a generalization of Eq. (II.1).( One must verify of course ai a a that for the particular case A = k, we have Pk =Pk 2'-' k). By expanding Eq. (III. 1) ,we get : It is clear that the quantity S LtiB — E] X„ X^--- X^ is completely symmetric in indices a , a , .