LI~TTERE AL NUOV0 CIMENTO VOL. 26, N, 18 29 Dicembre 1979

On the Infinite Momentum Limit of Relativistic Elementary Systems (*).

M. QUIR6S

Instituto de Estructura de la Materia - Serrano, 119 jrladrid-6 Departamento de Fisica, Universidad de Alcald - Alcald de Henares

J. RAMi~EZ MITTELBRUNN (**)

Max-Planck-Inst. ]. Phys. u. Astrophys - ffhhringer Ring, 6 M4nchen

(ricevuto iI 12 0t tobre 1979)

Relativistic elementary systems are usually described by unitary irreducible repre- sentations of the Poincard group (~). Let P~, J, K be the infinitesimal generators of the Poincar~ algebra. The canonical realization is constructed taking as a complete set of commuting observables (CSC0) the operators (P, $3}, where S a is the third component of the spin operator. Poincar6 generators are represented by

(~) l i ~ L §

K ipo V~ § (S • P)/(Po § m),

Po ~o(p) = (p2 § m2)�89

and a finite Lorentz transformation A is represented by D(")(Rm), where D (") is the (2s+ 1)- dimensional representation of SU 2 and Rm is the Wigner rotation associated to A.

However the canonical representation (1) is not suitablc to perform the high mo- mentum limit

(2) Pa = kv3 (k -~ c~).

In fact such a kinematical limit must be described by an Inhnii-Wigner contraction (2) of the Lie algebra, while the contraction of Lie groups and their representations has been studied by MICK~LSSON and NIEDERL]~ (3) with the following main result: group representations can be contracted only if they are defined in Hilbcrt spaces related

(*) P r e s e n t e d a t t h e I n t e r n a t i o n a l Con fe r ence on M a t h e m a t i c a l P h y s i c s L a u s a n n e (Swi t ze r l and ) , A u g u s t 20-25, 1979. (**) O n Ieave of a b s e n c e f r o m : I n s t i t u t o de E s t r u c t u r a de l a M a t e r i a , S e r r a n o , 119. M a d r i d - 6 . (x) V. BARGER a n d E. P . WIGNER: Proc. Nat . -4cad. Sci . U S A , 34, 211 (1948); E. P . WIGNER." A n n . Math . , 40, 149 (1939). (~) E . INi3N~ a n d E. P . WIGNER: Proc. N a t . . A c a d . Sci . U S A , 39, 510 (1953). (a) J . ~r a n d J . NIEDERLE: Commun. Math . Phys . , 27, 167 (1972).

633

634 ~. quIl~6S and a. RAMfREZ MITTELBRUNN

by a un i ta ry t ransformat ion . This procedure for cont rac t ing representa t ions has been appl ied to the nonre la t iv is t ic l imi t (c -~ c~) wi thou t fur ther complicat ions (4) because the space of square- integrable funct ions in a common representa t ion space both for the Poincar6 and the Galileo group. Our aim will be to cont rac t the Poincar6 group, using the techniques described in ref. 0), and to get an u l t ra re la t iv is t ic covar ianee group (5). Never theless the canonical funct ions ~v(p, a) are not appropr ia te to per form the high m o m e n t u m l imi t (2), essent ia l ly because they depend on the cont rac t ion varia- ble P3, and the cont rac ted funct ions would bc ill defined.

The problem to be solved, in order to find a real izat ion of the Poincar6 group suit- able to the high m o m e n t u m l imit , is threefold:

i) To obta in a CSCO remai~ring unchanged in the contract io l l of the Lic algebra.

ii) To compute the common eigenfunct ions of the CSCO and to project over t hem the canonical ftmctions.

iii) To represent the Poincar6 group over the new set of functions.

Whi le P i (i = 1, 2) are good generators for the new basis (unal tered by the l imi t (2)), Pa is not, and must bc replaced by a new operator. A glance at the commuta t i on rcla- t ions of the Poincar6 algebra shows tha t K3, the boost along the th i rd axis, is the only genera to r commu~ing with P i , SO t ha t we must replace the set {P} by the set {P i , Ka}. On the other hand, a quick look at the canonical rcpresen ta t ion (1) of the generators shows tha t $3 does not commute wi th K 3 so tha t one must ro ta te the canonical basis u(a) in order to get the four th c o m m u t i n g operator. This can be done bccause the l i t t le group of (m, 0) is S03, and it is often used to pass from the canonical to the he l ic i ty basis, where the quan t iza t ion axis is ro ta ted to the direct ion of the m o m e n t u m p .

In our case the s imples t choice is the opera tor

(3)

w h e r e

(4)

N = n S ,

1 n = p-- T ( - - P2, P1 , O)

is the new quant iza t ion direct ion. The ro ta t ion T(~, fi, y) passing f rom Sa to N has, as Euler parameters ,

(5) ~ = arc tg (Pl/P2) , fi -- n/2 , 7 arbi t rary .

Thus the quant iza t ion axis is ro ta t ed from OZ to the new direct ion, (4), lying on the t ransverse plane and or thogonal to PT-

The fol lowing step is to compute the common cigenfunct ions of {P,, K 3, N} wi th e igenvalues (q~, ~, a}. They can be wr i t ten as

(6) V)q,,~,o(P) -- ~)~,a(P) U(pT, (:r) (5(2)(p T - qT),

where U(pT, (r) Tu(6) is the ro ta ted basis and ~,~,z(p) is the eige~ffunction of K 3 wi th

(*) J-. LE6N, ~I. QuIR6S a n d g. RAMIREZ /~s NUOVO Cimenlo B, 46, 109 (1978); J. Math. Phys. , 20, 1068 (1979). (5) An ea r l i e r a t t e m p t c a n be found i n : J . LE6N, M. QUIRfS and ft. t~AMtREZ 1V~ITTELBRUNN-" 2YItovo Cimento .A, 41, 141 (1977); Phys. Lett. B, 68, 247 (1977).

ON THE INFINITE MOMENTUM LIMIT OF RELATIVISTIC ELEMENTARY SYSTEMS 6 3 5

eigenvalue 4. Using (1), a straightforward calculation gives

(7) ~ , . (p ) = ~:~ exp [2iaO(p)],

where ~/= pc~k, withp+ = P0 + P3 and k being a constant with dimension of momentum, and

(8) O(p) = are tg (PT/P+ § m) .

Let us remark the fact that the constant k will play the same role in the ultrarelati- vistic contraction as the constant c did in the nonrelativistic limit (% Our representation will display explicitly, unlike the canonical representation, the contraction parameter.

Projecting the canonical functions ~ over the basis of functions (6), we get the new functions

r

~P(PT, 4, ~) = l ' d~ exp [-- 2iaO(p)] ~i~-1 q~(p, ~) (9)

o

which are related to the old ones by a 3Iellin transformation (~) over the variable ~. The scalar product induced by the usual product

da~ ( lO) (g~l, ~o=) = 22 / - ~ W;(P, + ~o=(p, ~)

a d ~

over the functions (9) can be written as

(11) 1 ~ fd:p~d~ ~(PT, ~, ~) ~2(PT, 4, ~), (~, ~) =

where the inverse Mellin transformation of (9) has been used. Thus the functions (9) are square-integrable in the plane (PT, 4).

The physical meaning of the eigenvalue ~ can be traced as the variable canonical conjugate to the <~rapidity~) y = aresh (p3 /mT) -? in (mT/k) - - - - i ~ / ~ 2 . Thus two features widely used in high-energy physics, the rapidity variable and the Mellin transfor- mation techniques, do emerge spontaneously in our formalism from group-theoretical considerations. The convergence properties of ~ as ]p] -+ c~ translate into analyticity properties of the Mellin transformed ~v in the complex k-plane. In particular if ~o has a finite norm, then ~ is analytic on the s~rip - - c ~ < R e 2 < § c~, ] I m 2 l < e ( e > 0 ) .

The canonical representation of the Poincar4 generators J ~ induces, through the MelIin transformation and the rotation T, the new representation J~v over the functions to. We shall write, for simplicity, this representation in the infinite momentum limit (2) as

PT = PT , ~ = 4 , #3 = L3 , Po ~- t"3 ~- (kl2) e x p [ - y ] ,

(12) ,71 ~- - - I ~ ~ - - (k/2)( iV1 - - (pl/p~) 83) exp [-- y] ,

J~ _~ Z-~ _~ (~/2)(iv~ + (p~/p~) 83) cxp [ - y].

The operators j ~ act over the functions ~, (9), which are not altered by the high mo- mentum limit (2).

( t ) A. ERDELYI et al.: Table o/ Integral Trans]orms (New York, N. Y., 1953).

~ 3 6 M. QUIR6S a n d j . RAMIREZ MITTELBRUNN

Other topics which are being studied (7) within the present formalism are:

a) Local 2(2s + 1)-dimensional representations, wave functions and the interac- tion problem.

b) The relationship with the null-plant* basis (8) and the infinite moinentum frame.

c) The relationship with other high momentum transformations, as Cini-Touschek and Mclosh (9).

In fact it is amazing to remark that the quantization direction (4) coincides with the rotation axis of the Melosh transformatiol~, considered as a rotation from the cano- nical to the null-plane b~sis, while the phase exp [2i(r0] in (7) can be interpreted as a rotation of an angle 0 arom~d the quantization axis and the valu(', of 0 is precisely, as pointed out by ALD_a.yA and AZC-~RRA~;A (lo), the second 3[elosh transformation.

$ $ $

On(, of the authors (J.R.M.) would like to thank Prof. H. P. DiJRR and W. DRECHSLER for their warm hospitality at the Max-Planck Insti tut and the Max-Planck Society for fi~aucial support. The other a~tthor (3LQ.) is indebted to Prof. J. A. D~ Azc,i~r~AGa for useful discussions.

(~) • . QUIRbS a n d J . PtA.M~REZ ~][ITTELBRUNN: i n p r e p a r a t i o n . (~) D. ]E. SOPER: Phys. Rev. D, 5, 1956 (1972). (9) M. CINI a n d B. TOUSClaEK: Nuoco Cimvnlo, 7, 422 (1958); H . J . MELOSH: Phys. Rev. D, 9, 1095 (1976). ('~) V. ALDAYA a n d J . A. DE AZC~RRAGA; Phys. Rev. D, 14, 1049 (1976).