J. Kowalski-Glikman, J.W. van Holten, S. Aoyama and J. Lukierski- The Spinning Superparticle

8/3/2019 J. Kowalski-Glikman, J.W. van Holten, S. Aoyama and J. Lukierski- The Spinning Superparticle

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NATIONAAL INSTITUUT VOOR KERNFYSICA EN HOGE-ENERGIEFYSICASeptember 1987 N1KHSF-HI87-16

The Spinn ing SuperparticleJ . Kowalsk i -Gl ikman and J .W. van Hol ten,NIKHEF-H, P.O. box 41882, 1009 DB, AmsterdamThe Nether lands

S. Aoyama,Ins t i tu te fo r Theore t i ca l Phys ics ,Un ivers i ty o f Kar ls ruhe,P.O. Box 6380, 7500 Kar lsruheJ . Luk ie rsk i ,Inst i tu te of Theoret ica l Physics,Un ivers i ty o f Wroc law,ul. Cybulsk iego 36, Wroclaw, Poland

AbstractDoubly graded massless supersymmetric particle models with both woi'd-l ine local andspace-t ime global supersymmetry are considered. We describe the f i rst quantizat ion of themodel with four-dimensional space-t ime and N-1 world-l ine SUSY. Using th Gupta-Bleuler

method we obtain as the super wave-function a pair of D=4 chiral spinor superfiec>> with the on-shell spectrum containing scalar and vector multiplets.

aiKMEF&ecfig.H M s f U 4ift3, loo? bk AMsfER6AM


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The Spinning Superp article

T w o w a y s of i n c o r p o r a t i n g s p i n d e g r e e s off reedom into reparamctr izal ion invar iant par t ic leand s t r ing m ode l s have been cons ide red . Thefirst way, r ea l i zed in the sp inn ing pa r t i c l e orst r ing, consis ts in adding to the coord ina tes Xt he v e c t o r - l i k e G r a s s m a n n v a r i a b l e s* ( | 1 | - | 4 | ) . The l o c a l r e p a r a m c l r i z a t i o ni n v a r i a n c e is e x t e n d e d to l o c a l w o r l d - l i n e( w o r l d - s h e e t ) s u p c r s y m m e t r y (see | 1 ] , [ 4 ] ) . Th i smode l can be c o n s i d e r e d as a t heo ry of oned i m e n s i o n a l s u p e r g r a v i t y c o up le d to sca l a rm u l l i p l e t s ; its s t r i n g c o u n t e r p a r t , the NSRm o d e l is d e s c r i b e d in two d i m e n s i o n s by Ds c a l a r m u t t i p l c t s c o u p l e d to s u p e r c o n f o r m a lsuperg rav i ty (see [5|, [6|).

W e use the Gup t . -Bleu lc r qu an t i z a t ion , wh ichhas been shown to lead to a correct spect rum olt he f i r s t - quan t i zed theory for the superpar t i c l cmode l ((151; for a harmonic superspace t reatmentsee (16)) and we rest r ic t our c o n s i d e r a t i o n toD = 4 .The k inemat i cs of the f ree spinning superpar t icle is descr ibed in t e rms of sets of supcr spaccc o o r d i n a t e s ( X m ( - r ) , 9 (T ) , A m (x) , 0 (r)), w h e r e

m A A mX and o are c o m m u t i n g , 0 ana Aa n t i c o m m u l i n g ; m and A are vec to r and spinori n d i c e s in D - d i m e n s i o n a l fl a t s u p e r s p a c e . Asment ioned above we c h o o s e D=4, so m = 0, ...,3 ; A = 1, .. .. 4.T h e L a g r a n g i a n of the s y s t e m r e a d s asf o l l o w s ,

Space t ime super sym mct r i c mode l s arc ob ta inedb y c o n s i d e r i n g t r a j e c t o r i e s ( X (T) , 8 (x) ) ( super -particles) or (X ( T . O ) , 8 ( T , O ) ) (supcrstrings) intarget sup crsp acc [7] - [ 10] . The mass lcss supcr -particlc action is de te rmined by the SUSY invarl an l l i ne e l e m e n t 1 . Af te r Gu p ta -B leu lc r quan t izat ion the superpar t i c l e mode l is d e s c r i b e d inD 4 by ;i chiral supcr f ie ld . The s t r ing coun te rpj r t of the superpar t i c l c was found in [I I) .In this paper wc cons ide r a doub ly -g raded mode l ,w h i c h can be cons ide red e i the r as the g loba l ,s p a c e t i m e S U S Y e x t e n s i o n of the m a s s l c s ssp inn ing pa r t i c l e , or the local wor ld l ine SU SYe x t e n s i o n of the m a s s l c s s s u p c r p a r t i c l c . Ourmodel can be treated therefore as a d--2 > d= 1r educ t ion of a A-2 doub ly g r aded s igma-mode lwi th r ig id supcr spacc as a target formulatedrecent ly under the n a m e of local supcrsymmctry-squared s igma mode l in |12] and i n d e p e n d e n t l yby two of us 113], [14|.

L = l2 ( ^ o> ov, - A m A - y VA mo> ++ 2Y'"(>m - iy A m) + Vy y m 0 ) , (1)

w h e r e a = /dz a; 0. 9 are the Majorana sp inor s ,V and v arc the c inbe in and a o n e - d i m e n s i o n a lg rav i t ino r espec t ive ly , and

u> = X - i y *. ( 2 )

T h e l a g r a n g i a n (1) is i n v a r i a n t u n d e r thefollowing sets of t r ans fo rmat ions :(a) local one d i m e n s i o n a l s u p e r s y m m c t r y w ith

paramete r c(t)m m8o> = I(EA )' - 2ic0y B,

h\ = (o> - l\l/\ )C + E07 ,

0 = - vSV = 2iLV,V = c.

( 8 - v0 )E,

(3a)(3b)( 3 c )( 3 d )( 3 c )

The first supcrparticle model, presented in |7jdocs not provide first-class constraints leading afterquantization to the D irac equa tion, except after additionof the Wcss-Zumino term, proposed in (10J. which existfor N22.

(b , l oca l S i egc l i nvar i ancc (K (T) a n l i c o m m u li n g )

T h i s e x p r e s s i o n h o l d s for any D, in D 4, asw e l l as in D 10, th e l a s t t e r m v a n i s h e s forMajorat) o due to the Pier/ identi ty


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S^V = 4 I K ,8 r v = O,^X = i y ,p \ ,s,e" = Op*)*./ i 2 i - ft^ = - v KH.

w h e r ep = v ( to - vA + V y 0)

( 4 a )( 4 b )( 4 c )( 4 d )( 4 e )

( 5 )

( c ) t h e n e w b o s o n i c i n v a r i a n c c w i t h t h e go mmutiny . spinor parameter a (T)

V = - 4 V u o ,brfi = Opa) - 2o 6 a .

( 6 a )( 6 b )

I t p l a y s a s i m i l a r r o l e i n r e m o v i n g d e g r e e s o ff r e e d o m f r o m a s S i e g e l i n v a r i a n c e i n r e m o v i n gd o t f r o m 6 . In a d d i t i o n f r o m t h e c o m m u t a t o r so f t h e s y m m e t r i e s g i v e n a b o v e o n e f i n d s m o r el o c a l s y m m e t r i e s , w h i c h , h o w e v e r d o n o t r e m o v eany extra degrees o f f reedom.( d ) G l o b a l s p a c e - l i m e s u p c r s y m m c t r y w i t h p a r a

Ameter cS,.X = icy e ,

A Aft = .( 7 a )(7b)

T h e f i r s t s t e p i n p e r f o r m i n g t h e c a n o n i c a lq u a n t i z a t i o n o f t he s p i n n i n g s u p e r - p a r t i c l em o d e l i s t o i d e n t i f y t h e p h a s e s p a c e v a r i a b l e s .It i s c o n v e n i e n t t o u s e t w o c o m p o n e n t s p i n o rnotat ion for 6 :

" it m - ti ix m - ftH - r - - (H a ap H- H a ' ) . ( 8 )O n e f i n d s f or t h e c a n o n i c a l m o m e n t ac o r r e s p o n d i n g t o ( 1 )

( 9 b )

( 9 c )

.11.. = v ( w m ~ '^m += - ' / . / = "?

. i> - *

0 = n

i A = 7tV

0 = 7C ,0 = 7 1 *

(9d)( 9 c )( 9 0( 9 . )

Eq uat io n (9a) can be so l ve d for X , the equat ion( 9 b - g ) g i v e s r i s e t o t h e p r. m a r y c o n s t r a i n t s .O n e c a n e a s i l y f in d t h e c a n o n i c a l H a m i l t o m a no f t h e s y s t e m u s i n g m o m e n t a d e f i n i t i o n s g i v e nby form ulas (9a g ) and g e t s

H , . ,


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t = *alin" - O, ( 2 4 b )U s i n g ( 1 8 ) o n e s e e s t h a t ( 1 6 ) c a n b e r e w r i t t e na s

p J = . (1 9 )O n e c a n t h e n c h e c k t h a t t h e r e a r e n o o t h e rc o n s t r a i n t s , s i n c e t h e b r a c k e t s o f ( 1 2 ) , ( 1 3 ) a n d( 1 8 ) w i t h Hp l e a d o n l y t o c o n d i t i o n s o n t h eL a g r a n g e m u l t i p l i e r s p a n d f.

O n t h e m a s s - s h e l l ( p = 0 ) t h e se t o f c o n s t r a i n t s( G ^ G a) i s hal f f i r s t - and hal f s econd c las s .The f irst-class part is

( 2 0 . )( 2 0 0

Then one can sp l i t the fu l l s e t of 6 co ns tr ai nt s( 1 2 ) , ( 1 3 ) , ( 2 0 a , b ) i n t o t w o c o m p l e x c o n j u g a t e ds e t s

A , = ( G , D ) ,A ; - ( C , D ) . ( 2 1 )

N o w , i n t h e G u p t a B l e u l e r q u a n t i z a t i o n w e o n l yn e e d t o a s s u m e t h a t t h e m a t r i x e l e m e n t o f t h eo p e r a t o r s A . , c o r r e s p o n d i n g t o A / ( b e t w e e np h y s i c a l s t a t e s v a n i s h e s , s i n c e t h e m a t r i xe l e m e n t s o f A , t h e n a l s o b e c o m e z e r o .The o var iab le cons traints (9d) and (18) can a l sob e w r i t t e n in a t w o - c o m p o n e n t s W c y l f o r m a n ds p l i t i n t o t w o c o m p l e x - c o n j u g a t e d s e t s o f f i r s tc l a s s c o n s t r a i n t s :

( 2 2 a )( 2 2 b )

w h e r e i n a n y f r a m e i n w h i c h pm = (p, , p 0 ) , p mi s def ined as p m = ( p , , - p 0 ) . O b s e r v e t h a t f r o mF a = 0, C a = 0 , ( 2 4 a , b ) it f o l l o w s t ha t " = 0 ,Tt = 0 , w hic h can be made s tron g equat io n aftera p p l y i n g D i r a c b r a c k e t p r o c e d u r e . T h e r e f o r e ,

- P - * Pafter qu an t iza t ion , on e can treat 0 and re asz e r o o p e r a t o r s , in p a r t i c u l a r t h e w a v e f u n c t i o n- P - Pd e p e n d s n e i t h e r o n 7t n o r ; a n d ( 2 3 )c o n s i s t s o n l y o f f i r s t c l a s s c o n s t r a i n t s :

B = (Tt 0). ( 2 5 )Thus we end up wi th the fu l l s e t of f i r s t -c las sc o n s t r a i n t s t o b e u s e d i n Q u a n t i z a t i o n

Tt o,/ - o,p A = 0,G a = 0 ,D Q -TCV - 0 ,

0 ,

If 0 .

( 2 6 )( 2 7 )( 2 8 )( 2 9 )( 3 0 )( 3 1 )( 3 2 )

In order to qua nt iz e the theory on e mu st rep lacet h e D i r a c b r a c k e t s b e t w e e n p h a s e s p a c e v a r i a b l e sb y ( i ) t i m e s a n ( a n t i - ) c o m m u t a l o r . T h e c o o r d i n a t e r e p r e s e n t a t i o n o f t h i s ( a n t i - ) c o m m u t a t o ra l g e b r a i s

( 3 3 )( 3 4 )( 3 5 )

pm~

K y - -

r)" a x *d

i da w

K * ~ - *- 3TC ' - 1

a

T h e m a x i m a l s e t o f f i r s t - c l a s s c o n s t r a i n t s ,w h i c h c a n b e e m p l o y e d i n t h e G u p t a B l c u l c rq u a n t i z a t i o n i s

B = (**\ F ^ ^ . o " , c = j ra / " ) . ( 2 3 )T h e g a u g e t r a n s f o r m a t i o n s g e n e r a t e d b y Fa and

C K c a n b e f i x e d ( s e c [ 1 7 | ) b y c o n d i t i o n s


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in w h i c h c a s e t h e w a v e f u n c t i o n s b e c o m e( c o m p l e x ) 4 - s p i n o r s * , ; o r o n e u s e s 2 a n l i c o m -m u t i n g c - n u m b e r s :

A , ~ ? ( \ + c ' > ,

A , - , ^ ( , " , - , c ' ) ,

A , - . ' ( * + C 2 ) ,

v 2 rfC ( 3 8 )

T h en t h e w a v e f u n c t i o n s a r e f u n c t i o n s o f t h eG r a s s m a n n v a r i a b l e s C , 1 = 1 , 2 , w i th 4 c o m p o n e n t s [ 1 8 | :

4H.C') = % , C 7 * , . C Z * 2 c'c?4> l2 . ( 3 9 )T h e se 4 c o m p o n e n t s a r c i n 1- 1 c o r r e s p o n d e n c e

w i t h th e s p i n o r c o m p o n e n t s i n t h e m a t r ixr e p r e s e n t a t i o n ( 3 7 ) . W e c h o o s e t o w o r k w i t h t h el a t t e r o n e .

It f o l l o w s f r o m t h e d i s c u s s i o n a b o v e , i np a r t i c u l a r t h e c o n s t r a i n t s ( 2 6 ) , ( 3 1 ) , ( 3 2 ) t h a tt h e p h y s i c a l s u b s p a c c o f t h e H u b e r t s p a c ec o n s i s t s o f t h e s p i n o r v a l u e d f u n c t i o n s /t(Xm,

AH ) s a t i s f y i n g t h e c o n d i t i o n sa * * = 0 .

v "" r ^ : ' (z , 6 ) d e s c r i b e s a m a s s l c s sv e c t o r m u l l i p l e t .O n t h e o t h e r h a n d , i f w e h a v e a w a v e f u n c t i o n


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5W r i t i n g U n p = W m o ^ t h e e q u a t i o n o n U c a nb e w r i t t e n

w i t h t h e s o l u t i o nW m = d m * Q # = 0 . ( 5 2 )

T h u s t h e p a r t i c l e c o n t e n t o f th e s t a l e s * (* ,>0 ) i s t h a t o f a m a s s l c s s ( c h i r a l ) s c a l a rm u l t i p l c t .

W c c o n c l u d e t h a t t h e t h e o r y d e s c r i b e s m a s s l e s sN = l s c a l a r a n d v e c t o r m u l t i p l c t s , w h i c h i sd i f f e r e n t f r o m t h e s p e c t r u m o f t h e s u p c r p a r l i c l cm o d e l s w i t h s i m p l e S U S Y a n a l y z e d i n t h e p a s t.I t w o u l d b e v e r y i n t e r e s t i n g t o i d e n t i f y t h es p e c t r u m o f t h e s p i n n i n g s u p c r p a r l i c l c i n D = 1 0a n d t o e x t e n d t h e a n a l y s i s t o s p i n n i n g s u p e r -s t r i n g m o d e l .

ACKNOWUZOCJFMlWrsJ . L . w o u l d l i k e t o t h a n k prof. A b d u s S a l a m a n dt h e I n t e r n a t i o n a l A t o m i c E n e r g y A g e n c y f o r t h eh o s p i t a l i t y a t I C T P , T r i e s t e .F o r t w o o f u s , J . K . a n d J . W . v . H . , t h i s w o r k s i sp a r t o f t h e r e s e a r c h p r o g r a m o f t h e S t i c h t i n gF O M .

REFERENCES[ 1 ] L . B r i n k , P . d i V c c c h i a , P . H o w e , N u c l .

P h y s . B l 1 8 ( 1 9 7 7 ) 7 6 .| 2 | A . B a r d u c c i , R . C a s a l b u o n i a n d L . L u s a n n a ,

N u o v o C i m . 3 5 A ( 1 9 7 6 ) 3 7 7 .[ 3 ] F . A . B c r c / . in a n d M . S . M a r i n o w , A n n . P h y s .

1 0 4 ( 1 9 7 7 ) 3 3 6 .[ 4 ] L . B r i n k , S . D e s c r , B . Z u m m o , P . d i

V e c c h i a , P . H o w e , P h y s . L e t t. 6 4 B ( 1 9 7 6 )4 3 5 .[ 5 ] L . B r i n k , P . D i V c c c h i a a n d P . H o w e , P h y s .L e tt . 6 5 B ( 1 9 7 6 ) 4 7 1 ;S . D e s c r , B . Z u m i n o , P h y s . L el t. 6 5 B ( 1 9 7 6 )3 6 9 .

[ 6 ) M B . G r e e n , J . H . S c h w a r z a n d E . W i t t en ,S u p c r s t r i n g T h e o r y , V o l . 1 , C a m b r i d g eU n i v e r s i t y P r e s s , 1 9 8 7 .

| 7 J R . C a s a l b u o n i , N u o v o C i m . 3 5 A ( 1 9 7 6 ) 2 8 9 .[ 8 ) L . B r i n k , J . H . S c h w a r z , P h y s . L e t t. 1 Q 0 B

( 1 9 8 1 ) 3 1 0 .[ 9 ] D . V . V o l k o v , A . I . P a s h n e v , T e o r . M a t . F i z. ,

M ( 1 9 8 0 ) 3 2 1 .[ 1 0 ] J . A . d e A z c a r r a g a , J . L u k i e r s k i , P h y s . L e tt .

1 1 3 B ( 1 9 8 2 ) 1 7 0 .[ 1 1 ] M B . G r e e n , J . H . S c h w a r z , P h y s . L et t. 1_5_2B

( 1 9 8 5 ) 3 6 7 .[ 1 2 ] R . B r o o k s , F . M u h a m m e d , S . J . G a t e s Jr .,

N u c l . P h y s . B 2 6 8 ( 1 9 8 6 ) 5 9 9 ; C l a s s . Q u a n t .G r a v . i ( 1 9 8 6 ) 7 4 5 .

[ 1 3 ] J . K o w a l s k i - G l i k m a n , J . W . v a n H o l t e n ,N u c l . P h y s . B 2 8 3 ( 1 9 8 7 ) 3 0 5 .

[ 1 4 ] J. K o w a l s k i - G l i k m a n , P h y s . L e tt . 1 8 0 B( 1 9 8 6 ) 3 5 9 .

[ 1 5 ] A . F r yd r y s z a k , P h y s . R e v . 3 0 D ( 1 9 8 4 )2 1 7 2 .

[ 1 6 ] E . N i s s i m o v , S . P a c h e w a , S . S a l o m o n ,W e i z m a n n I n s t. P r e p r in t W I S - 8 7 / 2 8 / P H .

[ 1 7 ] K . S u n d e m e y e r , C o n s t r a i n e d D y n a m i c s w it hA p p l i c a t i o n t o Y a n g - M i l l s t h e o r y , G e n e r a lR e l a t i v i t y , C l a s s i c a l S p i n , D u a l S t r i n gM o d e l , L e c t u r e N o t e s i n P h y s i c s 1 6 9 ,S p r i n g c r - V c r l a g 1 9 8 2 .

[ 1 8 ] J . O W i n n b e r g , J. M a t h . P h y s . 18 ( 1 9 7 7 )6 2 5 .

J. Kowalski-Glikman, J.W. van Holten, S. Aoyama and J. Lukierski- The Spinning Superparticle

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