Volume 31B, number 4 P H Y S I C S L E T T E R S 16 Februa ry 1970

LOCAL CURRENT A L G E B R A M O D E L W I T H L I N E A R

H. L E U T W Y L E R $ CERN - Geneva

Received 22 December 1969

T R A J E C T O R I E S

We cons t ruc t a model containing an infinite number of par t ic les with a rb i t r a r i l y high mass and spin corresponding to a family of l inear Regge t r a j ec to r i e s . The model does not contain s ta tes with space - like momentum and gives r i s e to local cu r ren t s obeying the s tandard equal t ime commutat ion r e l a - t ions. The bas ic s t ruc tu re is r emin i scen t of the osc i l la tor r epresen ta t ion of the Veneziano model; in pa r t i cu la r ghost s ta tes with negative norm occur / f~

The m o d e l m a y be f o r m u l a t e d in t e r m s of an i n f i n i t e c o m p o n e n t f i e ld g~v) s a t i s f y i n g the wave e q u a t i o n

{c1[~ + i c 2 F ~ a ~ + M 2 } ~ ( x ) = 0 (1)

w h e r e c 1 and c 2 a r e two r e a l p a r a m e t e r s w h e r e - a s F ~ and M 2 a r e m a t r i c e s o p e r a t i n g on the c o m - p o n e n t s of ¢4v). In o r d e r to s p e c i f y t h e s e m a t r i c - e s we i n t r o d u c e a s e t of f o u r h a r m o n i c o s c i l l a t o r m o d e s d e s c r i b e d by c r e a t i o n and a n n i h i l a t i o n

+ a ~ ( g = 0 , 1 , 2 , 3 ) : o p e r a t o r s a ~ , ,

(2) : : 0 ,

g g v = diag(+ 1 , - 1 , - 1 , - 1) i s the L o r e n t z m e t r i c . The r e p r e s e n t a t i o n of t h e s e o p e r a t o r s i s s p e c i - f i ed by the r e q u i r e m e n t t h a t t h e r e e x i s t s a g r o u n d s t a t e w i t h the p r o p e r t y a g ]0) = 0. A c o m p l e t e s e t of b a s i s v e c t o r s i s g e n e r a t e d by v e c t o r s of the type

+ + a + ]0) (3) ~°N = a v l a v 2 " '" Vn

w h e r e N s t a n d s f o r the c o l l e c t i o n (Vl, v 2, . . . , Vn). C l e a r l y , the m e t r i c in t h i s v e c t o r s p a c e i s i n - d e f i n i t e , e . g . , (0 t a , a+vtO) = - g g v . A c c o r d i n g l y the a d j o i n t a ~ i s de~-ined by T/-la~.TT/, w h e r e the o p e r a t o r 77 s t a n d s f o r the i n d e f i m t e m e t r i c : 7/a k =

= ' k ~ , ,7,0 -- -ao,7, ~ I0) : l o). The c o m p o n e n t s of o u r b a s i c f i e ld @(x) a r e in

o n e - t o - o n e c o r r e s p o n d e n c e w i th the a b o v e b a s i s v e c t o r s and m a y b e l a b e l l e d giN(X) = ~ V l Vn(X)" T h e m a t r i c e s a , and a ~ m a y be a p p l i e f f ~ b ' t h i s f i e l d in a n o b v i o u s f a s h i o n , e .g . , a ~ @ N =

= ~P~.Vl... v n"

On leave of absence f rom Inst i tut ftlr Theore t i sche Physik, Universi t t t t Bern , Bern , Switzerland.

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+ the m a t r i c e s F/1 and M 2 In t e r m s of a ~ and a / l a r e d e f i n e d by

Fbc : i (a~ - a /~ ) (4)

M 2 = a - t i a r a ~ ±

w h e r e a and fl a r e two r e a l p a r a m e t e r s . T h i s c o m p l e t e s the s p e c i f i c a t i o n of the wave e q u a t i o n and we now p r o c e e d to i t s so lu t i on . C l e a r l y , if c ~ = 0 the p l a n e wave s o l u t i o n s a r e e i g e n s t a t e s of M and m a y b e i d e n t i f i e d w i t h the c o m p l e t e s e t of b a s i s v e c t o r s q~N:

M2~ON = (a +fin)~p s (5)

In the g e n e r a l c a s e c 2 ¢ 0 a c o m p l e t e s e t of p l a n e wave s o l u t i o n s i s o b t a i n e d by t i l t i n g t h e s e b a s i s v e c t o r s :

¢/(x) = exp (- ipx) exp (- iypA) ~o N (6)

w h e r e A . = a . + a + , and 7 i s c h o s e n s u c h t h a t c 2 = fly. ~ o t e ~ h a t h e f o u r p a r a m e t e r s Cl, c2, a , fl o c c u r r i n g in the wave e q u a t i o n m a y be m u l t i - p l i e d by a c o m m o n f a c t o r w i thou t a f f e c t i n g the m o d e l . We f ind i t c o n v e n i e n t to m a k e u s e of t h i s f r e e d o m to n o r m a l i z e the c o e f f i c i e n t c 1 by c 1 = = 1 + f i t 2. Wi th t h i s c o n v e n t i o n we a r e l e f t w i th a , fl and 7 a s i n t r i n s i c p a r a m e t e r s of the m o d e l and one e a s i l y v e r i f i e s t h a t the m a s s of the so - l u t i o n (6) i s g i v e n by pgpg = a + ~n. A c o m p l e t e s e t of p o s i t i v e e n e r g y s o l u t i o n s r e s u l t s if ~ N r u n s t h r o u g h the c o m p l e t e s e t of b a s i s v e c t o r s i n t r o d u c e d above . Of c o u r s e , t h e r e e x i s t s a

~J" The appearance of ghost s ta tes with negative norm is the p r ice we pay for the fact the model is local and avoids spacel ike momenta [2]. The l i t e r a tu re on models of local cu r ren t a lgebra can be t raced back f rom ref. 3.

Volume 31B, number 4 P H Y S I C S L E T T E R S 16 Feb rua ry 1970

c o m p l e m e n t a r y s e t of n e g a t i v e e n e r g y s o l u t i o n s - t h e r e a r e , h o w e v e r , no s p a c e l i k e s o l u t i o n s pup u -<< O.

T h e w a v e e q u a t i o n i s e x p l i c i t l y L o r e n t z i n - v a r i a n t p r o v i d e d the f o u r m a t r i c e s a u c o n s t i t u t e a L o r e n t z v e c t o r . T h i s t r a n s f o r m a t i o n r u l e s p e c - i f i e s t he s p i n c o n t e n t of the m o d e l : t he s e t of p o s - i t i v e e n e r g y s o l u t i o n s b e l o n g i n g to a g i v e n m a s s v a l u e m 2 = c~ + f in c o n s i s t s of a d e g e n e r a t e f a m i l y of p a r t i c l e s w i t h s p i n j = n , n - 1, . . . , 0. T h i s s p e c t r u m c o r r e s p o n d s to a f a m i l y of l i n e a r R e g g e t r a j e c t o r i e s ; the l e a d i n g t r a j e c t o r y and i t s f i r s t d a u g h t e r a r e s i m p l e , the s e c o n d and t h i r d d a u g h t e r s a r e o c c u p i e d t w i c e , e tc . T h e l e a d i n g t r a j e c t o r y and a l l e v e n d a u g h t e r s h a v e p o s i t i v e n o r m , w h e r e a s a l l odd d a u g h t e r s h a v e n e g a t i v e n o r m .

We now t u r n to t he c o n s t r u c t i o n of l o c a l c u r - r e n t s and o b s e r v e t h a t the m o d e l a d m i t s of a c o n - s e r v e d c u r r e n t

J g = ½ i ~ { c l ~ g + ic2rU}~ (7)

w h e r e ~-= ~h*TT? d e n o t e s the a d j o i n t of ~. In o r d e r t h a t t he m o d e l g i v e s r i s e to a s e t of l o c a l c u r r e n t s i t h a s to b e e x t e n d e d in two r e s p e c t s : t he p a r - t i c l e s m u s t b e e q u i p p e d w i t h i n t e r n a l d e g r e e s of f r e e d o m and the f i e l d ¢~x) m u s t b e q u a n t i z e d . C o n c e r n i n g the i n t r o d u c t i o n of i n t e r n a l d e g r e e s of f r e e d o m we o b s e r v e t h a t i t i s p o s s i b l e to t r e a t t he c a s e of a b r o k e n s y m m e t r y by a l l o w i n g the m a s s o p e r a t o r M 2 to sp l i t t he i n t e r n a l d e g e n e r a c y . We s h a l l , h o w e v e r , r e s t r i c t o u r s e l v e s to a d i s - c u s s i o n of the s i m p l e s t c a s e e x h i b i t i n g fu l l s y m - m e t r y . C o n c e r n i n g the n e e d f o r q u a n t i z a t i o n we m e n t i o n t h a t the e x t e n s i o n to a l o c a l m a n y - p a r - t i c l e t h e o r y c a n b e a v o i d e d if one r e s t r i c t s o n e - s e l f to c u r r e n t a l g e b r a r e l a t i o n s a t i n f i n i t e m o - m e n t u m w h i c h a r e s a t i s f i e d by one p a r t i c l e s t a t e s . F r o m t h i s p o i n t of v i e w the i n t r o d u c t i o n of q u a n t - i z e d l o c a l f i e l d s i s m e r e l y a c o n v e n i e n t f o r m a l d e v i c e .

We d e n o t e the q u a n t i z e d i n f i n i t e c o m p o n e n t f i e l d by g/a(X), t he i n d e x a l a b e l l i n g the i n t e r n a l d e g r e e s of f r e e d o m . U n d e r an i n f i n i t e s i m a l t r a n s - f o r m a t i o n of the i n t e r n a l g r o u p t h i s f i e l d t r a n s - f o r m s a c c o r d i n g to ~a ~ ~a + i e k ¢'b(Qk)ba; t he m a t r i c e s Qk a r e t he g e n e r a t o r s of t he r e l e v a n t r e p r e s e n t a t i o n of t he i n t e r n a l g r o u p and s a t i s f y the c o r r e s p o n d i n g b r a c k e t r e l a t i o n s . T h e c u r - r e n t s a r e now d e f i n e d a s $

j~(x) = ½i:~a(X)Vkab{Cl'~ + i c 2 F U } ff/b(X): (8)

:~ Note that the me t r i c in the many par t ic le space is also indefinite; accordingly ~ a denotes the adjoint with r e spec t to this indefinite me t r i c .

One e a s i l y v e r i f i e s t h a t t h e s e c u r r e n t s a r e c o n - s e r v e d and t h a t t h e i r t i m e c o m p o n e n t s s a t i s f y t he s t a n d a r d c o m m u t a t i o n r e l a t i o n s p r o v i d e d the f i e l d and i t s c o n j u g a t e m o m e n t u m

na : ~ a ( Cl'~ o - ½iC2ro} (9) a r e q u a n t i z e d a c c o r d i n g to the c a n o n i c a l e q u a l t i m e c o m m u t a t i o n r u l e s

[%(x), ~b(y)] : [~a(~), ~b(y)] : i~ab ~ - 2 ) (10) (a l l o t h e r c o m m u t a t o r s b e t w e e n ~ a , ~a , ga and ~a v a n i s h a t e q u a l t i m e s ) .

F i n a l l y we i n d i c a t e the one p a r t i c l e m a t r i x e l e m e n t s of the a b o v e c u r r e n t s :

(p,N,a]j~(O)[ q, M, b)

= ½(2n) -3 ~kab q~N exp (i~Ap) {Cl(Pg + qg ) - c 2 F ~ } ×

× exp(-i~Aq)q~ M (11)

In p a r t i c u l a r , if b o t h N and M d e n o t e t he g r o u n d s t a t e we f i nd

(p, O, al j ~ (0)Iq, O,b) =

= ½(2n)-3 Qkab (Pg + q~) exp ( ~ 2 t ) (12)

w h e r e t = ( p - q)2 i s the i n v a r i a n t m o m e n t u m t r a n s f e r . T h i s r e s u l t s h o w s t h a t the m o d e l c o r - r e s p o n d s to f o r m f a c t o r s t h a t d e c r e a s e e x p o n e n - t i a l l y in the s p a c e l i k e r e g i o n **. We h a v e c h e c k e d t h a t the o n e - p a r t i c l e m a t r i x e l e m e n t s (11) s a t u r - a t e the c u r r e n t a l g e b r a r e l a t i o n s a t i n f i n i t e m o - m e n t u m .

** The model involves only finite dimensional sp inor represen ta t ions of the Lorentz group. It contains , however, an infinite tower of such rep resen ta t ions with a r b i t r a r i l y high spin. The mixing of this in - finite set of s ta tes through the t i l t ing operat ion exp ( - i pA) is respons ib le for the fact that the form factors a re not t r iv ia l in this model.

References 1. Y. Nambu, Inter . Conf. on Symmetr ies and quark

models , Wayne Univers i ty (1969); L. Susskind and G. Frye , Yeshiva Univers i ty P r e - pr in ts (1969); S. Fubini , D. Gordon and G. Veneziano, Phys. Le t te r s 29B (1969) 679.

2. I .T . Grodsky and R. F. S t rea te r , Phys. Rev. Le t te r s 20 (1968) 695.

3. S. Chang, R. Dashen and L. O'Raifear ta igh, Phys. Rev. Le t t e r s 21 0-968) 1026; C. Fronsdal and A. M. Harun Ar Rashid, T r i e s t e In- t e rna l Report IC/69/117 (1969).

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