quantum dynmaics of a massless relativistic string- goddard

8/12/2019 Quantum Dynmaics of a Massless Relativistic String- Goddard

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N u c l ea r P h y s ic s B 5 6 ( 1 9 7 3 ) 1 0 9 - 1 3 5 . N o r t h - H o ll a n d P u b li s hi n g C o m p a n y

Q U A N T U M D Y N A M I C S O F A M A S S L E S S R E L AT I V I S T I C S T R I N G

P G O D D A R DDe partm ent o f Mathematics, University o f Durham, Durham City

J . G O L D S T O N EDe partm ent o f Ap plied M athema tics and Theoretical Physics,

University o f Cambridge, Cambridge

C . R E B B I a n d C .B . T H O R NCERN, Geneva

R e c e i v e d 6 N o v e m b e r 1 9 7 2

A b s t r a c t : We d e v e l o p t h e c l a s si c a l a n d q u a n t u m m e c h a n i c s o f a m a s s l e s s r e l a ti v i st i c s t r in g ,t h e l i g h t s t r in g , w h i c h i s c h a r a c t e r i z e d b y a n a c t i o n p r o p o r t i o n a l t o t h e a r e a o f t h e w o r l ds h e e t s w e p t o u t b y t h e s t r i n g i n s p a c e t im e . We s h o w t h a t , c l a s si c a ll y, th e r e a r e o n l y D - 2d y n a m i c a l l y i n d e p e n d e n t c o m p o n e n t s a m o n g t h e D f u n c t i o n sxU o,r) w h i c h r e p r e s e n tt h e w o r l d s h e e t ( D i s t h e d i m e n s i o n o f s p a c e t im e ) . Q u a n t i z i n g o n l y t h e s e i n d e p e n d e n t

c o m p o n e n t s , w e f i n d th a t t h e a n g u l a r m o m e n t u m o p e r a t o r s s u g g e s te d b y t h e c o r r e s p o n -d e n c e p r i n c ip l e g e n e r a t e O ( D - I , 1 ) o n l y w h e n t h e f ir s t e x c i t e d s t a t e is a p h o t o n , i .e ., as p i n - o n e m a s s l e s s s t a t e , a n d w h e n D = 2 6 . B y a l l o w i n g a d d i t i o n a l d e g r e e s o f f r e e d o m i nt h e q u a n t u m m e c h a n i c s , w e ar e ab l e to q u a n t i z e t h e s t ri n g i n a L o r e n t z c o v a r i a n t m a n n e rf o r a n y v a l u e o f D a n d a n y m a s s f o r t h e f i r st e x c i t e d s t a t e . In t h i s la t t e r s c h e m e t h e f u l lF o c k s p a c e c o n t a i n s n e g a t i v e n o r m s t a t e s . H o w e v e r , w h e n D


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110 P. Godd ard et al., Massless relativistic string

t i m e c o m p o n e n t o f t h e s tr i ng v ib r a t e d a l o n g w i t h t h e s p a c e c o m p o n e n t s . O n l y r e-c e n t l y, f o l l o w i n g an o r i g in a l id e a o f N a m b u [ 2 ] h a s p r o g re s s b e e n m a d e i n g i vi n g

t h e s tr i n g a s p a c e - t im e i n t e r p r e t a t i o n [ 3 ] . T h e s u g g e s t io n w a s m a d e t h a t t h e a c t i o no f t h e r e l at iv i s ti c s t r in g s h o u l d b e p r o p o r t i o n a l t o t h e a r e a o f t h e w o r l d s h e e t s w e p to u t b y t h e s t r i n g i n sp a c e t im e . T h e Vi r a s o r o g a u g e c o n d i t i o n s [ 4 ] w h i c h a p p e a r i nd u a l m o d e l s a r is e n a t u r a l l y i n t h is d e s c r i p t i o n b e c a u s e o f t h e a r b i tr a r in e s s o f p a r a m -e t r i z a t i o n o f th e s u r f a c e s w e p t o u t b y t h e s t r in g : o n e c a n a l w a y s c h o o s e a nor tho-n o r m a l p a r a m e t r i z a t i o n s u c h t h a t

3 x 3 x 03 -~ 3 ~o = 0 \ 3 7 ] + \ 3 o ]

T h e F o u r i e r c o m p o n e n t s o f t h e s e e q u a t i o n s j u s t s a y th a t t h e Vi r a s o r o g au g e s,L N ,

a re ze ro .T h i s f r e e d o m o f p a r a m e t r i z a t i o n c a n b e e x p l o i t e d t o c h o o s e , as o n e p a r a m e t e r,

t h e t i m e , x , i n s o m e L o r e n t z f r a m e . B y c h o o s i n g t h e o t h e r p a r a m e t e r, o , t o b e t h ef r a c t i o n o f t h e t o t a l e n e rg y i n c l u d e d b e t w e e n o n e e n d o f t h e s t ri n g a n d t h e p o i n t i nq u e s t i o n , t h e e q u a t i o n s o f m o t i o n s i m p li f y t o

1 3 2 ) x o , , ) - - 0(27rE) 2302 ,

a n d t h e s u b s i d i a r y c o n d i t i o n s

3X 3X = (3X ~2 1 (3X ) 23 t 3 0 O , \ 3 t ] + - = 0( 2 h E ) 2 ~

w h e r e E is th e t o t a l e n e rg y o f th e s t r in g . O n e c o n s e q u e n c e o f t he s e c o n d i t i o n s ist h a t t h e e n d s o f t h e s t r i n g m o v e a t t h e s p e e d o f l ig h t. A n o t h e r is t h e f a c t t h a t o n l yt h e m o t i o n o f t h e s t r in g p e r p e n d i c u l a r t o i t s e l f is d y n a m i c a l l y s i g n if i ca n t . T h u s t h en u m b e r o f i n d e p e n d e n t v i b r a t i o n s o f t h e s t ri n g is o n l y t w o : t h e s t r in g is t ra n s ve r se .T h e s t r i n g c a n t h e n b e q u a n t i z e d b y m a k i n gx ( o , t ) q u a n t u m m e c h a n i c a l o p e r a t o r sa n d s u b j e c t i n g t h e p h y s i c a l s t a t e s t o t h e s u b s i d i a r y c o n d i t i o n s . T h i s p r o c e d u r e c l e a rly uses a pos i t ive de f in i t e Hi lbe r t space so the re i s no p rob lem wi th nega t ive norms t at e s. H o w e v e r, m a n i f e s t L o r e n t z c o v a r i a n c e is l o s t a n d m u s t b e p r o v e n b e f o r e t h et h e o r y is a c c e p t a b l e .

I n t h e m e a n t i m e a n e x t en s i v e s t u d y o f t h e p h y s i c a l s t a te s i n t h e d u a l r e s o n a n c em o d e l l e d t o p r o o f s o f th e a b s e n c e o f g h o s t s ( n e g a t i v e - n o r m s t a t e s ) u n d e r c e r t a i nc o n d i t i o n s [ 5 , 6 ] . T h e s e c o n d i t i o n s a r e t h a t t h e i n t e r c e p t o f t h e l ea d i ng R e g g e t ra -j e c t o r y m u s t b e u n i t y a n d t h e d i m e n s i o n o f sp a c e- ti m e c a n n o t b e g r e at e r t h a n 2 6 .F u r t h e r, t h e se p r o o f s e s t a b l is h e d t h a t w h e n t h e d i m e n s i o n o f s p a ce t i m e is 2 6 , t h espe c t ru m of phys ica l s t a tes is pure ly t r ansverse . S ince a se t o f tr ansverse s t a tes hada l r e a d y b e e n e x p l i c i t l y c o n s t r u c t e d [ 7 ] t h e s e c o u l d b e t a k e n a s a c o m p l e t e s e t o fp h y s i c a l s ta t e s. T h e O ( 2 5 , 1 ) t r a n s f o r m a t i o n p r o p e r t i e s o f p h y s i c a l s t a te s i n 2 6 d i-m e n s i o n s w e r e s tu d i e d [ 8 ] a n d a r e p r e s e n t a t i o n o f O ( 2 5 ) , t h e l i tt l e g r o u p o f t h e


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P. Go ddard et al ., Ma ssless relat iv t ic s tr ing 111

t o t a l m o m e n t u m o f a p a r t i c u l a r s t a te o n t h e t r a ns v e rs e s u b s p a c e a l o n e c o n s t r u c t e dexp l ic i t ly. Th i s cons t ruc t ion fa i l ed in l e ss than 26 d imens ions because the phys ica l

s t a tes a re then no t t r ansverse .I t i s our a im in th i s paper to unders tand the resu l t s desc r ibed in the p rev ious

paragra ph in t e rms o f the qua n t iz a t io n o f the re lat iv i st i c s tr ing . S ince the t r ansversep h y s i c a l s t a te s m e n t i o n e d b e f o r e a r e si m p l e in th e l i m i t o f i n f in i t e m o m e n t u m , w ea re le d t o c h o o s e o n e p a r a m e t e r o f th e s u r f a c e s w e p t o u t b y t h e s t ri n g p r o p o r t i o n a lto x+ = x / ~ (x 0 + x 3 ) r a th e r than s imply x 0 . We shal l show ho w prev iou s resu l ts ont h e r e p r e s e n t a t i o n s o f t h e l i tt le g r o u p i n 2 6 d i m e n s i o n s a ri se n a t u r a l l y f r o m t h ischo ice o f pa ra m et r i za t io n . In the p rocess we sha ll r ev iew com ple t e ly the c lass ica la n d q u a n t u m m e c h a n i c s o f th e r e l a ti v is t ic s t r i n g a n d s h a ll a t t e m p t t o u n i f y t h e v ar -i o u s t r e a t m e n t s .

2 . C lass ical mec han ics o f the l igh t s t ring

2 .1 . E q u a t i o n s o f m o t i o n

We charac te r i ze a s t r ing mathemat ica l ly as a f in i t e curve in space which in genera lis a l l o w e d t o c h a n g e i t s s h a p e a n d p o s i t i o n a s a f u n c t i o n o f t i m e . I n a p a r t i c u l a r L o -ren tz f ram e we can desc r ibe th e s t r ing a t a time x = t by a set o f func t ions

x i o , t ) ,w h e r e x i ( o , t ) is t h e i t h s p a t i a l c o o r d i n a t e o f t h e p o i n t o n t h e s t ri n g la b e l e d b y t h ep a r a m e t e r a . I n s p a c e - t i m e t h e p r o p a g a t i n g s t r i n g c a n b e c h a r a c t e r i z e d b y t h e t w o -d i m e n s i o n a l s u r f a c ex U ( o , t ) :

x O ( o , t ) = c t , x i ( o , t )= x i o ,t ) . (1 )

C l e a r l y, t h e k n o w l e d g e o f t h is s u r f a c e g iv e s a c o m p l e t e d e s c r i p t i o n o f th e m o t i o n o ft h e s t ri n g : w e o b t a i n t h e c o n f i g u r a t i o n o f t h e s t r in g a t a n y t i m e t b y i n t e r s e c ti n gt h e s u r f a c e w i t h t h e h y p e r p l a n e x 0 =c t .

T h e o n l ya p r i o r i res t r i c t ion we p lace on the su r face spanned by the s t r ing i st h a t i n t h e n e i g h b o u r h o o d o f e a c h o f i ts p o i n t s t h e r e e x i s t s a n i n f in i t e si m a l d i sp l ac e -ment a long the su r face which po in t s in the t ime- l ike o r l igh t - l ike d i rec t ion ; i . e . , wer e q u i r e e a c h p o i n t o f t h e s t r i n g t o m o v e a t a v e l o c i t y l es s t h a n o r e q u a l t o t h e v e l o c -i ty o f ligh i . We sha ll ca ll such a su r face t ime- l ike . Eq . (1 ) g ive a poss ib le pa ram et r i cr e p r e s e n t a t i o n o f t h e s u r f a c e , b u t w e p r e f e r t o u s e a c o m p l e t e l y g e n e r a l p a r a m e t r i -z a t i o n .

x u = x U ( o , r ) . ( 2 )

L e t u s i n t r o d u c e t h e d i f f e re n t i a l f o r m s

d x u 3 x u 3 x u= ~ - a d a + - ~ - d~ , ( 3)


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112 P. Godda rd et al., Ma ssless relativistic string

(3x u 3x v 3x u 3 x V ]d F u v = d o d r \ - ~ o 3 r 3 r ~ o ]

= dx u A ( ix v . (4)

T h e n t h e c o n d i t i o n t h a t t h e s u r f a c e b e t i m e - l ik e i s j u s t *

- d F V d F = ( d o d r ) 2 (~ )u v 3 r ] k 3 o ] \ 3 r 2 1_ 1

T h e a r e a e l e m e n t s p a n n e d b y t h e t w o i n f i n i t e s i m a l d i s p l a c e m e n t s

3 x u 3 x u3 o d o , ~ d r ,

i s g i v e n b y

d A = { - d F u vd F v } 7 . ( 6 )

F o r o u r p u r p o s e s i t w i l l b e c o n v e n i e n t t o c h o o s e t h e p a r a m e t e r s s o t h a t t h e e n d so f t h e s t r i n g c o r r e s p o n d t o o = 0 a n d a = 7r a n d s o t h a t t h e i n i t i a l a n d f i n a l c o n f i g u -r a t i o n s o f t h e s t ri n g , a ft e r s o m e m o t i o n i n s p ac e t i m e , c o r r e s p o n d t o f i x e d v a lu e s o fr , r i a n d r f . I n a l m o s t a l l p h y s i c a l p r o b l e m s t h e i n i ti a l a n d f i n a l c o n f i g u r a t i o n s o ft h e s t r i n g w i l l b e t h o s e s e e n b y a d e f i n i t e o b s e r v e r a t a gi v e n i n s t a n t o f t i m e i n h i sL o r e n t z f r a m e : t h is m e a n s t h a t , i n i t i a l l y a n d f i n a l l y, t h e l i n e s o f c o n s t a n t r w i l l b e

l in e s o f c o n s t a n t t i m e i n t h a t f r a m e . B u t w e s h a l l b e m o r e g e n e r a l a n d s h a l l a l l o w a si n i ti a l a n d f in a l c o n f i g u r a t i o n s o f t h e s t ri n g a n y t w o n o n - i n t e r s e c t i n g s p a c e - l ik ec u r v e s , x ~ ( o ) ,x ~ ( o ) , w h i c h c a n b e c o n n e c t e d b y a t i m e - li k e s u r fa c e .

We s h a ll t a k e t h e d y n a m i c s o f t h e s t ri n g t o b e g iv e n b y c h o o s i n g t h e a c t i o n p r o -p o r t i o n a l t o t h e a r e a o f t h e s u r fa c e s w e p t o u t b y t h e s t r i n g [ 9 ]

) }; x= 1 f rr 3 x 2 ~ 3 x ~ 2 ( 3 x ~ 2d r d o 3 - o e r - \ 3 o ] \ 3 r ]2 7 r ~ c 2 r i 0

r i

T h i s i s t h e s i m p l e s t a c t i o n o n e c a n w r i t e d o w n w h i c h i s i n t r in s i c t o t h e s u r f a c e , a n dt h e c h o i c e i s c l o s e l y a n a l o g o u s t o c h o o s i n g t h e a c t i o n o f a s tr u c t u r e l e s s p o i n t p a r -t ic l e p r o p o r t i o n a l t o t h e l e n g t h o f i ts w o r l d l in e . T h e c o n s t a n t a ' h a s d i m e n s i o n s( e n e r g y ) - 2 a n d e n s u r e s t h a t S h a s th e d i m e n s i o n s o f a c t io n . O u r t h e o r y t h u s h a s af u n d a m e n t a l l e n g t h , w h i c h , a s w e s h a ll se e m e a s u r e s t h e s l o p e o f R e g g e t r a je c t o r i e s :

d d = ~ ' 4 1 .d E 2

* O ur m etric is _gOO = +gdi = 1.


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P. Go dda rd e t a l ., M ass less re la t iv i s t i c s t r ing l 13

We s ha l l h e n c e f o r t h c h o o s e o u r u n i t s s o t h a t ~ = K - - - c = 1 , s o t h a t a l l p h y s i c a lq u a n t i t i e s w i l l b e d i m e n s i o n l e s s . A t a n y p o i n t , o f c o u r s e , w e c a n r e s t o r e t h e s e f a c -

t o r s a n d c o n v e r t t o c o n v e n t i o n a l u n i t s .T h e e q u a t i o n s o f m o t i o n f o r th e s t r in g f o l l o w f r o m t h e p r i n c ip l e o f l e a s t a c t i o n :

w e r e q u i r e t h a t S b e s t a t i o n a r y u n d e r a n y s m a l l v a r i a t i o n o f t h e s u r f a c e t h a t j o i n st h e i n i t ia l a n d f i n a l c o n f i g u r a t i o n s o f t h e s t r i n g( 2 - a x / a t , x = a x / a o ) :

d o - - - f ixJ o x a x

r f ( 6 0 4 ) = T r / (5 a ~ ) r = ri r = rff d r x u + d o x = 0 ( 8 )

r i = 0for a l l6 x U ( o , r )s u c h t h a tg x U ( c r,r i ) = 6 x U ( o , r f ) = 0 .

T h u s w e f i nd t h e e q u a t i o n s o f m o t i o n *

a a a a+ - 0 , ( 9 a )

O r a k u a o a x , ~

a n d t h e b o u n d a r y c o n d i t i o n

a L ( 0 , r ) a L (z r r ) = 0 ( 9 b )~)X ~ ~)X ~

T h e s e e q u a t i o n s h a v e a v e r y s im p l e p h y s i c a l i n t e r p r e t a t i o n . I n s o m e p a r t i c u l a rL o r e n t z f r a m e t a k e t h e p a r a m e t e r r t o b e t h e t i m e t , s o t h a t t h e m o t i o n i s d e s c r i b e db y x =x ( t , o ) , x a t h r e e - v e c t o r. L e t d s =I O x / O o l d ob e t h e e l e m e n t o f l e n g t h a l o n g t hes t r ing a nd u =3 x / ~ t - ( O x / ~ s ) [ ( a x / a t ) a x / a s ) ] t h e v e l o c i t y o f th e s t ri n g p e r p e n -d i c u l a r t o i t s e l f . T h e n ( 7 ) b e c o m e s

S = i f L d t ,

t

z = - r o f d o - (7 )l

w h e r e T o =1 / ( 2 7 r a * t c )h a s t h e d i m e n s i o n s o f f o rc e . I f w e s i m p l i f y f u r t h e r b yc h o o s i n g o s o t h a t( a x / a t ) ( a x / a o )= 0 i. e. , s o t h a t t h e p a t h o f a p o i n t o f c o n s t a n to i s a l w a y s p e r p e n d i c u l a r t o t h e s t ri n g , th e n ( 9 ) b e c o m e s

* I f o u r s u r f a c e w e r e s p a c e - l i k e , e q . 9 a ) w o u l d b e t h e e q u a t i o n f o r a s u r f a c e o f m i n i m a l a r e a ,a n d i f x = t , t h e m i n i m a l E u c l i d i a n a r e a. I n t h e E u c l i d i a n c a s e , h o w e v e r , n o s o l u t i o n e x i s t su n l e s s th e b o u n d a r y is c l o s e d : t h e r e i s n o s o l u t i o n s t a t i s fy i n g 9 b ) . I n o u r c a s e th e e q s. 9 b )c a n b e s a t i sf i e d , a n d r e q u i r e t h e e n d s o f t h e s t r i n g to m o v e w i t h t h e v e l o c i t y o f l ig h t .


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114 P. Go ddard e t a l . Mass l e s s r e l a t iv i s t i c s t r ing

E T o i 1a t - fi l v { I c 2 ) To = g o - c 2 I . ( 9 a )

T h e o b v i o u s i n t e r p r e t a t i o n i s t h a t w e h a v e a s t r in g w i t h z e r o r e s t m a s s a n d r e s t te n -s i o n T 0 . T h e m a s s p e r u n i t l e n g t h i s( To / C 2 ) ( 1 - o 2 / c 2 ) - ~a n d t h e e f f e c t iv e t e n s i o nis T = TO( 1 - o 2 / c 2 ) ~ .O n l y t h e t r a n sv e r s e m o t i o n i s s i g n i f ic a n t . T h e r e i s n o c o n -s e r v a t i o n o f m a t e r i a l o f t h e s t r i n g ; s t r e t c h i n g i t m a k e s i t m o r e m a s s i ve . T h e o r i g in a li n t e r p r e t a t i o n i n t e r m s o f t h e a r e a o f t h e w o r l d s h e e t s h o w s t h a t t h e t h e o r y i s c o -v a r i a n t .

T h e b o u n d a r y c o n d i t i o n s b ec o m e

d o I d o 2( o ) 1 = ( 0 ) 2 = C , d t - d t = 0 , ( 9 b )

s o t h a t a f r e e e n d m o v e s w i t h t h e s p e e d o f l i g h t a t r i g h t a n g le s t o t h e s t r i n g . T h i s i sn e c e s s a r y t o m a k e T = 0 a t t h e e n d s . I t is n o w s i m p l e t o v e r if y t h e o b v i o u s e x p r e s -s io n s fo r m o m e n t u m , e n e r g y an d a n g u l a r m o m e n t u m o f th e s t ri n g, b u t w e d o t h i su s i n g t h e g e n e r a l c o v a r i a n t e x p r e s s i o n s i n th e n e x t p a r a g r a p h . I t i s a l s o e a s y t o v e r i f yt h a t a s i m p l e s e t o f p o s s i b le m o t i o n s o f t h e s t r i n g a r e ri g id r o t a t i o n s o f a s t r a i g h ts t ri n g o f l e n g t h 2 a w i t h a n g u l a r v e l o c i t y c o, w h e r e c o a = c. I n s u c h a m o t i o n t h et o t a l e n e r g y E ~ a a n d t h e a n g u l a r m o m e n t u m J ~ a 2 s o t h a t E 2 ~ J . I n f a c t E =rrToa

= a E . We s h a ll s e e t h a t t h i s c o r r e s p o n d s= Q r To / 2 c ) a 2 s o t h a t J / ~=( lz r r l ~ c To ) E 2 2t o t h e l e a d i n g R e g g e t r a j e c t o r y o f t h e d u a l r e s o n a n c e m o d e l . A t e n s i o n o f 1 3 t o n sg iv e s a s l o p e o f 1 G e V - 2 .

R e t u r n i n g t o t h e g e n e r a l f o r m u l a t i o n , w e n o t e t h a t b e c a u s e t h e L a g r a n g i a n d e n -s i t y L i s i n v a r i a n t u n d e r t h e P o i n c a r 6 g r o u p ( s i n ce i t i s a f u n c t i o n o f L o r e n t z s c a la r sf o r m e d o u t o f t h eder iva t iveso f x U ) , t h e r e w i ll b e l o c a l l y c o n s e r v e d q u a n t i t i e s a s s o -c i a t e d w i t h 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 s i n th i s g r o u p . B y p e r f o r m i n g a n i n f i n i te -s i m a l t r ans la t ionw e s e e t h a t t h e e n e r g y m o m e n t u m f l o w i n g a c r o ss an i n f i n it e s i m a ll in e e l e m e n t i s j u s t

~ L O Ld P U = ~ - x d o + d r .

a a

T h u s o n t h e s u r f a c e w e c a n d e f i n e an e n e r g y m o m e n t u m c u r r e n t P i ( i = r , o ) w i t hc o m p o n e n t s

L = pu p~ , - ~ L (10 )= ' a x '

T h e n e q . (9 a ) e x p r e s s e s t h e l o c a l c o n s e r v a t i o n o f e n e r g y m o m e n t u m a l o n g th e s u r-f a ce , a n d e q . ( 9 b ) e x p r e s s e s th e f a c t t h a t n o e n e r g y m o m e n t u m f l o w s f r o m t h e e n d so f th e s t r in g . I n p a r t i c u l a r , t h e t o t a l m o m e n t u m ,


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P. Godda rd et al., Ma ssless relativistic string 115

r

P u = f ( d o P r ~+ d r P ~ o ) = f d o p e , ( 1 1 )

c 0w h e r e C is a n y c u r v e f r o m t h e b o u n d a r y l i n e x u ( O , r ) t o t h e b o u n d a r y l in e x U( zr, r ) ,is i n d e p e n d e n t o f t h e c h o ic e o f C a n d r ep r e s en t s t h e t o t a l e n e rg y m o m e n t u mo

t h e s t r in g . I n a s im i l a r w a y w e c a n p e r f o r m i n f in i t e si m a lL o r e n t z t r a n s f o r m a t i o n sa n d s e e t h a t t h e a n g u l a r m o m e n t u m f l o w i n g a c ro s s a n i n fi n i te s i m a l l i ne e l e m e n t i s

d M u v = ( x U p ~ - x ~'P U z) d o + ( x pVo - x V p ~ )d ~ . ( 1 2 )

S o w e c a n d e f i n e a n a n g u l a r m o m e n t u m c u r r e n tM u v i = x U p ~ , - x v p ~w hich i s lo -c a l ly c o n s e r v e d

a ~ a U ~ o3 r + 3 a - 0

M ~ Vo= 0 , a t a = 0 , Tr. ( 1 3)

E q s . ( 1 3 ) a r e a l so a d i r e c t c o n s e q u e n c e o f e q s. (9 ) . T h e ( c o n s e r v e d ) t o t a l a n g u l a rm o m e n t u m o f t h e s t ri n g is, o f c o u rs e ,

M ~ = f d oM~ Vr ( 1 4 )0

B e c a u s e o u r a c t i o n i s i n d e p e n d e n t o f t h e w a y w e p a r a m e t r i z e t h e s u r f a c e , e q . ( 9 a )is f o r m i n v a r i a n t u n d e r a n a r b i t r a r y n o n - s i n g u la r r e p a r a m e t r i z a t i o n o f t h e s u r f ac ea n d ( 9 b ) i s f o r m i n v a r i a n t u n d e r t h o s e r e p a r a m e t r i z a t i o n s f o r w h i c h t h e l in e sx ( ~ = O, ~ ) , x ( ~ = 7r, ~ ) co inc id e w i th the l inesx ( o = O , r ) , x ( o = n , r ) .T h i s m e a n st h a t i f t h e f u n c t i o n s

x ~ o , r )

s a t i s f y ( 9 ) , t h e n s o d o t h e f u n c t i o n s

x" (~ (o , ~) , 7(0 , r ) ) ,

p r o v i d e d o n l y o ( 0 , r ) = O , o O T, r ) = n . T h i s i n v a r i a n c e r e f le c t s t h e f a c t t h a t t he s et w o f u n c t i o n s d e f i n e t h e s a m e s u r f a c e a n d s h o u l d t h e r e f o r e b e r e g a r d e d a s e q u i va -l e n t. T h e r e is a c o n t i n u o u s i n f i n i t y o f f u n c t i o n s w h i c h s a t is f y e q . ( 9 ) g i v e n th e i ni -t i a l a n d f i n a l c o n f i g u r a t i o n s . We m u s t t h e r e f o r e s p e c i f y t h e p a r a m e t r i z a t i o n i n s o m ew a y b e f o r e w e c a n s o l v e t h e m .

I n f i x i ng o u r p a r a m e t r i z a t i o n w e w o u l d a l so li k e t o d o i t i n s u c h a w a y a s t o s i m -p l i f y e q s. ( 9 ). I t i s a w e l l - k n o w n r e s u lt f r o m t h e t h e o r y o f E u c l i d i a n s u r f a c e s [ 1 0 ]t h a t o n e c a n l in e ar iz e t h e s e e q u a t i o n s b y c h o o s i n g an o r t h o n o r m a l p a r a m e t r i z a t io n

x

a o a ~ = ' \a o - \ o r = '


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l 16 P. Godda rd et al., Massless relativistic string

w h i c h c a n b e d o n e f o r a n y E u c l i d e a n s u r fa c e . S in c e o u r s u r fa c e i s a s s u m e d t o b et i m e : l i k e , t h e s e c o n d o f th e s e e q u a t i o n s i s i m p o s s i b l e . H o w e v e r , w e c a n a l w a y s

c h o o s e a p a r a m e t r i z a t i o n s o t h a tx

c ~ o c 3 - ~= 0 , \ ~ o 1 + \ O r ] = 0 . ( 1 5 )

Wi t h t h i s p a r a m e t r i z a t i o n , e q . ( 9 a ) b e c o m e s s i m p l y t h e o n e - d i m e n s i o n a l w a v e e q u a -t i o n

~12 ~2

We c o u l d t h u s i m p o s e e q s . (1 5 ) a s a d d i t i o n a l c o n s t r a i n t e q u a t i o n s m e r e l y o n g e o -

m e t r i c a l g r o u n d s. H o w e v e r , w e p r e f e r to p r o c e e d f r o m a m o r e p h y s i c a l p o i n t o fv i ew a n d f i x a p a r a m e t r i z a t i o n a c c o r d i n g t o t h e a r g u m e n t s t h a t f o l l o w. T h e o r t h o -n o r m a l i t y c o n d i t i o n s , e q s . ( 1 5 ) w i l l st il l r e s u l t, t o g e t h e r w i t h a f u r t h e r c o n s t r a i n t .

I n o u r p r o b l e m w e w o u l d l i k e to i d e n t i f y z w i t h s o m e t i m e c o o r d i n a t e , s o w es t a r t b y s e t t in g

n - x = 2 ( n - P ) z , ( 1 6 )

w h e r e P i s t h e t o t a l m o m e n t u m o f th e s t r i n g a n d n i s a c o n s t a n t v e c t o r , s u c h t h a tn 2 ~ 0 . We d o n o t s p e c i f y n = ( 1 , 0 , 0 , 0 ) b e c a u s e w e w a n t t o l e a v e s o m e a r b i tr a r i -n e s s in t h e c h o i c e o f f r a m e a n d a l s o b e c a u s e , a s w e s h a l l s e e , w h e n n 2 = 0 , w e g e t

s p e c i al s i m p l i f i c a t io n s . T h e c o m p o n e n t o f m o m e n t u m a l o n g n , ( n . P ) , i s i n s e r t e db e c a u s e i t s im p l i f ie s s o m e f u r t h e r e q u a t i o n s , b u t t h is is n o t f u n d a m e n t a l . I f w ew a n t t o i d e n t i f y o w i t h s o m e p h y s i c a l q u a n t i t y, a n d s t il l k e e p i t v a r y i n g f r o m 0 t o 7r,w e m u s t r e l a t e i t to a c o n s e r v e d q u a n t i t y. T h i s s u g ge s ts t h a t w e t a k e o p r o p o r t i o n a lt o t h e r el a ti v e m o m e n t u m a l on g n i n c l u d e d b e t w e e n t h e b o u n d a r y a n d t h e p o i n tc o n s i d e r e d , i .e . ,

( n ' P ) o = l r f d o n P r0fixed r

T h i s e q u a t i o n a m o u n t s t o s a y in g th a t n .i s c o n s e r v e d , i t m u s t a l so b e i n d e p e n d e n t o f r , i .e . ,

n ' PF I ~ 7 . = 7 T

T h e n t h e e q u a t io n s o f m o t i o n ( 9 a ), (9 b ) b e c o m e

0n . P o = 0 ' P o = 0 a t o = 0 , rr

a o

( 1 7 )

P r i s c o n s t a n t a l o n g o a n d s i n c e i ts i n t e g r a l

( 1 8 )

s o t h a t n P o = O . O n t h e o t h e r h a n d ,


9/27

P. God dard et al., Ma ssless relativistic string 117

~ . X t

n ~ cc 1

((J X' )2 _ 52 X'2 }gSO tha t 2 . x ' = 0 . Then (18) impl ies 22 + x ,2 = 0 , so we f ind the or tho-n orm al i tycondi t ions (15) .

Summ arizing , our sys tem o f equat ions has become

L * a x 0 x 2 0 x 20 o O r O , \ 0 o ] + \ O r ] = 0 , ( 1 9 a )

0(~-r-r2 0 2 ~ x u : 0 (1 9b)0 o 2 1

Ox ~ - 0 at o = O r (19c)3o

n x = 2 ( n . P ) r. ( 1 9d )

Tak ing advantage o f (19a) we can eva lua te the m om entum and angu lar mom entumcurrents

O x u a x up u - -

2 7 r Or P~ 27r 00

M ~ = 2~ -aTr ar I ' = - 2~ gao ao I

Eq. (18) i s therefore a consequence o f (19d) and (19a) .By speci fy ing our parameter r as in (19d) we have g iven up m anifes t Lorentz

covad ance since we have singled o ut a part icular direct ion nU. The qu an ti tyn . x / n . P plays the ro le of a parameter ra ther than a dynam ical variable . I f we onlyrequire eqs. (19a) - (c) , wh ich are ma nifest ly covariant , then the para m etrizat ionis no t f ixed: one can sti ll perform a reparam etrizat ion,

o = o ( a , 7 ) , T = ~ g d , 7 ) ,

wh ich preserves (19a), pro vided th at

a o a r a o a r a oa '~ a T ' a 7 a T a T 0 , a t ' ~ - - O , ~ r.

These las t two equ at ions de termine o(~ , T) (up to an addit ive cons tant ) f romr( ~, '7) and fur the r requi re tha t r (~ , 7) sati sf ies

(a ~2 ~ 2 ) r (d ' , ~ ) = 0 , -f-r = 0 , a t o = O , r r. (20 )a

In te rms o f the new parameters only (19a) - (c ) a re t rue ; but T i s no longer propor-t ional to a t ime variable. Conv ersely, i f we had ini t ial lly required o nly (19a) then we


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118 P. God dard et al. Ma ssless relativistic string

c o u l d a l w a y s f u r t h e r s p e ci a li z e r t o b e p r o p o r t i o n a l t o n x s i nc e e a c h c o m p o n e n to f x s a ti sf ie s 2 0 b ) . B e c a u s e t h e sp e c ia l r e p a r a m e t r i z a t i o n m u s t s a ti s f y t h e w a v e

e q u a t i o n a n d b o u n d a r y c o n d i t i o n s 2 0 b ) , i t is c o m p l e t e l y f ix e d o n c e w e s p e c i fy tw os p a ce - li k e n o n - i n t e r s e c t i n g e q u a l - r l i ne s o n o u r s u r f a c e . I n p a r t i c u l a r, o n c e w e s p e c i f yt h e i n i t i a l a n d f i n a l c o n f i g u r a t i o n s t o b e e q u a l - r l i n e s , t h e n t h e p a r a m e t r i z a t i o n o ft h e su r f a c e h a s b e e n c o m p l e t e l y s p e c if i ed b y r e q u i ri n g o r t h o n o r m a l i t y 1 9 a ) .

A t t h is p o i n t w e c a n d r a w a n a n a l o g y w i t h e l e c t r o m a g n e t i s m . I n t h is t h e o r y, s p e c -i f y in g th e L o r e n t z c o n d i t i o n 3 u A u = 0 d o e s n o t c o m p l e t e l y d e t e r m i n e t h e p o t e n -t ia l, s i n c e t h e g a u g e t r a n s f o r m a t i o n

A u - - , A M + 3 ~ X ,

prese rves3 u A ~ = 0 i f 3 2 X = 0 . H o w e v e r, o n c e w e s p e c i f y t h e p o t e n t i a l s i n i ti a ll y a n d

f inal ly e .g . , A 0 = 0 a t t = +- oo), the n this las t specia l gauge f re ed om is co m pl ete lyr e m o v e d . I n b o t h c a s e s , t h e m a n i f e s t l y c o v a r i a n t g a u g e c o n d i t i o n s a r e s u f f i c i e n t t oe n a b l e o n e t o s o lv e t h e e q u a t i o n s o f m o t i o n .

2 . 2 . Canonica l fo rm al i sm

F o r o u r d i s c u s s i o n o f t h e q u a n t u m m e c h a n i c s o f t h e r e l at iv i st ic s t ri n g w e s ha lln e e d e x p r e s s i o n s f o r t h e P o i s s o n b r a c k e t s o f o u r d y n a m i c a l va r ia b le s . O n e c o u l dt h i n k o f ta k i n g x U a n d P r a s c a n o n i c a l l y c o n j u g a t e v a ri a b le s : h o w e v e r, t h e s e q u a n -t i t ie s are no t indep end en t . Us ing eq . 10) an d the express ion fo r g i t i s s imple to

c h e c k t h a t

3x 2 1 [3x~ 23 ~ P r = 0 P r + = 0 (2 1)

2702 k 3 ]

so tha t our phase space is cons t ra ined . T he p resen ce o f eqs . 21) i s r e la ted to thea r b i tr a r in e s s i n t h e c h o i c e o f t h e p a r a m e t r i z a t i o n .

I n t h e p r e s e n c e o f c o n s t r a i n t s , o n e c a n e s t ab l i sh a c a n o n i c a l f o r m a l i s m e i t h e r b yc o m p u t i n g a ll P o i s s o n b r a c k e t s b e f o r e t h e c o n s t r a i n t s a re a p p li e d a s s u m i n g i n t h isc a s e c a n o n i c a l P o i s s o n b r a c k e t s f o r a l l t h e c o m p o n e n t so f x U and pu) , and im-p o s i n g a f t e r w a r d s t h e c o n s t r a i n t e q u a t i o n s o n t o t h e d y n a m i c a l s y s t e m [ 11 ] , o r b ys o lv i n g e x p l i c i t l y t h e c o n s t r a i n t s t o e l i m i n a t e s o m e o f t h e v a r ia b l es f r o m t h e e q u a -t io n s o f m o t i o n .

T h e q u a n t i z a t i o n p r o c e d u r e s t h a t f o l l o w t h e s e t w o t r e a t m e n t s o f th e c la s si c alsys tem a re d i ffe ren t , and we sha l l d i scuss them bo th . At the c lass ica l l eve l we sha l ls t u d y i n d e t ai l o n l y t h e f o r m a l i s m b a s e d o n t h e e l i m i n a t i o n o f th e r e d u n d a n t v a ri a-b le s . F o r t h e o t h e r f o r m a l i s m s e e r e f. [ 11 ] an d a l so a p p e n d i x A . ) O u r p r o c e d u r e ist o f ir s t s p e c i f y c o m p l e t e l y t h e p a r a m e t r i z a t i o n a n d t h e n u s e eq s . 2 1 ) t o e x p r e s s allo f t h e d y n a m i c a l v a r ia b l e s in t e r m s o f a c e r t a i n s e t o f i n d e p e n d e n t o n e s .

A l t h o u g h t h i s m e t h o d c o u l d b e f o l lo w e d f o r a n y c h o ic e o f p a r a m e t r i z a t io n , w ef i n d i t p a r t i c u l a r l y c o n v e n i e n t t o f i x t h e p a r a m e t r i z a t i o n a c c o r d i n g t o e q s . 1 6 ) a n d18) , wi th n l igh t- l ike . We in t rod uce the no ta t i on n._ = x /~ n 0 _+ n 3 ) fo r every


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P. G oddard et al., Massless relativistic string 119

L o r e n t z v e c t o r nU , a n d d e n o t e b y n it s r e si d u al s p a c e c o m p o n e n t s . F o r d e f i n i te n e s s ,w e t a k e n o f t h e f o r m l / x /~ - ( 1 , O , - 1 ) ; i .e . , n _ = 1 , n + = n = 0 . E q s . ( 1 8 ) a n d ( 1 6 )

t el l u s th a t ( t h e p a r a m e t r i z a t i o n is s u c h t h a t) t h e d e n s i t y o f m o m e n t u m /9+ d o e s n o td e p e n d o n o , s o t h a t i t is p r o p o r t i o n a l t o t h e t o t a l m o m e n t u m P + :

1+ (22a)] 9 - - 7 . ,

a n d t h a t x + i s p r o p o r t i o n a l t o r :

x + = 2 P + r . ( 2 2 b )

T h e n e q s . ( 2 1 ) g i v e

e + x_ = x P, ( 2 3 a )7 r

21+ X 2p _ = p 2 .~ ( 2 3 b )71 (2rr)2

We i n t r o d u c e t h e b a r i c e n t r i c c o o r d i n a t e

1 dox_ o,r)_ r ) = ~0

We c a n t h e n s o lv e e q s . ( 2 3 ) e x p l i c i t ly t o o b t a i n *

o , ~ - ) = q _ r ) + 7r / ' d o , - 0 ( ~ ' - o ) x P ( 24 a )

P _ ( o , r ) = 7 p 2 + x ' 2 I ( 2 4 b )(21r) 2/

We s e e t h a t a l l d y n a m i c a l v a r i a b l e s c a n b e e x p r e s s e d i n t e r m s o f t h e t r a n s v e r s e v a r ia -b l e s x , P, a n d t h e a d d i t i o n a l t w o q u a n t i t i e s P + a n d q _ .

We s h o u l d e s t a b l i s h P o i s s o n b r a c k e t s a m o n g t h e s e q u a n t i t i e s s o t h a t t h e e q u a -t i o n s o f m o t i o n f o l lo w in th e H a m i l t o n i a n f o r m

j ~= { f , H } + ~ f r .

N o t i c e t h a t H s h o u l d b e g iv e n b y/ r

0 (27r) 2 ] '

( 2 5 )

( 2 6 )

* q _ t ) must be introduced into eq. (24a) because eq. (23a) contains onlya x - / a o .


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120 P. Goddard e t a l . , Mass less re la t iv i s t i c s t r ing

s ince P _ i s the gen era to r o f in f in i t e s imal t r a ns la t ions in the x+ d i rec t ion . The equa-t i o n s o f m o t i o n a re t h e f o l l o w i n g o n e s :

X rr

/5 = 2-~-' (2 7a )

= 27r p , (27 b)

P + = c o n s t a n t , ( 2 7 c )

H_ = 2 P _ - p + . ( 2 7 d )

T h e s e e q u a t i o n s , t o g e t h e r w i t h e q s . ( 2 5 ) a n d ( 2 6 ) , d e m a n d t h a t

{ x i, x / ) = { p i , p / } = O r ( x i ( o ) , P / ( o ' ) } =8 0 8 ( 0 - o ' ) , ( 2 8 a )

a n d t h a t

Hq - = q o - + - - 7 ,P +

w h e r e q o - is a c o n s t a n t o f th e m o t i o n . We c a n t h e n a s s um e a c a n o n i c a l P o i s s o nb r a c k e t b e tw e e n q o - a n d P + .

{ q o - , P + } = - 1. ( 2 8 b )

This l a s t ansa tz is cons i s t en t wi th the fac t tha t P+ genera tes d i sp lac em ents in them i n u s d i re c ti o n, an d , t o g e th e r w i t h { P + , x } = { P +, P } = ( q o _ , X } = { q o _ , P } = 0 ,c o m p l e t e s t h e s p e c if i c at i o n o f t he P o i s so n b r a c k e ts a m o n g t h e i n d e p e n d e n t d y n a m -ical var iables .

T h e P o i s s o n b r a c k e t s o f th e x _ v a r ia b l es a m o n g t h e m s e l v e s a n d w i t h o t h e r v a r ia -b l es c a n b e c o m p u t e d f r o m ( 2 4 ) a n d ( 2 8 ) . I t is m o r e i n s t ru c t i v e , h o w e v e r, t o e x p a n da ll o f t h e v a ri a b le s i n t o n o r m a l m o d e s a n d d e r i v e t h e P o i s s o n b r a c k e t s f o r t h e se . B yv i rt u e o f t h e w a v e e q u a t i o n a n d b o u n d a r y c o n d i t i o n s w e c a n e x p a n dx U ( o , r )asfo l lows :

n e - i n r ] . ( 2 9 )= COS H Ox U ( o , r ) q U + ~ [a uo + i n

n ~ O

The n q~o, an wi ll a ll be co ns ta n t s o f the m ot i on . Re a l i ty o f x ~ impl ies qo = qo* ,a o - '- a ~ , a * = a nE q s . ( 2 2 ) a n d ( 2 4 ) g i v e t h e f o l l o w i n g c o n s t r a i n t s o n t h e n o r m a lmodes:

+ = a + = 0 , n 4 : 0 , a + = V ~ P + 30 )0 n o

1 ~ 31)a n = 1 + L n where L = -~ a _ k . a n + k ;a k = ~

o


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P. Go dda rd e t al., M assless relativistic string 121

i i +an , qo , aoan d qo a r e i n d epe n d e n t va r iab l es . The Po i s son b r acke t s am ong t he se va ri -ab les fo l low f ro m 28) and 29) :

(a in , ~ } : _ i n 6 n , _ m 6 i / , { qio , 4 } : v ~ 6 i / ,{ q i , q ] o } : O , { q g , a } : - ~ 3 2 )

F ro m t h e se t he a lgeb ra o f t he depe nden t mo des fo l lows :

{ C n , t m } = - i ( n - m ) L n m , { L n , a / m } = i m a / m + n , { i o , l - m } = X / 2 a i m . ( 3 3 )

I n t e r m s o f t he n o rm a l mod e s t he H ami l t on i an is

H = 2 P P _ = L o = p2 + ~ an . a*n 34 )n = l

and the invar ian t m ass ) 2 i s

M 2 = 2 P + P _ p 2 = ~ l a n [ 2 . 3 5 )n = l

We c a n al so w o r k o u t t h e t o t a l m o m e n t u m a n d a n gu l ar m o m e n t u m :

p u = ~ 2 2 a u 36)

a .u t / t

2 U a u _ a v a u ) + i n - n 37 )M U U = q o o - o o - n

n ~ O

I f we cons ider the case of the s t r ing ro ta t ing ab ou t the 3 ax is in the n th m ode( a 2 = i a l ) ,t h en t he t o t a l a ngu l a r m om en tum and i nva r ian t mas s a r e:

j = M 1 2 = i n n= 2 [anl _n n n

M = l a n l

s o t ha t

J = M 2 . 38)g /

We have a lead ing l inear Regge t ra jec tory n = 1) wi th s lope 1 in our un i t s . I f wechoose conv ent iona l un i t s the s lope is jus t c , o f course.

2 .3 . L o r e n t z c o v a r ia n c e p r o p e r t ie s

Having deve loped a canonica l fo rm al i sm for the m ot io n of the s t r ing in ligh t -conev a r ia b l es we cou ld p roce ed im med ia t e ly to t he q uan t i z a t i on o f t he sy s t em th roughthe cor respondence pr inc ip le :

i { Po i s son b r ack e t } -+ [ c o m m u ta to r ]


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122 P. Goddard et al . , Massless relat ivist ic s tr ing

H o w e v e r , i t is u s e f u l t o f i rs t a n a l y z e t h e c o v a r i a n c e p r o p e r t i e s o f o u r d y n a m i c a lv a r i a b le s u n d e r L o r e n t z t r a n s f o r m a t i o n s . T h i s is n o n - t r iv i a l s in c e w e h a v e g i v e n u p

m a n i f e s t L o r e n t z c o v a r i a n c e t h r o u g h e q s . ( 2 2 ) . T h e s e e q u a t i o n s w i ll n o l o n g e r h o l di n t h e f r a m e o f a d i f f e r e n t o b s e r v e r w h o s e x + c o o r d i n a t e w i ll b e

x + = A + v x V , ( 3 9 )

w h e r e A is t h e L o r e n t z t r a n s f o r m a t i o n b e t w e e n h i s f r a m e a n d t h e o r ig i n a l o n e ; ~ ' +w i ll in g e n e ra l d e p e n d o n b o t h r a n d o , a n d w i ll n o t s a t is f y t h e t r a n s f o r m e d e q . ( 2 2 b )

~ '+ =~ 2 P + r.

N e v e r t h e l e s s w e c a n a l w a y s c h a n g e t h e p a r a m e t r i z a t i o n b y 7 = 7 ( o , r ) , ~ = ~ ' ( o , r )s o th a t , i n t e r m s o f t h e n e w v a r i a b l e s

~ ' + ( ~ ,' ~ ) = 2 P + r. ( 4 0 )

T h e p o s s i b i l it y o f r e d e f i n i n g r s o t h a t ( 4 0 ) is t r u e d e p e n d s o n t h e f a c t t h a t , a s af u n c t i o n o f o a n d r, ~' + = A +x uV s a t is f ie s t h e w a v e e q u a t i o n a s w e l l a s t h e b o u n d a r yc o n d i t i o n ~ ' + / ~ o = 0 . We sh a ll s t u d y t h i s p r o c e d u r e i n d e t a il f o r a n i n fi n i te s i m a lL o r e n t z t r a n s f o r m a t i o n :

~ U ( o , 7 ) = x U ( a , 7 ) + e M U V x v ( a , 7 ) , fft a = p ~ + e ~ U U p v ( 4 1 )

E q . ( 4 0 ) d e m a n d s

7 ( o , r ) = r + --~ -e ~ x V ( o , r ) - e T ~ 4 + P . ( 4 2 )2 P v v

T h e n ~ m u s t b e c o m p u t e d f r o m ~ o ' /~ r = a T / ~ o , a ~ ' / O a = ~ 7 / O r. We o b t a i na

+ C f i - - v P= o ~ -p+ d o ~ x ( o , r ) - e o ~ l + P v . ( 4 3 )U

T h e n e w o b s e r v e r w i ll t h e r e f o r e d e s c r i b e t h e s y s t e m t h r o u g h t h e f u n c t i o n s

7 ( 0 7 ) , 7 5 ) ,

s o t h a t t h e d i f f e r e n c e i n d e s c r i p t i o n s w i ll b e g i v e n b y :

5 x = ~ ' ( c r ( ~ , 7 - ), r ( ~ ', 7 ) ) - x ( ~ ', 7 / ( 4 4 )o

( 'u { f d o 12 v _ 2 o p v } + 2 u { x u _ 2 r P V} ) ] ,-- 7 ) - - 2 / + i x 0

I n t e r m s o f t he n o r m a l m o d e s w e h a v e:

5 a ~ = e l m I r a v V m 2 m s n U n i .rn a S n = _ ~ i n ~ / ~ q o a

n~sO


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P. Goddard et al., Massless relativistic string 123

n n

6qUo = e MUvqo - i iqVo a ,n

a ? l =o

n ~ O

6 a u = e ~ U a v . 4 5 )0 V 0

We se e t h a t t h e d e s c r i p t i o n s o f t h e e v o l u t i o n o f t h e s t r in g j n t w o d i f f e r e n t f r a m e sa re n o t r e la t e d b y a m e r e L o r e n t z t r a n s f o r m a t i o n ; t h e L o r e n t z t r a n s f o r m a t i o n in -d u c e d b y t h e c h a n g e o f fr a m e i s f o l lo w e d b y r e p a r a m e t r iz a t i o n , b y w h i c h t h e n e wo b s e r v e r a d j a st s t h e d e s c r i p t i o n t o h is o w n l ig h t - c o n e p a r a m e t e r s . O f c o u r s e , eq s . 4 4 )p r o v i d e a r e p r e s e n t a t i o n o f t h e L o r e n t z g r o u p an o n - l i n e a r and n o n - t r i v i a l o n e )w i t h i n t h e s p a ce o f f u n c t i o n sx ( o , r ) , 2 ( 0 , r )a n d t h e n u m b e r s q o a n d a + .

I f o u r p r e s c r i p t i o n f o r P o is s o n b r a c k e t s is c o n s i s t e n t , t h e n t h e t r a n s f o r m a t i o n s4 4 ) o r 4 5 ) s h o u l d b e g e n e r a t e d b y t a k i n g t h e P o i s s o n b r a c k e t w i t h t h e t o t a l an g u l a r

m o m e n t u m 3 7 ) . We s ha ll s h o w in a p p e n d i x B t h a t w h e n t h e e x p r es s io n s 3 0 ) , 3 I )f o r q + , a n , q o , a n a re s u b s t it u t e d i n t o t h e e x p r es s io n f o r t h e a n g u la r m o m e n t u m

i ~lUVM u vg e n e r a t e s t h e t r a n s f o r m a t i o n 4 4 ) , i. e. ,3 7 ) , t h e n t h e e x p re s s io n M = - 3

6 x U ( o , r ) = e ( M , x U ( o ,r ) } . 46)

I t t h e n f o l l o w s t h a tMUV o b e y s t h e L o r e n t z a l g e b r a :

( M uv, M oO } = (gUOMV - g~OMU - gUM V + g~ M U ) . 4 7 )

3 . Q u a n t u m m e c h a n i c s o f t h e l i g h t st ri n g

3 .1 . G e n e r a l c o n s i d e r a t i o n s

I n c o n s t r u c t i n g t h e q u a n t u m t h e o r y o f th e s t ri n g w e r e l y o n t h e c o r r e s p o n d e n c epr inc ip le . Tha t i s, we regard the dy nam ica l va r iab les as ope ra to r s w hose equ a l - rc o m m u t a t o r s a r e o b t a i n e d b y t h e r u l e

i P o i s so n b r a c k e t ) ~ [ c o m m u t a t o r ] . 4 8 )

T h e p r e s e n c e o f c o n s t r a i n t s a m o n g t h e c la s si ca l q u a n t i t i e sx U ( o , r ) a n d W ( o , r )r e q u i re s s o m e c a r e in t h e c o n s t r u c t i o n o f t h e q u a n t u m t h e o r y . We h a v e se e n th a t ,a l ready a t the c lass ica l l eve l , we can e i the r e l imina te some of the va r iab les f rom thee q u a t i o n s o f m o t i o n , a n d i m p o s e c a n o n i c a l P o is s o n b ra c k e t s o n t h e r e m a i n in g o n e s ,o r i n t r o d u c e c a n o n i c a l P o i s s o n b r a c k e t s f o r a l lxU a n d P u , b u t i m p o s e t h e c o n s t r a i n t si n t h e w e a k s e n se o f D i ra c , n a m e l y, a f t e r a ll t h e P o i s s o n b r a c k e t s h a v e b e e n c o m -p u t e d .

C o r r e s p o n d i n g l y, a t th e q u a n t u m l ev e l w e c a n a ss u m e c a n o n i c al c o m m u t a t i o nr e l a t io n s f o r t h e t r a n s v er s e o p e r a t o r sx i a n d P i , p l u s t h e a d d i t i o n a l o p e r a t o r s q o -

a n d P + , i f w e c h o o s e t o e l i m i n at e t h e r e d u n d a n t v a ri ab le s . T h e o t h e r o p e r a t o r s x _ ,P _ , x + a n d P + w i ll b e e x p r e s s e d i n t e r m s o f t h e se . I n p a r t i c u l a r, a c c o r d i n g t o t h e


16/27

124 P. G od da r d e t aL , M ass l e s s r e l a t i v is t i c s t r i ng

c h o i c e o f p a r a m e t r i z a t i o n , x + w i ll b e g iv e n b y t h e t o t a l m o m e n t u m i n t h e + d i re c t i o nt i m e s a c - n u m b e r.

A l t e r n a t i v e l y, w e c a n q u a n t i z e c o v a r i a n t lyi x u , p v ] = i g U V 8 ( o - o ) , ( 4 9 )

a n d i m p o s e e q s. ( 2 1 ) o n l y f o r m a t r i x e l e m e n t s b e t w e e np h y s i c a l sta tes:

i ~x X 2( J l -~ -o P [ ~ 2 ) = ( ~ 1 1 p 2 +(2r r )2 I t~ 2 ) = 0 . (50 )

T h i s la s t p r o c e d u r e i s a n a l o g o u s t o t h e G u p t a - B l e u l e r p r e s c r ip t i o n f o r i m p o s i n g t h eL o r e n t z g a u g e i n q u a n t u m e l e c t r o d y n a m i c s , w h e r e a s t h e o t h e r o n e r e s e m b l e s t h em e t h o d b y w h i c h e l e c t r o d y n a m i c s w a s f i rs t q u a n ti z e d .

O f c o u r s e , i n t h e f u l l y c o v a r i a n t m e t h o d o f q u a n t i z a t i o n i t is d i f f ic u l t t o a s c r ib ea m e a n in g t o t h e t ime v a r i ab l e a l ong t he s t ri ng Xo(O , r ) , o r t o t he equ iva l en t l i g h t-c o n e v a r ia b l e x + ( a , r ) w h i c h b e c o m e o p e r a t o r s . I n o u r o p i n i o n , t h is d i f f i c u l t y h a sb e e n t he m a in ob s t ac l e i n v i s u a li z ing t he q ua n tu m s t r ing i n t rod uc ed i n r e f. [1 ] a s i m-b e d d e d i n o r d i n a r y s p a c e - t im e . T h e d i f f ic u l t y is n o t p r e s e n t i f w e q u a n t i z e o n l y t h ei n d e p e n d e n t v a ri a bl e s.

H ow e ve r, as w e s h a ll s ee , i t is pos s ib l e , un de r c e r t a i n con d i t i ons , t o e l im ina t e x +a s a d y na mica l va r i a b l e a l so i n t h e fu l l y cova r i an t f o rma l i sm , by r equ i r i ng t ha tx+ = 2P +T a s a w e ak equ a t i on , i . e . ,

( ~b 11 (x+ - 2 P + r ) l t~ 2 =0 , ( 5 1 )w he n I~b 1) and I~b 2) are ph ys ica l s ta tes .

We sh a l l de s c r i be f i r s t t h e no n - cova r i an t quan t i z a t i on , and t hen d i s cus s t he o th e rp o s s i b il i ty. I n b o t h c a se s , t h e t r a n s i t i o n f r o m t h e c la s si ca l to t h e q u a n t u m t h e o r yi n v o lv e s a m b i g u i ti e s r e l a ti v e t o t h e o r d e r o f o p e r a t o r s i n th e d e f i n i t i o n o f s o m eq u an t u m v a r i ab l e s . We sha l l show how the se amb igu i t i e s a r e r e so lved , i n t he non -c o v a r i a n t f o r m a l i s m , b y t h e r e q u i r e m e n t t h a t t h e q u a n t u m t h e o r y b e i n f a c t L o r e n t zc o v a r i a n t ; w h e r e a s t h e y a r e r e s o lv e d b y t h e r e q u i r e m e n t t h a t e q . (5 1 ) b e c o m p a t i b l e ,i n t h e c o v a r i a n t q u a n t i z a t i o n p r o c e d u r e .

3 . 2 . N o n - c o v a r i a n t q u a n t i z a t i o n

We h a v e s e e n in s u b s e c t . 2 . 2 . t h a t w e c a n c o n s i d e r a s in d e p e n d e n t d y n a m i c a l v ar i-a b le s x , P, q o a n d P + , a n d e x p r e s s a ll o t h e r s i n te r m s o f t h e se . We t h e n u s e t h e c a n o ni c a l q u a n t i z a t i o n p r o c e d u r e o n t h e s e i n d e p e n d e n t d y n a m i c a l v a r i a b l e s :

[ x i ( o ) , p/(cr ' ) ] = i 6 i J~ ) (o -o ), [ x i ( o ) ,x/ (o ) ] = [ ] i (o ) , ]o / (o ' ) ] = 0 ,

[ q o ' e + ] = - i , [ q o x i ]= [q o ' p i ] = [S + ,x i] = [p+ , p i ] = O . ( 52a )


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I t w i ll b e m u c h c l e ar e r t o w o r k w i t h n o r m a l m o d e s s o t h a t w e d o n o t h a v e t o w o r r ya b o u t c o n t i n u o u s i n d i ce s . I n t e r m s o f th e s e t h e q u a n t u m c o n d i t i o n s a re :

[ain, a ~ ] = n 6 n ; rn ~ i] , [q io , a /o ] = - i x / ~ 6 i J , [ q io , q ]o ] = 0 ,

[ q o , a + o ] = - i x / ~ , [ q o , a i n ] = [ q o , q io ] = [a ~ ,a in ] = [a + o , q i o ] = O .( 5 2 b )

( We s h a l l r e g a r da n n > 0 a s a n n i h i l a t i o n o p e r a t o r s ; a n =a_ n a r e t h e n c r e a t i o n o p e r -+

a t o r s f o r n > 0 . ) T h e d e p e n d e n t o p e r a t o r sa n, qo a r e g i v e n i n t e r m s o f t h e s e b ye q s . ( 3 0 ) a n d ( 3 1 ) , b u t n o w w e h a v e a n a m b i g u i t y in o r d e r i n g t h e o p e r a t o r s . I n f a c t ,t h e o n l y p l a c e th i s a m b i g u i t y o c c u r s i s i n th e o p e r a t o ra o , s i n c e th i s o n e i n v o l v e sp r o d u c t s o f o p e r a to r s w h i c h d o n o t c o m m u t e . F o r t h e m o m e n t w e s h al l l ea v e t h isa m b i g u i t y o p e n a n d w r i t e

_ 1 [Lo_ l 5 3 )a O = ~a

o

w h e r e b y L o w e m e a n t h en o r m a l o r d e r e d e x p r e s s i o n .T h e p h y s i c a l m e a n i n g o f a oc a n b e s ee n f ro m t h e q u a n t u m m e c h a n i c a l a n a lo g u e o f ( 3 5 ) w h i c h b e c o m e s

o0

M 2 = L o _ a o _ P 2 = ~ a + . a n _ s o ( 5 4 )n = l

s o t h a t ( - a o ) i s j u s t t h e ( m a s s ) 2 o f t h e g r o u n d s t a te . We d e n o t e b y 10, k ) th e g r o u n d

s t a t e v e c t o r w i t h m o m e n t u m k . T h e f i rs t e x c i t e d s t a tea ~ i l o , k )has M 2 = 1 - a o . O nt h e o t h e r h a n d i t h a s o n l y t r a n s v e rs e d e g r e e s o f f r e e d o m , s o i f t h e t h e o r y i s t o b eL o r e n t z c o v a r i a n t t h i s s t a t e m u s t b e m a s s l e s s , i . e. , a o = 1 . We s h a l l se e t h i s r e s u l ta g a in in o u r d e t a i l e d s t u d y o f t h e L o r e n t z c o v a r i a n c e o f o u r t h e o r y, t o w h i c h w en o w t u r n .

I n w r it i n g d o w n t h e a n g u la r m o m e n t u m o p e r a t o r s , w h i c h g e n e r a te L o r e n t z t r a ns -f o r m a t i o n s , w e m u s t a g a in be c a r e fu l a b o u t h o w w e o r d e r t h e o p e r a t o r s w h i c h o c c u ri n p r o d u c t s . F o r a s t a r t w e s y m m e t r i z e t h e c l a s si c a l e x p r e s s i o n i n x a n d P t o e n s u r et h a t M U Vi s a H e r m i t a n o p e r a t o r , s o w e w r i t e :

1 '~m u v = ~ f d o ( x ~ P v + p V xU - x U p u - p U x V ) . 55)0

W o r k i n g t h e s e o u t in n o r m a l m o d e s s e p a r a t e l y fo r t h e v a ri o u s c h o i c e s o f a n d v, w eh a v e :

M f f :~ l (q io J o _ q jo a io )_ i ~ a n a ln - a I n a ln, ( 5 6 a )n = l n

~+ 1 1 - i(Lo- oW o - qo .oi - = - M - i =2X /~ t_ a o o

- L ~ a L n L n - L -n d n(x+ n = l n

0

( 5 6 b )


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126 P. Go ddar d e t a l. , Ma ss less r e la t iv i s t i c s t r ing

M i + = _ M + i = 1 i + 5 6 c)X ~ q o a o ,

M + _ = _ M _ + _ 1 + + _2XT/~ qoa o + a o q o ) . 5 6 d )I t is n o t h a r d t o c h e c k , u s i ng th e k n o w n a l g e b ra o f t h e o p e r a t o r s o c c u r r i n g i n 5 6 ) ,t h a t a c h a n g e o f t h e o r d e r i n g o f a n y o f t h e o p e r a t o r s l e ad s e i th e r t o n o c h a n g e i nt h e e x p r e s s i o n o r t o a n e x p r e s s i o n w h i c h i s n o l o n g e r H e r m i t i a n , s o t h a t h e r m i t i c i t ya l o n e f i x e s t h e o r d e r i n g o f t h e o p e r a t o r s .

I f w e c o n s i d e r f o r e x a m p l e t h e s t a t e w i t h o n l y n t h m o d e e x c i t a t i o n s a n d d e f i n i t ec o m p o n e n t s o f a n g u la r m o m e n t u m i n t h e 3 d i r e c ti o n :

I ~ ) = a ; 1+ i a + 2 ) K l o >

t h e n

1 +1 +2 xt~ l _i a 2 ) l q J n. _J3 I t~n) =M l21~n > ~( a n + i= a n ) tu n

- I ~ n ) = + [ ~ n ) .

We see tha t the re i s a l ead ing Regge t ra je c to r y n = 1 ) g iven by

s ( s ) = s + .

We s ti ll m u s t p r o v e t h a t t h e e x p r e s si o n s 5 6 ) f o r m a r e p r e s e n t a t i o n o f t h e L o r e n t z

g r o u p . We h a v e a l r e a d y s e e n t h e r e is n o h o p e u n le s s s o = 1 b u t l e t u s c o m p u t e t h ea l g eb r a o f t h e e x p r e s s i o n s 5 6 ) f i rs t w i t h s o a r b i tr a r y. T h e a l g eb r a o f M 6 a m o n gthemse lves i s a l l r igh t s ince they invo lve no non-canonica l express ions . However,t h e e x p r e s s i o n f o r M i i n v o lv e s t h e n o n - c a n o n i c a l o p e r a t o r sL n which have an a lge -bra s imi la r bu t no t qu i t e the sam e as the c lass ica l va r iab les :

( n 3 n ) 5 7 a )[ L n , L m ] = ( n - m ) L n + m + i 5 ( O -2 ) 8 n , _ m - ,

[ L n , a i ] = - m a / r e + n , 5 7 b )

[q i , L . ] iV ~a in , 57c)

w h e r e D i s t h e d i m e n s i o n o f s p a ce t i m e . T h e a p p e a r a n c e o f t h e s e c o n d t e r m i n 5 7 a )is a p u r e l y q u a n t u m m e c h a n i c a l e f f e c t w h i c h is d u e t o t h e f a c t t h a t t h e o p e r a t o r L ois i n n o r m a l o r d e r e d f o r m a n d o n c o m m u t i n g s a yL n w i t h L _ n o n e g e t s t e r m s w h i c ha re n o t n o r m a l l y o r d e re d . T h e t e r m is p r o p o r t i o n a l t o D - 2 , w h i c h is t h e t o t a l n u m -b e r o f tr a n sv e r se o p e r a t o r s o f a g i v en m o d e , b e c a u s e e a c h c o m p o n e n t c l e a r l y c o n -t r i b u t e s t h e s a m e a m o u n t .

I n a p p e n d i x B w e sh al l c o m p u t e[ M i - , M / - ] w h i c h s h o u l d b e z e r o if o u r t h e o r yi s Loren tz covar ian t . In genera l i t i s non-ze ro , in fac t :

a i i[ M - , M i - ] - 4 a1 20 ~ [ m 1 - - z -~ D --2 ) + 1 1 D _ 2 ) _ a o ) ] m a l m _ a J m a m ) .rn=l m

58)


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P. Godda rd et al., Ma ssless relativistic string 127

F o r a r b i t r a r y v a l u e s o f D a n d a o , t h e n , t h e t h e o r y i s n o t c o v a r i a n t . F o r t h e p a r t i c u -lar values

Oeo = 1 ,

D = 2 6 , ( 5 9 )

a n d o n l y f o r t h e se v a l u e s d o e s[M - , M / -] = 0 . N o n e o f th e o t h e r c o m m u t a t o r sa m o n g t h e e x p r e s s i o n s ( 5 6 ) g i ve t r o u b l e a s c a n b e e a s il y se e n b y d i r e c t c o m p u t a -t i o n . T h u s t h e q u a n t u m m e c h a n i c s g i v en b y t h e c o r r e s p o n d e n c e p r i n c ip l e i s c o n s is -t e n t o n l y w h e n t h e i n t e r c e p t o f t h e l e a d in g t r a j e c t o r y a n d t h e d i m e n s i o n o f s p a c et i m e a r e q u a n t i z e d a c c o r d i n g t o ( 5 9 ) . A p a r t i c u l a r c o n s e q u e n c e o f ( 5 9 ) i s t h a t t h eg r o u n d s t a t e i s a t a c h y o n , M 2 = - 1 .

A l l t h e s e r e su l t s h a v e b e e n o b t a i n e d i n r ef . [ 8 ] w h e r e w e s t u d ie d t h e L o r e n t zc o v a r i a n c e o f p h y s i c a l s t a te s i n t h e d u a l r e s o n a n c e m o d e l . I n t h a t p a p e r w e c o n -s t r u c te d t h e g e n e r a t o r s o f O ( 2 5 ) , ( th e l it tl e g r o u p o f t h e t o t a l m o m e n t u m o f t h es t a t e ) , o n t h e t r a n s v e r s e p h y s i c a l s t a t e s i n 2 6 s p a c e - t i m e d i m e n s i o n s c o n s t r u c t e d b yD e l G i u d i c e , D i Ve c c h i a a n d F u b i n i [ 7 ] w h i c h , in f a c t , c o i n c i d e w i t h w h a t o n ew o u l d g e t f r o m t h e e x p r e s s i o n s f o r t h e s e g e n e r a t o r s c o n s t r u c t e d f r o m ( 5 6 ) . Wef o u n d , f r o m e s s e n t i a l l y t h e s a m e a l g e b r a u s e d h e r e , t h a t t h e c o n s t r u c t i o n o n l yw o r k e d f o r D = 2 6 . H o w e v e r, th e f o r m t h e g e n e r a t o r s t o o k a n d t h e i r a c t i o n o n t h et r a n s v e r s e s t a t e s w e r e m y s t e r i o u s a n d w i t h t h e p r e s e n t w o r k w e s e e h o w t h e y c o m eo u t n a t u r a l l y i n t e r m s o f a p a r t i c u l a r p a r a m e t r i z a t i o n o f t h e r e l a ti v is t ic s t r in g . We

a ls o h a ve c o n s t r u c t e d t h e r e m a i n i ng g e n e r a to r s o f 0 ( 2 5 , 1 ) w h i c h w e h a d n o t d o n ei n o u r p r e v i o u s w o r k .

3 .3 . T h e c o v a r i a n t q u a n t i z a t i o n

T h e c o v a r i a n t q u a n t i z a t i o n is p e r f o r m e d b y a s su m i n g

[ x ~ (o ) , p ~ ( o ) ] - = i g ~ a ( o - o )equ l r

a n d i m p o s i n g e q . ( 5 0 ) o n t h e p h y s i c a l s t at e s.R e s o l v i n g e q s . ( 4 9 ) a n d ( 5 0 ) i n t o n o r m a l m o d e s , w e s ee t h a t t h e y a r e e q u i v a le n t

to

[an arm ] = gu~6 n , - m lq~o ao ] = / x / 2g~V , (60 )

6 11 L w l ~ 2 ) = O , N : / : 0 , 6 1 a)

( 41 ILol~b2) = C~o(~ 1 hb 2) , (6 1b )

w h e r e

L N = ~ : a - l a l + N: (62 )


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128 P. Godda rd e t aL, Mass less re la t iv i s tic s t r ing

The operators L N satisfy the algebra

[LN, rM l = X - LN+ M + 1 D 6 N _ M ( N 3 - i V ) , (63a)

[L N , aUn ] = - n aUn+N (63b)

[qUo, L N ] = i x / 2 a ~ v . (63c)

We therefore impose the subsidiary condi tion s *

L N [ ~ ) = 0 , N > 0 , (64a)

Lohb) = ao [~ ). (64b)

Because of the indefi nite metric in (60) we are not sure that the so lutions of eq. (64)contain no negative norm state. The solutions of these equations have been studiedby Brower [5] and two of us [6]. The result s are that for a o = 1 and 1 ~< D ~< 26and for a o < 1 and 1 ~< D ~< 25 there are no negative norm sol utio ns to (64 a) and(64b). For the dual resonance model in which the physical states satisfy eq. (64)wit h a o = 1 these results have the significance that there are no negative wid th res-onan ces (ghosts) coupl ing to physica l states whe n 1 ~< D ~< 26.

So, if we only require the positiveness of the physical subspace we can relax thecon dit ion s D = 26, a o = 1. The price we pay, as can be seen from refs. [5, 6] is thatthe nu mber of indep endent degrees of freedom in the q uant um system is larger than

in the classical system. The qua nt um system no longer reflects that last degree o ffreedom allowed after impos ing (19a) in the classical system. An anal ogy from QEDis helpful here. In the covariant quantization the Lorentz gauge condition becomesin mo me nt um space ( on shell k 2 = 0)

~ a ~ : ) l~ > = 0 .

Now, this equation has three solutions:

k a+10), a~[ 0),

where a2 are the transverse components. However, k. a+10) has zero norm:

(0[a . h a + . k[0)cc k 2 = 0,

and decouples from the theory. The existence of such zero norm states reflects anarbitrariness in the de fini tion of physical states, nam ely the state

The operator L o which appears in eq. (64b) can be considered the Hamiltonian of the system,so that, in the Heisenberg picture,

i3c~= Ix WLo] .

This identification specifies a particular parametrization, and corresponds to the constraints(19a) in the classical formulation. The fact that the choice of a parametrization (of a gauge

for the string) is equivalent to the specification of a particular form for the Hamiltonian with-in a class of equivalent ones is illustrated in ref. [ 11 ] and, for our case, also in appendix A.


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P. Godda rd e t ak, M assless relativistic string 129

I@ ') = I ~ > + X k ' a + 1 0 > ,

w h e r e I f ) i s p h y s i c a l i s a l s o p h y s i c a l a n d f u r t h e r h a s i d e n t i c a l c o u p l i n g s t o I f ). E f -f e c t i v e ly t h e o n l y p h y s i c a l st a t e s w h i c h c o u p l e i n t h e t h e o r y a r e th e f i n i t e n o r mo n e s a n d t h e r e a r e o n l y t w o s u c h s t a t e s s o th e p h o t o n is re a l ly tr a n s v e rs e .

I n o u r t h e o r y a s im i l ar p h e n o m e n o n m a y h a p p e n . I t m a y b e t h a t t h e r e a r e z e r on o r m s o l u t i o n s t o ( 6 4 ) . I f th e r e a r e t h e n t h e s t a t e s o f p h y s i c a l i n t e r e s t , t h e f i n it en o r m e d s t a t e s , a r e n o t u n i q u e l y d e t e r m i n e d . I t is t h is a m b i g u i t y w h i c h r e f l e c t s t h ef u r th e r f r e e d o m o f p a r a m e t r i z a t i o n a l l o w e d a f te r ( 1 9 a ) a re i m p o s e d . We m a y t h e nf o r m u l a t e t h e p o s s i b i l i t y o f i m p o s i n g ( 1 9 d ) a s f o l l o w s : w e s h o u l d a l w a y s b e a b le t oc h o o s e t h e f i n i te n o r m e d s o l u t i o n s o f ( 6 4 ) i n s u c h a w a y t h a t

(~ 1F I n x I ~ F ) =2 ( ~ F ln P I ~ F ) r , ( 6 5 )

f o r a n y l i g h t- l ik e v e c t o rn ( n 2 = 0 ). I n t e r m s o f n o r m a l m o d e s , t h i s m e a n s

( ~ F I n a p l~ 2 F ) = 0 , 1 4 = 0 . ( 6 6 )

I n o t h e r w o r d s I ~ F ) m u s t s a t i s fy

n . a l l ~ F )= 0 , l > 0 , ( 6 7)

i n a d d i t i o n t o

L I I ~ F ) = 0 , l > 0 .

I n r e f. [ 6 ] i t h a s b e e n s h o w n t h a t a l l t h e s o l u t i o n s o f ( 6 7 ) a r e p r e c i s e l y th e s t a t e sc o n s t r u c t e d b y D D F f o r n = k . T h e r e s u l t s o f r e fs . [ 5, 6] s h o w t h a t t h e s e s t a t e s s p a nt h e f i n i t e n o r m s u b s p a c e o f t h e s o l u t i o n s o f ( 6 4 ) i f a n d o n l y i f D = 2 6 a n d a o = 1 .

F o r D < 2 6 a n d cto = 1 e f f e c t i v e l y o n l y t h e m o d e s a 1 a r e t r a n s v e r s e a n d a l l o f t h eo t h e r s h av e t h r e e i n d e p e n d e n t c o m p o n e n t s . F o r D < 2 5 a n d a o < 1 a ll o f t h e m o d e se f f e c ti v e l y h a v e t h r e e i n d e p e n d e n t c o m p o n e n t s . T h e r e a re s o m e a n a lo g i es o f th em o r e g e n e r a l s o l u t io n s o f ( 6 4 ) a n d t h e H i g gs m e c h a n i s m f o r o r d i n a r y g a u g e t h e o ri e s .B y m e a n s o f th e H i g g s m e c h a n i s m o n e i s a b l e to g i v e a m a s s t o t h e p h o t o n i n s u c h aw a y a s t o p r e s e r v e e n o u g h o f t h e g a u g e in v a r i a n c e o f th e t h e o r y t o e n s u r e t h e a b -s e n ce o f g h o s t s . O f c o u r s e , i n th e p r o c e s s o n e m u s t a l so s u p p l y a t h i r d d e g r e e o f

f r e e d o m b e c a u s e a m a s s iv e v e c t o r p a r t i c l e c a n n o t b e t ra n s v e r se . S i m i l a r l y, a s o n ed e p a r t s f r o m D = 2 6 , a o = 1 , m a i n t a i n i n g (6 4 a ) , o n e m u s t i n t r o d u c e i n t o t h e t h e o r ye x t r a d e g r e e s o f f r e e d o m i n a w e l l - d e f i n e d w a y. O f c o u r s e , at p r e s e n t w e o n l y h a v ea n S - m a t r i x f o r t h e c a se a o = 1 , t h e c o n v e n t i o n a l d u a l r e s o n a n c e m o d e l . T h e p o s s i -b i l i t y re m a i n s o p e n o f c o n s t r u c t i n g a n S - m a t r i x f o r d i f f e r e n t v a l u e s o f a o .

We w o u l d l i k e to t h a n k K o r k u t B a r d a k q i f o r h e l p f u l d i s c u s s i o n s e s p e c i a l l y w i t hr e g a rd t o q u a n t i z a t i o n o n l ig h t - li k e s u r fa c e s . We h av e a l s o e n j o y e d s t i m u l a t i n g d i s-c u s s s io n s w i t h S e r g i o F u b i n i a n d m e m b e r s o f t h e T h e o r e t i c a l S t u d y D i v i s io n a tC E R N . W e th a n k D a n i e l e A m a t i a n d D a v i d O l i ve f o r c r i t i c a ll y r e a d i n g t h e m a n u -

s c r i p t .


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130 P. G oddard et al., Ma ssless relativistic string

A p p e n d N A

I n t h is a p p e n d i x w e w o u l d l ik e t o d e s c r ib e a m e t h o d f o r t r e a t i n g t h e v a r i a t io n a lp r o b l e m i n t e r m s o f th e p h a s e s p a c e v a ri a b l e sx**, p , r a t h e r t h a n i n t e r m sof x**,2 * . T h e a c t i o n c a n b e w r i t t e n s i m p l y

S = f d o d r 2 . P. ( A . I )

H o w e v e r , w e c a n n o t v a r y x a n d p i n d e p e n d e n t l y b e c a u s e o f th e p h a s e s p a c e co n -s t r a in t s ( 2 l ) . We , t h e r e f o r e , i n t r o d u c e t w o L a g r a n g e m u l t i p l ie r s X l , ) ,2 , d e f in e

( 2 rr ) 2 - ~ X2 x - P ] . ( A . 2 )

a n d r e q u i r e S ' t o b e s t a t i o n a r y u n d e r i n d e p e n d e n t v a r i a t i o n so f x , p, ~-1 and X2.Va r y i n g X1 a n d ~ ' 2 w e g e t t h e c o n s t r a i n t s

p 2 + x ' 2 = 0 , ( A . 3 )( 2 n ) 2

x P = 0 . ( A . 4)

Va r y i n g P w e o b t a i n

1 t2 ' * - X I t ) * * - g XzX = 0 . ( A .5 )

F i n a l l y v a r y i n g x , w e o b t a i n

0 Xl

X1 1

x + ~ - X 2 P = 0 , a t t he e n d s;( 2 r 0 2

~'1 a n d X 2 a r e d e t e r m i n e d b y s u b s t i t u t i n g P a s d e t e r m i n e d f r o m ( A . 5 ) i n t o ( A . 3 )

a n d ( A . 4 ) . O n d o i n g t h i s , o n e f i n d s ( A . 5 ) a n d ( A . 6 ) b e c o m e

p _ L ( 2 , x ' )32 '

a a L a a L+ - 0 ,

a T a 2 O o a x '

a L0 a t t h e e n d s ,

Ox '

i n a g r e e m e n t w i t h ( 9 a ) a n d ( 9 b ) .

* We than k K . John son for an instructive conversation on this form ulation of the variationalp rob lem.


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P. Go ddard et al., Masslessrelativistic string 131

One v i r tue of form ula t ing the var ia tiona l problem th is way i s tha t w e can imposeour par t icu lar paramet r iza t ion choice (22)befo re f ind ing the s ta t ionary p oin ts of

S ' . Sub s t i tu t ing , we f indr2 *r p+

s = f d - fd o [ 2 ( e+ +/5 +r) P +rl 0

, p +- 2 . P - -~ X2 (x - - x . P ) ] .g

~'1 (2/~+ p - - p 2 (2/7)2)x'_~_

(A.7)

5 7 - x = 0 , x ' = 0 , a t

P+ ' = x ' 2P+- - x p , _ _ p - = p 2 + _ _

Tg I r

equa t ions

P + = 0 , x = 27r p , 2 = 2 7 r p - ,

x ,2

(21r)2 '

O = 0 7 r

in agreem ent wi th our p revious resu lt s (23) and (25) .Going over to the H amil tonian formal i sm f rom (A.2) we f ind a Ham il tonian

+ x ' 2 ~ + X 2 x P I ( A . 8)/4 =fdo I~Xlp 2 (2rr)2 ,

wh ere ) t 1 and X2 are a rb i t ra ry. Fo l lowing Dirac [11] one then takescanocicalPoisson brackets

{x (o ) , pv (o ' ) }= g U V 8 ( o -o ' ) , ( A . 9 )

and imposes the cons t ra in ts

P 2 + , x '2 = 0 , x ' . P = 0 , (A .1 0)

(2rr) 2o n l y af ter f ind ing Hami l ton ' s equa t ions o f mo t ion

2 u = {xU , H } , p u ={ pU, H '} .

The equa t ions o f mo t ion depend , o f cou r se , on the a rb i t r a ry func t ions X1 and X2ref lec t ing the fac t tha t there a re m an y fun c t ionsxU(o , r )represent ing the sames ta te o f mo t ion of the sys tem. I f one e l imina tes h 1 and X2 us ing (A.10) one re turnsto t he equa t ions o f m o t ion (9a ).

Di rac ' s qua nt iza t ion p rescr ip t ion is som ew hat d i ffe ren t f rom the one w e de-scr ibed in the tex t , eqs . (44) and (50) . He would impose the canonica l commuta t ionre la tions (40) and would requi re on phys ica l s ta tes thephase-space cons t ra in ts(A. 10)

Requ i r ing th a t S ' be s ta t ion ary un der a rb i t ra ry small var ia t ion inP + ( r ) , x - ( o , r ) ,P - ( o , r) , x(o , r) , P (o, r) ,X1 and X2, i t is s t raightforw ard algebra to verify the


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132 P. Godd ard e t aL, Massless relativistic string

( ~ l l ( P 2 + x 2 ] l ~ 2 )= O l o ( ~ l [~ 2) @ l [X P [ ~ 2 ) = O(21r)2]

His equat ions of mo t ion wo uld s till involve the a rb i t ra ry quan t i ties X1 and X2. Bychoos ing the H am il tonian to be L o we have res t r ic ted ourse lves to those par t icu larsolut ions for which X2 = 0 and ~1 = 1. These two restr ict ions enforce the ortho-norm al i ty condi t ions (15) in the w eak sense of Di rac .

Append ix B

In th is append ix we shal l ver i fy some equ at ions quo ted in the tex t . We turnf ir s t to eq . (46) . I t i s eas ies t to ver i fy tha t the no rmal m odes t ran sform c orrec t ly(45) . The dem onst ra t ion is s t ra ight forward and w e sha ll i llus t ra te the pro cedu re byconsidering the case where o nly.~ r + is non -zero. The n

M = i gi ;

M - is given by eq. (37):

l : i 1 a / + iM / - = ~ t q o ~ - ~ L o - q o - ~ = _ ~

nv~0Comput ing :

(M i- ,a i rn }=s iJ6 m ,o la L o

= 6 i / k m 1

a + a + n= _~o o

F r o m ( 4 5 )

e M '+am] = - a + moo

0n ~ O

i.e.,

6 a lm = e ( M , a lm } .

Againa i ix /~i o _ _ _

{ M i - , q i o } = ~ 7 o - a T + 5 ' I q o a n:_oo

n =O

( B . I )

a i n L - n (B.2)7

i m i ~ a / m + i ~

+ qa ao n : - - .n4:0

m i a j _ i m _ i 4 a i mn m - n - r - ~ q o a

0

am_nanJ qi \7i a i ] l

n X /2 ml .J

ia na l_ n

n

im ainaim n - i n S i j 6 n - m L - n

(B.3)


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P . God dard et al., Massless relativistic string 133

w h e r e a s

6 q o = e J + q o -

so we hav e

- O n a o )a + x / T ~ iqZo a ,

r l = ~ r l

o

n ~ O

8 q i = e { M , q ] }

F r o m ( B . 3 ) a n d ( B . 4 ) i t f o l lo w s t h a t

5 x J o , r ) = e { M , x / o , r ) ) .

i 1 im q i 1 Lm{ M - , L m } = a m 7 L o + - ~ O a+

0 0

i ~ i (n + m ) Lm _n a i - i n an+mi L_n+ - -

+ na n = - ~0

n ~ O

i m i l m k-~r~ qO~o Lm + ~oaiO Lm -aTo n = - -

n#~Oi

( M - , } - a + 2 ,

,Lm _ n a in

soi ~ - a i

i m ' q o _ m a m - n n { M i - , a m } = - - ~ - - a

v ~ a+ a+o n= -= nn4:0

w h e r e a s

t~ a m = e a + m \ n v, . q a0

n ~ O

SO again

8 a m = e { M ,a m } .

F i n a ll y, w e c o m p u t e

a i L nn -{ ~ - , q o } i 1 L o _ J n

= - - q o a +~ o o n = - o .

n - - /= O

(B .4 )

(B .5 )

(B .6 )

(B .7 )


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134 P. Go ddard et al., Massless relativistic string

a s c o m p a r e d t o

/ / + o )qo ~/I i+ N/~ ~ an a i

a \ n=_ ~

n ~ O

which ver i f i e s

q o = e { M ,q o } B . 8 )

( B . 8 ) a n d ( B . 7 ) i m p l y

6 x - ( o , r ) = e { M , x - ( o ,r ) } . ( B . 9 )

F i n a l l y t h a t

6 x+(o , r ) = e ( M , x + ( o , r ) } , ( B . 1 0 )

i s r ead i ly ve r i f ied . Th us , w e have p rov ed (4 6) wh en on l y M i+ 4= 0 . The o th e r casesa re i n f a c t m u c h e a si e r, s o w e le a ve i t f o r th e r e a d e r t o c o m p l e t e t h e p r o o f o f (4 6 ) .

F i n a l l y, w e m u s t p r o v e e q . ( 5 8 ) . T h e q u a n t u m m e c h a n i c a l e x p r e s s i o n f o r m / -is g i v en in ( 5 6 b ) . T h e p r o o f i s r a t h e r t e d i o u s a n d m o s t o f t h e a l g e b r a ha s b e e n d o n ei n o u r p r e v i o u s p a p e r [ 8 ] . T h e m a i n d i f f i c u l t y c o m e s i n w o r k i n g o u t t h e c o m m u t a -tor.

n=l t i 'm= l m ( B . 11 )

I f w e r e m o v e t h e z e r o m o d e s f r o m L rn w r i t in g

a 2Ln= ,+ao n , , 0 , Lo=To+ oT h e n i n r ef . [ 8 ] w e w o r k e d o u t t h e c o m m u t a t o r

L - Z - h e ; o -= n ' ~,=1 m

= - 2 ( ~ - ~ ( D - 2 ) ) ~ a ' - n a l n - a l - n a ' n ~ ' 2 ( ~ ( D - 2 ) - l ) ~ n ( a / _ _ n a i n - a i n a i n )n = l n n = l ( B . 1 2 )

We s h a l l n o t p r o v e t h i s f o r m u l a a g a i n , b u t o n l y r e m a r k t h a t t h e t e r m s w i t h f a c t o r s~ ( D - 2 ) a n d ~ ( D - 2 ) - 1 c o m e f ro m th e f ac t t ha t w h e n o n e p e r fo r m s t he c o m m u -t a t o r t h e r e s u l t h a s t e r m s w h i c h w o u l d c a n c e l e a c h o t h e r e x c e p t f o r t h e o r d e r i n g o ff a c t o r s. O n e t h e n u s e s t h e c o m m u t a t i o n r e l a ti o n s t o a r ra n g e t h e f a c t o r s in t h e s a m eo r d e r s o t h e y c a n c e l l e a v i n g o n l y t h e c o n t r i b u t i o n s f r o m t h e c o m m u t a t o r s . T h e c a l -c u l a t i o n i s t e d i o u s b e c a u s e o n e m u s t s e p a r a t e t h e p o s i t i v e m o d e s f r o m t h e n e g a t i v em o d e s a s i n ( B . 1 2 ) t o m a k e s u re w e k e e p f a c t o rs n o r m a l o r d e r e d t h r o u g h o u t . O t h e r -

wise one can eas i ly add and sub t rac t in f in i t ie s and g e t the w ron g resu l t. U s ing (B . 12)i t is t h e n o n l y s t r a i g h t f o r w a r d a l g eb r a t o v e r i f y e q . ( 5 8 ) .


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P. Goddard et aL Massless relativistic string 135

e f e r e n c e s

[ 1 ] Y. N a m b u , P r o c . I n t . C o n f . o n s y m m e t r i e s a n d q u a r k m o d e l s , Wa y n e S t a t e U n i v e r s i ty,1 9 6 9 ;H. Nie l sen , 15 th In t . Conf . on h igh energy phys ics , Kiev, 1970;L . S u s s k i n d , N u o v o C i m e n t o 6 9 1 9 7 0 ) 2 1 0 .

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d u a l r e s o n a n c e m o d e l , p r e p r i n t 1 9 7 1 ) ;O. Hara , P rogr. The or. Phys . 46 197 1) 15 49;T. G o t o , P r o g r. T h e o r. P h y s . 4 6 1 9 7 1 ) 1 5 6 0 ;J . M a n s o u r i a n d Y. N a m b u , P h y s . L e t t e r s 3 8 B 1 9 7 2 ) 3 7 5 ;M. Minam i , P rogr. Theo r. Phys . 48 197 2) 1308 .

[4 ] M.A. Vi rasoro , Phys . Rev. D1 197 0) 2933 .

[ 5 ] R . C . B r o w e r, S p e c t r u m g e n e r a t i n g al g e b ra a n d t h e n o - g h o s t t h e o r e m f o r t h e d u a l m o d e l ,M I T p r e p r i n t 1 9 7 2 ) .

[ 6 ] P. G o d d a r d a n d C. B . T h o r n , P h y s. L e t t e r s 4 0B 1 9 7 2 ) 2 3 5 .[7 ] E , De l Giud ice , P. Di Vecch ia and S . Fub in i , MIT prep r in t , CTP 209 1971 ) .[ 8 ] P. G o d d a r d , C . R e b b i a n d C .B . T h o r n , N u o v o C i m e n t o 1 2 A 1 9 7 0 ) 4 2 5 .[9 P.A.M. Di rac, P roc . Roy . Soc . A2 68 196 2) 57 .

[ 1 0 ] J . D o u g l a s, A n n . M a t h . 4 0 1 9 3 9 ) 2 0 5 .[11] P,A.M. Di rac , Can . J . Math . 2 195 0) 12 9 ; Proc . Roy . Soc. A2 46 195 8) 326 .

quantum dynmaics of a massless relativistic string- goddard

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