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REVIVAL OF THE STRING MODEL
A. Neveu
To cite this version:
A. Neveu. REVIVAL OF THE STRING MODEL. Journal de Physique Colloques, 1982, 43 (C3),
pp.C3-260-C3-262. �10.1051/jphyscol:1982351�. �jpa-00221905�
JOURNAL DE PHYSIQUE
CoZZoque C 3 , supple'ment au n o 12, Tome 43, de'cembre 1982 page C3-260
REVIVAL OF THE STRING MODEL
Laboratoire de Physique The'orique de ZrEcoZe Normale Supe'rieure, 24 rue Lhomond, 75231 Paris Cedex 05, France
L a s t year v:e observed a sudden f l a r e o f renewed i n t e r e s t i n t h e o l d dual resonance r,iodel, which had been n e a r l y dormant f o r several years. The remark t h a t t r i g g e r e d t h i s i s due t o Polyakov. I t r e l a t e s some o f t h e d i f f i c u l t i e s o f t h e model t o cell-known problens o f two-dimensional f i e l d t h e o r i e s , i n an e f f o r t t o understand and r e s o l v e them. Ve s h a l l describe t h i s s t a r t i n g p o i n t and t h e v a r i o u s developments t o which t h i s i n i t i a l idea has l e d .
The o l d Nambu-Goto s t r i n g a c t i o n
i s j u s t t h e area o f t h e two-dimensional m a n i f o l d described by t h e
3
f u n c t i o n s xu(a,r) w i t h"
=a / a r ,
In,= 2/20.
It d e f i n e s a c o n s i s t e n t c l a s s i c a l theory i n any space- time d i m e n s i o n s "'.~roblems began t o show up when one t r i e d t o quantize i t . The canonical 1 ight-cone q u a n t i z a t i o n o f r e f . (1) cane o u t Lorentz i n v a r i a n t on1 y f o r8
= 26, and l e a d i n g i n t e r c e p t a. = 1, meaning t h a t t h e r e e x i s t s a tachyon i n t h e spectrum. The tachyon problem i s n o t t h e most serious : a s u i t a b l e v e r s i o n o f t h e supersymmetric s t r i n g e x i s t s which has no tachyon, and anyway, i t j u s t means t h a t one i s expanding around t h e wrong vacuuni ( b u t nobody has y e t been a b l e t o f i n d t h e r i g h t vacuulii). The problem w i t h t h e dimension o f space-time was more p u z z l i n g . Although Lorentz i n v a r i a n c e w i t h o u t negative m e t r i c s t a t e s can be achieved a t t h e t r e e l e v e l i n any3
< 26, s t a r t i n g from t h e a c t i o n o f eq. ( I ) , problems w i t h t h e dimension come back when one computes u n i t a r i t y c o r r e c t i o n s , f o r c i n g t h e t h e o r y back t o3
= 26.I t had been known f o r some t i n e ( * ) t h a t t h e r e e x i s t s another a c t i o n which i s c l a s s i c a l l y equivalent, and which i s q u a d r a t i c i n t h e xu v a r i a b l e s :
A'
=I
do d ~ gabaa
xuab
xu ( 2 )w i t h a, b = a, r, gab t h e m e t r i c o f t h e s u r f a c e spanned by xY(o, T)
,
and g = d e t gab.This a c t i o n shares w i t h A t h e basic p r o p e r t y o f r e p a r a m e t r i z a t i o n i n v a r i a n c e . gab i s an independent dynamical v a r i a b l e . From i t s c l a s s i c a l equation o f motion, i t i s p r o p o r t i o n a l t o
aa
xuab
xu,
so t h a t c l a s s i c a l l y A ' immediately reduces t o A. Another way o f seelng t h l s uses t h e geometrical f a c t t h a t i t i s always p o s s i b l e t o choose a (0, T ) p a r a m e t r i z a t i o n such t h a t gab i s p r o p o r t i o n a l t o 6,b :gab = e@
s~~
(3)and i n th i s parametrization, P.' does n o t c o n t a i n cp a t a1 1.
Quantum mechanicallj~,A1 i s a b e t t e r s t a r t i n g p o i n t than A, which i s an unusual f u n c t i o n a l o f xFI
.
Indeed A ' i s j u s t t h e a c t i o n f o r massless s c a l a r f i e l d s xu,u
= O , l , .. . a -
1, i n a curved two-dimensional m e t r i c gab.
The quantum theory i sthus d e f i n e d by t h e f u n c t i o n a l i n t e g r a l
g a b 8 xu e x p t i
j
do d r6
gabaa
xuab
xu+
iu! do d rfi
+ source terms
I
x gauge f i x i n g terms x r e g u l a t o r s (4)Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982351
A . Neveu C3-261
I n eq. ( 4 ) , t h e t'erm
uo
2 I d a d.r6
has been provided t o make t h e i n t e g r a l over 9 ab convergent. Another way(3) i s t o f i x t h e t o t a l area I d a d ~.
The gauge i n v a r - iance of t h e a c t i o n i s r e p a r a m e t r i z a t i o n invariance, and i s t r e a t e d l a Faddeev- Popov. Formally, one can choose t h e gauge c o n d i t i o n s as c o n d i t i o n s on xu,
and one can then do t h e gab i n t e g r a t i o n p o i n t by p o i n t ( 4 ) recovering e x a c t l y t h e s t a r t i n g p o i n t ( 5 ) of t h e quantum treatment o f t h e Nambu a c t i o n by f u n c t i o n a l i n t e g r a l s , as i t has been used t e n years ago t o d e r i v e t h e Veneziano model from eq. (1). One even recovers t h e unusual measurea
xp&pfi o f r e f . ( 5 ) ., P ,
~ o l ~ a k o v ~ ~ ) remarked t h a t t h e r e g u l a t o r s , necessary t o make t h e x' i n t e g r a l w e l l - defined, can be made r e p a r a m e t r i z a t i o n i n v a r i a n t , b u t s p o i l t h e @ independence o f t h e theory. T h i s quantum nechanical anomaly, which i s o f e x a c t l y t h e same nature as t h e A d l e r anomaly, t u r n s @ ( o r e q u i v a l e n t l y t h e m e t r i c
fi)
i n t o a dynamical v a r i a b l e , by p r o v i d i n g an e f f e c t i v e Lagrangian f o r @.
T h i s e f f e c t i v e Lagrangian contains a piece p r o p o r t i o n a l t o,
coning from t h e xp i n t e g r a l , and a piece independent o f a),
coming from t h e Faddeev-Popov ghost i n t e g r a t i o n . They combine t o g i v e2 2
where p i s an a r b i t r a r y renormalized parameter, which c o n t a i n s po
.
From t h i s r e s u l t , one understandsa
= 26 as t h e c a n c e l l a t i o n o f t h e conformal anomaly between xp and t h e Fadd ev -Popov ghost. I na
= 26, @ i s indeed n o t a dynanical v a r i a b l e , and i n5
~ 2 6 , i t provides an e x t r a , l o n g i t u d i n a l , degree o f freedom on t h e s t r i n g (which seems u n r e l a t e d t o those o f r e f . ( 7 ) ) . We s t r e s s t h a t t h e o l d treatment o f t h e o r b i t a l xu degrees o f freedom o f t h e s t r i n g remains v a l i d , and t h a t X e f f ( @ ) should j u s t p r o v i d e a small m o d i f i c a t i o n o f t h e various s t r i n g s c a t t e r i n g amplitudes, which should r e s t o r e u n i t a r i t y i n a l l dimensions.From t h i s i n i t i a l remark, e x p l o r a t i o n has proceeded i n v a r i o u s d i r e c t i o n s . F i r s t , one can extend t h i s r e s u l t t o t h e f e r m i o n i c string('), o b t a i n i n g a super- symmetric v e r s i o n o f eq. ( 3 ) , w i t h 26
- 3
replaced by 10- a .
Second, i n t h e case o f open s t r i n g s , one can compute t h e c c n t r i b u t i o n of t h e boundaries t o t h e e f f e c t i v e @ action('). However, most o f the e f f o r t has been centered on t h eq u a n t i z a t i o n o f t h e L i o u v i l l e a c t i o n ( 3 ) f o r
4 .
Using a semi-classical q u a n t i z a t i o n by keeping o n l y one @ f i e l d c o n f i g u r a t i o n , t h e authors o f r e f . (10) have recovered t h e o l d Veneziano model f o r t h e s c a t t e r i n g o f s t r i n g s asa
+,
which i s t h e v a l i d i t y range o f t h e semi-classical approximation. An exact q u a n t i z a t i o n o f t h e L i o u v i l l e theory has been considered i n r e f s . (11-13). The authors o f r e f . (11) quantize t h e L i o u v i l l e t h e o r y i n i n f i n i t e space. This i s n o t d i r e c t l y r e l e v a n t t o t h e s t r i n g problem, where t h e o parameter space i s always a f i n i t e segment w i t h a p p r o p r i a t e boundary c o n d i t i o n s a t t h e ends on t h e @ f i e l d . Furthermore, t h e L i o u v i l l e equation being c o n f o r m a l l y i n v a r i a n t i n two dimensions, t h e quantum t h e o r y i s plagued w i t h i n f r a r e d divergences when one t r i e s t o d e f i n e i t i n i n f i n i t e space.The closed s t r i n g , where @ i s p e r i o d i c i n
,
i s e a s i e r t o deal w i t h , and a complete c o n f o r m a l l y i n v a r i a n t q u a n t i z a t i o n has been given by t h e authors o f r e f . (12). It i s very i m p o r t a n t t o know t h a t a t l e a s t one such c o n f o r m a l l y i n v a r i a n t quantum t h e o r y e x i s t s f o r @.
Conformal invariance, w i t h t h e associated Virasoro generators and algebra, p l a y s a c r u c i a l r o l e i n dual resonance models. The c e n t r a l charge o f t h e Virasoro algebra now contains a a) -dependent p i e c e coming from t h e@ f i e l d , which has t o be added t o t h e piece coming from t h e o r b i t a l x" modes.
I n t h e q u a n t i z a t i o n method o f r e f . (12) t h e
3
dependence o f t h e c e n t r a l charge cancels o n l y when t h e @ f i e l d i s t r e a t e d s e m i - c l a s s i c a l l y , b u t a new3
dependence comes back when i t i s t r e a t e d quantum mechanically. As i t blows up a t a) = 26, t h i s new dependence i s unwelcome f o r t h e moment.The separation o f modes o f t h e @ f i e l d i n t o e q u a l l y spaced harmonic o s c i l l a t o r s i n t h e case o f t h e open s t r i n g has been achieved i n r e f . (13). The boundary conditions have been chosen i n such a way t h a t they g i v e r i s e t o a w e l l d e f i n e d v a r i a t i o n a l problem f o r
4
when t h e boundary terms found i n r e f . (9) a r e taken i n t o account.C3-262 JOURNAL DE PHYSIQUE
The canonical q u a n t i z a t i o n o f t h e modes found i n r e f . (13) leads t o a
bl
indepen- dent t o t a l c e n t r a l charge f o r t h e Virasoro algebra, and i s thus i n e q u i v a l e n t t o t h e q u a n t i z a t i o n o f r e f . (12).Separation o f modes f o r t h e supersymmetric L i o u v i l l e t h e o r y has been done i n r e f . (14).
The computation o f t h e improved dual amplitudes r e q u i r e s the c a l c u l a t i o n o f t h e a p p r o p r i a t e vacuum e x p e c t a t i o n values o f t h e 4 f i e l d . T h i s t a s k i s i n p r i n c i p l e s t r a i g h t f o r w a r d , a f t e r one has reduced t h e t h e o r y t o harmonic o s c i l l a t o r s , as i n r e f s . ( l 2 ) and (13). I t i s much e a s i e r than f o r other, more complicated, f i e l d t h e o r i e s l i k e sine-Gordon, where such a r e d u c t i o n i s n o t p o s s i b l e . The basic i n g r e d i e n t i s a v e r t e x w i t h s u i t a b l e conformal p r o p e r t i e s . We note i n t h i s r e s p e c t t h a t t h e source terms which t h e authors o f r e f s . ( 6 ) and (10) i n t r o d u c e d i n eq. ( 4 ) a r e i n general i n a p p r o p r i a t e t o t h i s purpose, s i n c e they l e a d t o unphysical s i n g u l a r i t i e s i n i n t e r - n a l channels when t h e e x t e r n a l l i n e s a r e n o t on-shell, a l o n g known disease i n t h e o l d dual-resonance model ( I 5 ) . So f a r , o n l y t h e spectrum of open o r closed s t r i n g s has been worked out, i n r e f s . (12) and (13). It i s p o s i t i v e d e f i n i t e f o r
3
< 26,and c o n s i s t s o f l i n e a r t r a j e c t o r i e s ; o n l y t h e daughter s t r u c t u r e i s changed r e l a t i - v e l y t o t h e o l d dual model, and t h e closed s t r i n g sector, i n r e f . (12), has a c o n t i - nuous spectrum, due t o zero modes o f t h e 4 f i e l d . Whether o r n o t t h i s continuous spectrum cancels t h e unwanted c u t n a t u r e o f t h e o l d dual pomeron i n
3
< 26 w i l l o n l y be seen when loops of open s t r i n g s a r e computed i n t h e new formalism.Progress has a l s o been made i n t h e o l d treatment o f t h e supersymmetric s t r i n g . While t h i s t h e o r y was, from t h e s t a r t , supersymmetric on t h e s u r f a c e spanned by t h e s t r i n g , i t has t u r n e d o u t t h a t an a p p r o p r i a t e s e c t o r o f i t was a l s o supersymmetric i n n = 10 dimensional space-time. This r e s u l t has a t l a s t been proved i n t h e remarkable papers o f r e f . (16), which l e a d t o a complete r e f o r m u l a t i o n o f t h e theory.
It i s n o t c l e a r t h a t such a supersymmetry, i n b o t h 2 and
a
dimensions would s u r v i v e a treatment 2 l a Polyakov. T h i s i s an i m p o r t a n t question, due t o t h e occur- rence o f s p i n n i n g s t r i n g s i n t h e 3-dimensional I s i n g model and i n 4-dimensional l a t t i c e gauge t h e o r i e s .REFERENCES
(1) Goddard P., Goldstone J., Rebbi C. and Thorn C., Nucl. Phys. B56 (1973) 109 (2) B r i n k L., D i Vecchia P. and Howe P., Phys. L e t t . 65B (1976) 4 r
(3) O n o f r i E. and Virasoro M.A., CERN p r e p r i n t 3233 ( m 2 ) (4) Gervais J.L. and Neveu A., unpublished
(5) Gervais J.L. and S a k i t a B., Phys. Rev. L e t t .
30
(1973) 716 (6) Polyakov A.M., Phys. L e t t . 103B (1981) 207(7) P a t r a s c i o i u A., Nucl. P h y s . m (1974) 525 (8) Polyakov M.a., Phys. L e t t . (1981) 211 (9) Alvarez O., unpublished
Friedan D., unpublished
Durhuus B., Olesen P. and Petersen J.L., Nucl. Phys. (1982) 157, 8201
(1982) 176
-
(10) Durhuus B., N i e l s e n H.B., Olesen P. and Petersen J.L., Nucl. Phys. (1982) 498
(11) dlHoker E. and Jackiw R., M I T p r e p r i n t 984 (1982)
(12) C u r t r i g h t T. and Thorn C.B., Phys. Rev. L e t t .
48
(1982) 1309Braaten E., C u r t r i g h t T. and Thorn C.B., U n i v e r s i t y o f F l o r i d a p r e p r i n t s 82-18 and 82-27 (1982)
(13) Gervais J.L. and Neveu A., Nucl. Phys. B199 (1982) 59 and Nucl. Phys. i n press (14) A r v i s J.F., Ecole Normale p r e p r i n t 8 2 / 1 m 8 2 )
(15) Drummond I.T., Nucl. Phys. 835 (1971) 269
(16) see M.B. Green's review a t m s conference, Queen P,lary College p r e p r i n t 82-12 and references t h e r e i n
