V01ume 216, num6er 3,4 PHY51C5 LE77ER5 8 12 January 1989

F R 0 M 7 H E 5UPERPAR71CLE 5 1 E 6 E L 5 Y M M E 7 R Y 7 0 7 H E 5 P 1 N N 1 N 6 PAR71CLE P R 0 P E R - 7 1 M E 5 U P E R 5 Y M M E 7 R Y

D.P. 5 0 R 0 K 1 N , V.1 .7KACH, D.V. V 0 L K 0 V and A.A. 2 H E L 7 U K H 1 N Khark0v 1n5t1tute 0f Phy51c5 and 7echn0109y, Academy 0f 5c1ence5 0f the Ukra1n1an 55R, Khark0v 310 108, U55R

Rece1ved 7 0ct06er 1988

7he c1a551ca1 e4u1va1ence 6etween the 1a9ran91an5 0f the N= 1, D= 10 ma551e55 5uperpart1c1e and 5p1nn1n9 part1c1e 15 e5ta6- 115hed. 7h15 15 d0ne 6y f1x1n9 the 9au9e 0f 5even 1ndependent 51e9e1 tran5f0rmat10n5, 1dent1fy1n9 the rema1n1n9 L0rent2-c0var1ant 51e9e1 9enerat0r w1th that 0f n= 1 10ca1 pr0per-t1me 5uper5ymmetry 0f the 5p1n • part1c1e and redef1n1n9 the 5uperpart1c1e fer- m10n1c de9ree5 0f freed0m 6y mean5 0f a 9enera112ed tw15t0r-11ke appr0ach.

1 . 7 h e c0rre5p0ndence 6etween the dynam1c5 0f 5uper5tr1n95 and 5p1nn1n9 5tr1n95 ~, wh1ch man1fe5t5 1t5e1f1n the e4u1va1ence 0f the phy51ca15pectra 0f the f1r5t 4uant12ed 5tr1n9 the0r1e5, 15 n0t c1ear1y under- 5t00d 0n the c1a551ca1 1eve1 51nce the 5pace-t1me 5tructure 0f the ferm10n1c de9ree5 0f freed0m 15 d1f- ferent f0r the5e tw0 type5 0f 5tr1n95. 5uch a c1a551ca1 c0rre5p0ndence can man1fe5t 1t5e1f 1n 51mp1er the0- r1e5 0f re1at1v15t1c ma551e55 5p1n • part1c1e5 [2,3 ] and 5uperpart1c1e5 [4] , wh1ch can 6e re9arded a5 5tr1n9- 1en9th-2er0 11m1t5. 50me c1a551ca1 e4u1va1ence 6e- tween the5e tw0 part1c1e the0r1e5 ex15t5 [ 5 ], a1th0u9h they are character12ed 6y d1fferent type5 0f 9106a1 and 10ca1 5ymmetr1e5. F0r examp1e, the N-extended 5u- perpart1c1e the0ry 1n D = 2 , 3, 4, 6, 10 d1men510n5 15 1nvar1ant under the 10ca1 ferm10n1c (51e9e1) tran5f0r- mat10n5 [ 6,7 ], 9enerated 6y N ( D - 2) 1ndependent ferm10n1c f1r5t-c1a55 c0n5tra1nt5, wh11e the 5p1n • par- t1c1e the0ry p055e55e5 0n1y n = 1 10ca1 pr0per-t1me 5u- per5ymmetry (5U5Y) .

1n ref. [ 8 ] 1t ha5 6een 5h0wn f0r the ca5e5 0 f N = 1, D = (2) 3, 4 5uperpart1c1e5 that the 51e9e1 5ymmetry can 6e 1nterpreted a5 a c0nvent10na1 10ca1 c0nf0rma1 5U5Y 0f the pr0per 5upert1me (r, tf), where t/~ are the 0dd 5uperpartner5 0f the even-t1me parameter 7, and the1r num6er e4ua15 that 0f the 1ndependent 51e9e1 9enerat0r5 (1 = 1, f0r N = 1, D = (2) 3, and 1= 1, 2 (0r 0ne c0mp1ex ~/= r/~ + 142) f0r N = 1, D = 4).

~ F0r a rev1ew and reference5, 5ee ref. [ 1 ].

70 06ta1n the5e re5u1t5, a new f0rmu1at10n 0f 5u- perpart1c1e dynam1c5 6a5ed 0n the u5e 0f c0mmuta- t1ve tw15t0r-11ke 5p1n0r var1a61e5 ha5 6een pr0p05ed [ 8 ]. 7h15 the0ry p055e55e5 60th 9106a15pace-t1me and 10ca1 (2, t/t) 5U5Y5. 1t adm1t5 the (r, t/1)-5uperf1e1d f0rmu1at10n 50 that the 51e9e1 a19e6ra 15 f0rmed 6y an 1rreduc161e 5et 0f L0rent2 c0var1ant c0n5tra1nt5 and 15 c105ed 0ffthe ma55 5he11. 7he 9enera112at10n 0fthe5e re5u1t5 t0 the 5pace-t1me d1men510n5 D = 6, 10 (where the tw15t0r-11ke f0rmu1at10n 15 p055161e a5 we11) en- c0unter5 519n1f1cant c0mp11cat10n5 c0nnected w1th the nece551ty t0 c0n5truct the (r, t/;)-5uperf1e1d f0rmu1a- t10n 0f N = 1 5uperpart1c1e the0ry, 1nvar1ant under n = 4 (1n D = 6 ) 0r n = 8 (1n D = 1 0 ) 10ca1 5U5Y tran5f0rmat10n5 0f the pr0per-t1me 5uper5pace (2, t/~) ( 1 = 1,..., D - 2 ). 1f 5uch a f0rmu1at10n ex15t5, we may h0pe t0 5tate the fu11 c0rre5p0ndence 6etween 51e9e1 5ymmetry (w1th D - 2 1ndependent 9enerat0r5) and n = D - 2 extended 10ca1 pr0per-t1me 5U5Y (1n D = 6, 10). 1t 5eem5 that 4uatern10n1c H (1n D = 6 ) and 0c- t0n10n1c 0 (1n D = 10) de5cr1pt10n5 0f the L0rent2 9r0up repre5entat10n5 6a5ed 0n the 9r0up 150m0rph- 15m5 5 0 ( 1 , 5 ) ~ 5 L 2 ( H ) , 5 0 ( 1 , 9 ) ~ 5 L 2 ( 0 ) 1n D = 6, 10, re5pect1ve1y, may 6e he1pfu1 f0r the 501u- t10n 0f the a60ve-ment10ned pr061em5 [ 8 ].

7he 501ut10n 0f the pr061em c0ncerned w1th the re- 1at10n5h1p 0f ma551e55 N = 1 5uperpart1c1e5 and 5p1n 1 part1c1e5 1n D = 10 (a5 we11 a5 1n D = 6 , 4) 5eem5 51m- p1er 6ecau5e 0ne need n0t kn0w the c0mp1ete c0rre- 5p0ndence 6etween the 51e9e1 5ymmetry and the

302

v01ume 216, num6er 3,4 PHY51C5 LE77ER5 8 12 January 1989

n = D - 2 extended 10ca1 5U5Y 6ut 0n1y the c0rre- 5p0ndence 6etween 0ne 0f the n = D - 2 1ndependent 51e9e1 tran5f0rmat10n5 and the 10ca1 n = 1 5U5Y 0f the 5p1n • part1c1e.

Here we 9enera112e the re5u1t5 0f ref. [ 5 ] and tran5- f0rm the c1a551ca1 1a9ran91an 0f the N = 1, D = 10 5u- perpart1c1e 1n 0rder t0 pr0ve 1t5 e4u1va1ence t0 the 5p1nn1n9 part1c1e 1a9ran91an. 7h15 15 d0ne 6y f1x1n9 the 9au9e f0r 5even 1ndependent 51e9e1 tran5f0rma- t10n5 and 6y 1dent1fy1n9 the rema1n1n9 L0rent2-c0- var1ant 51e9e19enerat0r w1th the 9enerat0r 0fthe n = 1 10ca1 pr0per-t1me 5U5Y. 7he 5p1n • part1c1e 1a9ran- 91an ar15e5 after the redef1n1t10n 0f ferm10n1c var1- a61e5 6y mean5 0f the 9enera112ed tw15t0r-11ke appr0ach pr0p05ed 1n ref. [5] f0r the re501ut10n 0f ma55-5he11 and D1rac c0n5tra1nt5 0n ma551e55 5p1n- n1n9 part1c1e dynam1c5. 7he5e re5u1t5 rema1n va11d f0r 5pace-t1me d1men510n5 D = 2, 3, 4 and 6 a5 we11.

2 .7he N = 1, D = 10 ma551e55 5uperpart1c1e act10n ha5 the f0110w1n9 f1r5t-0rder f0rm:

d r [ p , , , ( 2 ~ - 1 ~ ) • ••• 5= -~ep,,,p 1, (1)

where x••• ( m = 0 , 1 ... . , 9) are the 6050n1c c00rd1- nate5, and 0~ are the Maj0rana-Wey1 0dd 5p1n0r var1a61e5 (c~= 1 .... ,24); Pm 15 the m0mentum c0nju- 9ate t0 x•••, e ( r ) 15 the 1a9ran91an mu1t1p11er hav1n9 the phy51ca1 mean1n9 0f the 0ne-d1men510na1 ••9rav1- t0n••. 7he d0t den0te5 the r-der1vat1ve. 7he 5pace- t1me 519nature 15 ch05en t0 6e ( --, + , ..., + ).

1t 15 we11 kn0wn [4,6,7] that the f1r5t-c1a55 c0n5tra1nt5

p2 = 0 , (2)

(P,,,7~d),~ =-P,,,7~[ ~/3--1(~/np~0) ~] =0 (3)

(7r~ 15 the can0n1ca1 m0mentum, c0nju9ate t0 0~) 9enerate r-reparametr12at10n tran5f0rmat10n5 and the 10ca1 ferm10n1c (51e9e1) tran5f0rmat10n5 [ 6, 7 ] 0f dy- nam1c var1a61e5. N0te that e45. (3) are 06ta1ned fr0m the c0n5tra1nt5 d,~=-2r-1(7~p~0),~=0, wh1ch are a m1xture 0f the 5ec0nd- and f1r5t-c1a55 0ne5.1n c0n5e- 4uence 0f the 51n9u1ar1ty 0f the matr1x p,,, 7 ~ 0n the ma55 5he11 (2) (det p,,,7~= 0) 0n1y e19ht c0n5tra1nt5 0ut 0f 51xteen (3) are 1ndependent. 7he L0rent2 c0- var1ant extract10n 0f the5e 1ndependent c0n5tra1nt5 pre5ent5 a pr061em. An 1ntere5t1n9 way 0ut wa5 pr0- p05ed 1n a 5er1e5 0f art1c1e5 [ 9-14 ], where 5uperpar-

t1c1e and 5uper5tr1n9 dynam1c5 were c0n51dered 1n 119ht-c0ne harm0n1c 5uper5pace5.

7he recent1y pr0p05ed tw15t0r-11ke f0rmu1at10n 0f N = 1, D = 2, 3, 4 5uperpart1c1e5 [8 ] a110wed t0 501ve 60th pr061em5 c0ncerned w1th the 51e9e1 5ymmetry: the c0n5truct10n 0f the 1rreduc161e 5et 0f L0rent2 c0- var1ant 51e9e19enerat0r5 and the c105ure 0fthe 51e9e1 a19e6ra 0ff-5he11. 7he 0r191n 0f the 51e9e1 5ymmetry wa5 a150 c1ar1f1ed. 1t wa5 5h0wn t0 6e the man1fe5ta- t10n 0fthe u5ua1 10ca1 pr0per-t1me 5U5Y.

3. Here we u5e the tw15t0r-11ke appr0ach 0f ref5. [5,8] t0 e5ta6115h the c1a551ca1 re1at10n5h1p 6etween the ma551e55 N = 1, D = 10 5uperpart1c1e and D = 10 5p1n • part1c1e dynam1c5. 51nce the 1atter the0ry p05- 5e55e5 the n = 1 10ca1 pr0per-t1me 5U5Y, the f1r5t 5tep t0ward5 the 501ut10n 0f th15 pr061em 15 t0 extract 1n a L0rent2-c0var1ant manner 0ne 0f the e19ht 1ndepen- dent 51e9e1 9enerat0r5 and t0 1dent1fy 1t w1th the 0dd 9enerat0r 0fthe n = 1 10ca1 5U5Y. We ach1eve th15 6y 1nc0rp0rat1n9 1nt0 the the0ry the c0mmut1n9 Maj0r- ana-Wey1 5p1n0r var1a61e5 2~ (r) w1th the 5ame ch1r- a11ty a5 0f 0~(r).

7hen the ma55-5he11 c0nd1t10n (2) can 6e 501ved exp11c1t1y a5

Pm = - [4 / e ( r ) ] (,17,,,2) • (4)

7he m1nu5 519n 1n e4. (4) 15 ch05en f0r the m0men- tum p,,, t0 6e future d1rected (P0> 0).

7he re1at10n (4) can 6e 06ta1ned fr0m the act10n [81

d~ p, , , (2m-1~m6+ 2ymf1) , (5) 5•=

wh1ch 15 c1a551ca11y e4u1va1ent t0 the 0r191na1 act10n ( 1 ). 70 06ta1n e4. (4), 0ne mu5t 501ve the e4uat10n5 0fm0t10n f0r the aux111ary var1a61e5 f1~:

(p,,~/~2),~ = 0 . (6)

N0te that e ( r ) 15 n0t pre5ent 1n e4. (5), 6ut the ac- t10n 5• 15 5t111 reparametr12at10n-1nvar1ant 6ecau5e 0f the f0110w1n9 tran5f0rmat10n 1aw: 6 2 ~ = a ( r ) 2~+ ~1(v)2~ ( a ( r ) 6e1n9 the parameter 0f the r- reparametr12at10n).

We mu5t a150 n0te that e4. (6) 15 5tr0n9er than the ma55-5he11 c0nd1t10n (2): the 9enera1 n0ntr1v1a1 50- 1ut10n 0f (6) 1n term5 0f2~ 15 the re1at10n (4), wh11e that 0f (2) 15 p, .=-(4/e) .97, , ,2+1~u(~)(~m2),

303

V01ume 216, num6er 3,4 PHY51C5 LE77ER5 8 12 January 1989

where ~u(r) 15 an ar61trary 0dd parameter. 7hu5 the m0menta 0f the tw0 f0rmu1at10n5 0f the N = 1 5uper- part1c1e the0ry can d1ffer 6y the n11p0tent term 1~u(r) •,,,2.

5u65t1tut1n9 e4. (4) 1nt0 the 51e9e1 c0n5tra1nt5 ( 3 ) and u51n9 the F1er2 rearran9ement 0f the matr1x (~,,,7~)-/~r,~, 0ne can extract 0ne 0fthe e19ht 1ndepen- dent 51e9e19enerat0r5 a5 f0110w5:

Q=£"d, = 0 . (7)

Q 15 the L0rent2 5ca1ar, and 1t5 ant1-P01550n 6racket5 1ead t0 the r-reparametr12at10n 9enerat0r R = p,,,2y•••2 = 0 (wh1ch 15 e4ua1 t0 the c0n5tra1nt (2) a5 a c0n5e4uence 0fe4. (4)) :

{Q, Q},. = - 21p,, 2-7•••2. (8)

1t 15 ea511y 5een that the 5tructure ••c0n5tant•• 1n ( 8 ) 15 1ndeed c0n5tant 0ff-5he11.7hu5 Q and R f0rm the a19e6ra 0f 10ca1 n= 1 5uperc0nf0rma1 tran5f0rma- t10n5 1n a ••5ma11•• 5uper5pace (r, 4):

8r=a(r)+14a(r), 8 4 = a ( r ) + ~1(r) r/ (9)

( a ( r ) 15 the 0dd n= 1 5U5Y parameter), wh1ch 15 at the 5ame t1me the 5u6a19e6ra 0f the 51e9e1 tran5f0r- mat10n a19e6ra.

70 c105e the 5upera19e6ra (8) 0ff-5he11, when act- 1n9 0n the dynam1c var1a61e5 x,,,, p,~, 0,, 2~, 0ne mu5t 91ve the (r, 4) 5uperf1e1d f0rmu1at10n 0f the the0ry, 1.e., take 1nt0 acc0unt the c0ntr16ut10n 0f aux111ary f1e1d5 1nt0 the tran5f0rmat10n 1aw5 f0r the dynam1c var1a61e5.

70 th15 end, we u5e the 5uperf1e1d5

P,,,=p,,,+1~p,~, X,,,=x,,,+1~12 ....

0~ =0. +~;~., (10)

where p,,, and 2,, are the 6ra55mann 5uperpartner5 0f p,,, and x .... re5pect1ve1y. 7he n = 1 5U5Y tran5f0r- mat10n 1aw5 f0r 5uperf1e1d (10) c0mp0nent5 can 6e f0und 1n ref5. [2,5,8].

7he 5uperf1e1d act10n de5cr161n9 the dynam1c5 0f the 5uperpart1c1e w1th man1fe5t N = 1 9106a1 5pace- t1me 5U5Y and n= 1 10ca1 5U5Y ha5 the f0110w1n9 f0rm [8]:

55.v=-1~ drd4P,,,(DX~+107~D0), (11)

where D = 0104+1r10102. After the 1nte9rat10n 0f 55.v. 0ver 4 0ne 06ta1n5 the

act10n (5) w1th the add1t10na1 term 5aux. c0nta1n1n9 the aux111ary var1a61e5 p,,,, 2,,:

5 .... =1 f drp, , , (2~+ ~ 2 ) . (12) ,)

1t f0110w5 fr0m e4. ( 12 ) that the e4uat10n5 0f••m0- t10n•• f0r p,,,, 2,,, are

p , , ,=0 , 2,,,= --07,,,2. (13,14)

N0te that e4. (14) a5 we11 a5 the e4uat10n 0f m0t10n f0r x,n, 2,, ,-1•,, ,2= -X7,,,2 are the c0n5e4uence5 0f the 5uperf1e1d e4uat10n 0f m0t10n DXm+ 107•••D0= 0 ar151n9 after the var1at10n 0f the act10n ( 11 ) 0ver the 5uperf1e1d P,,,. Hence, we have 06ta1ned the 1n- ver5e-H1995-effect c0nd1t10n [15] wh1ch wa5 1m- p05ed 1n ref. [5 ] 0n the 5uperc0var1ant f0rm ~2,,, - DX,, + 107,,,D0 t0 06ta1n the tw15t0r-11ke 501ut10n5 0f the ma55-5he11 and the D1rac c0n5tra1nt5 0n 5p1n • part1c1e dynam1c5.

0ne can ver1fy, u51n9 re1at10n (4) and the permu- tat10n pr0pert1e5 0f the D=3 , 4, 6, 10 7-matr1ce5

0), that 0ne can 5ee that ( 14 ) 5at15f1e5 the D1rac c0n- 5tra1nt f0r the ma551e55 5p1n • part1c1e [2,3 ]

p,,,2••=0. (15)

7hu5, re1at10n (14) 15 the centra1 p01nt 1n the pr00f 0f the c1a551ca1 e4u1va1ence 0f the N = 1 5uperpart1- c1e5 and 5p1n • part1c1e5 1n the 5pace-t1me d1men- 510n5 D=2 , 3, 4, 6, 10.

N0te that 1n the ca5e 0fthe D = 2, 3 N = 1 5uperpar- t1c1e, the 5uperf1e1d act10n (1 1 ) e5ta6115he5 a c0m- p1ete c0rre5p0ndence 6etween the 51e9e1 5ymmetry (wh1ch 15 0ne-d1men510na1 1n D = 2, 3) and the 10ca1 pr0per-t1me 5U5Y, 60th 9enerated 6y the c0n5tra1nt Q (7) [8].

4.51nce the 5p1n • part1c1e p055e55e5 0n1y the n= 1 10ca15U5Y, t0 make a 5ec0nd 5tep t0ward5 the e5ta6- 115hment 0f c1a551ca1 e4u1va1ence 0fthe tw0 5er1e5 we mu5t f1x the 9au9e f0r a11 51e9e1 tran5f0rmat10n5 ex- cept 0ne. 50 1et u5 c0n51der the e4uat10n 0f m0t10n f0r 0~

(pm7m0)~ = 0 . (16)

A5 a c0n5e4uence 0fthe 51e9e1 1nvar1ance, the 5uper- part1c1e ve10c1ty 0, 1n the 6ra55mann d1rect10n5 0f

304

V01ume 216, num6er 3.4 PHY51C5 LE77ER5 8 12 January 1989

5uper5pace-t1me 15 character12ed 6y e19ht 1ndepen- dent parameter5:

]27,,,)ff 7 ~)~ (17) =P,,,7.a~ ~ . = - [4 / e ( r )

where ~"( r ) 15 an ar61trary Maj0rana-Wey1 5p1n0r w1th a ch1ra11ty 0pp051te t0 0.. 0n1y e19ht c0mp0- nent5 0f {" c0ntr16ute t0 e4. (17), 51nce the rank 0f the matr1x P,,,7~ 15 e4ua1 t0 e19ht 0n the ma55-5he11 5urface. Hence, the traject0ry 0f the N = 1 5uperpar- t1c1e 1n the D = 10 5uper5pace (xm, 0 . ) 15 n0t a 11ne, 6ut an ar61trary 5uper5urface w1th 0ne 6050n1c and e19ht ferm10n1c d1rect10n5 [ 16,17 ].

We f1x the 9au9e f0r 5even 51e9e1 tran5f0rmat10n5 1n 5uch a way that th15 5uper5urface 15 parametr12ed 6y 0n1y 0ne 6050n1c and 0ne ferm10n1c var1a61e5 and demand the ve10c1ty t). t0 6e character12ed 6y 0n1y 0ne 1ndependent 0dd L0rent2-5ca1ar parameter ~( r):

0. = e - 1 / 2 ~ ( r ) 2 , (18)

where ~u(r) 15 pr0p0rt10na1 t0 (2-{) 6ecau5e e4. ( 18 ) ar15e5 a5 the f1r5t term 1n the F1er2 rearran9ement 0f the r19ht-hand 51de 0fe4. (17).

7hu5 we have reta1ned 0n1y 0ne 10ca1 ferm10n1c 5ymmetry, wh1ch (a5 we have 5h0wn 1n 5ect10n 3) 15 n0th1n9 6ut the n = 1 10ca1 pr0per-t1me 5U5Y.

7he 1a5t 5tep 15 t0 carry 0ut the tran51t10n fr0m the N = 1 5uperpart1c1e act10n (5), (12) (0r e4u1va1ent act10n ( 1 ) ) t0 the 5p1n • part1c1e act10n 6y mean5 0f the dua11ty-11ke re1at10n5 (4), (14) and the e4uat10n 0fm0t10n f0r the 2 . var1a61e

;~. = 6 ( r ) 2 . (19)

(where 6(r) 15 an ar61trary rea1 parameter). E4. (19) f0110w5 fr0m the e4uat10n 0f m0t10n f0r p,,, (p,,, = 0) and the re1at10n (4). N0te that a5 a c0n5e4uence 0f e45. (4), (19), the "9rav1t0n•• e ( r ) ha5 the f0110w1n9 e4uat10n 0f m0t10n:

~ ( r ) = 2 6 ( r ) e ( r ) .

Hence, 6 ( r ) can 6e 1dent1f1ed w1th the parameter 0f the r-reparametr12at10n.

U51n9, where nece55ary, e45. (4), (14) and (19), 0ne can chan9e a11 the 6111near 5p1n0r c0m61nat10n5 £7,,0 1n act10n (5) (0r ( 1 ) ) 6y the 6ra55mann vec- t0r ;~,,, wh1ch 6ec0me5 the pr0pa9at1n9 ferm10n1c de- 9ree 0f freed0m. A5 a re5u1t, the we11-kn0wn act10n de5cr161n9 the dynam1c5 0f the 5p1nn1n9 re1at1v15t1c ma551e55 part1c1e [2,3 ] ar15e5 a5 f0110w5:

5~/2 J d r ( P , , , 5 c ~ - ~ 1 X , , , ~ - 1 ~ / p , , , X ~ 1 = - ~ e p , , , p ) ,

(20)

where the f0110w1n9 f1e1d redef1n1t10n ha5 6een made

2 , , , = 4 e - • / 2 2 .... (21)

N0te that the ••9rav1t1n0•• f1e1d ~( r ) 1n the act10n (20) 15 the 5ame a5 that 1n the e4uat10n 0fm0t10n ( 18 ) f0r 0.. 7h15 can 6e ver1f1ed 6y rewr1t1n9 the e4uat10n 0f m0t10n

2,,, = ~ , ( r ) p , .

(wh1ch re5u1t5 fr0m act10n (20) 6y 2m-var1at10n) u5- 1n9 the f1e1d redef1n1t10n5 (4), (14) and (21).

Let u5 n0w c0mpare the m0menta 0fthe tw0 the0- r1e5, wh05e c1a551ca1 e4u1va1ence ha5 6een dem0n- 5trated, u51n9 the dua11ty-11ke c0nd1t10n (14). A5 we have a1ready ment10ned 1n 5ect10n 2, the m0menta p , . 0ftw0 c1a551ca11y e4u1va1ent dynam1c 5y5tem5 can 6e e4ua1 up t0 a n11p0tent term 0f the f0rm: 1~u(r) •.,2. 0ne can ea511y 5ee that an ana1090u5 re- 5u1t 0ccur5 when the m0mentum 0f the N = 1 5uper- part1c1e (p~ff~= ~) = e - ~ (5¢. - 1•,,, 0) ) and that 0f the 5p1n • part1c1e (p~ /2~ = e - ~ ( ~ 9 . ~ 1 ¢ / 2 . , ) ) are c0m- pared u51n9 e45. (14), ( 18 )and (21):

p},]12, ~p~ff,=,, = (31fe) ~ ( r ) •,,,2.

7he tw0 m0menta 6ec0me the 5ame when the n = 1 10ca1 5U5Y 9au9e 15 f1xed 1n 5uch a way that the ••9rav1t1n0•• f1e1d ¢/(r) turn5 1nt0 2er0.

5. 1n c0nc1u510n, we have e5ta6115hed the c1a551ca1 re1at10n5h1p 6etween the ma551e55 N = 1 5uperpart1c1e dynam1c5 and that 0f the ma551e55 5p1n ~ part1c1e 1n D=2 , 3, 4, 6 and 10 5pace-t1me d1men510n5. 7he 5uperf1e1d act10n ( 1 1 ) p055e551n9 d0u61e 5uper1nvar- 1ance 15 the 6a51c act10n fr0m wh1ch 60th the N = 1 5uperpart1c1e and 5p1n ~ part1c1e 1a9ran91an5 are 06ta1ned.

Yet, the f1r5t 4uant12at10n 0f the tw0 the0r1e5 c0n- 51dered 1ead5 t0 d1fferent phy51ca1 5tate5. 7he 4uan- tum 5pectrum 0f the N=1 , D = 1 0 5uperpart1c1e c0nta1n5, f0r examp1e, the a6e11an 9au9e N = 1, D = 10 5upermu1t1p1et w1th e19ht 6050n1c and e19ht fer- m10n1c phy51ca1 de9ree5 0f freed0m [ 1 ]. At the 5ame t1me, the D1rac 5p1n0r f1e1d ar15e5 after 5p1n • part1c1e dynam1c5 4uant12at10n. When 1t 15 re5tr1cted t0 5at-

305

V01ume 216, num6er 3,4 PHY51C5 LE77ER5 8 12 January 1989

15fy1n9 the M a j 0 r a n a c0nd1t10n, the 5p1nn1n9 part1c1e

4 u a n t u m 5pec t rum c0nta1n5 51xteen ferm10n1c de-

9ree5 0 f f r eed0m, wh1ch 15 e4ua1 t0 the n u m 6 e r 0 f

N = 1, D = 10 5uperpart1c1e phy51ca1 5tate5 (8 + 8 =

16).

7 h e w0rk 0n the e5ta6115hment 0 f the c1a551ca1 re-

1at10n5h1p 6e tween 5uper5tr1n9 and 5p1nn1n9 5tr1n9

dynam1c5, 60 th p055e551n9 e4u1va1ent 4 u a n t u m

phy51ca15pectra, 15 1n pr09re55.

Ackn0w1ed9ement

7 h e auth0r5 are thankfu1 t0 V.P. Aku10v, V.D.

6e r5hun , R.E. Ka1105h, A.L Pa5hnev, V.A. 50r0ka and

A . 5 . 5 c h w a r 2 f0r va1ua61e d15cu5510n5.

Reference5

[ 1 ] M.8.6reen, J.H. 5chwar2 and E. w1tten, 5uper5tr1n9 the0ry, V015. 1, 2 (Cam6r1d9e U.P., Cam6r1d9e, 1987).

[2] L, 8r1nk, 5. De5er, 8.2um1n0, P. D1Vecch1a and P. H0we, Phy5. Lett. 8 64 (1976) 435.

[3] V.D. 6er5hun and v.1.7kach, P15•ma 2h, 7e0r. Ek5p. F12. 29 (1979) 320.

[4] L. 8r1nk and J.H. 5chwar2, Phy5. Lett. 8 100 ( 1981 ) 310. [ 5 ] D.v. v01k0v and A.A. 2he1tukh1n, Khark0v prepr1nt KhF71

88-29 ( 1988); P15•ma 2h. 7e0r. Ek5p. F12. 48 (1988) 61; D.P. 50r0k1n, v.1.7kach, D.V. v01k0v and A.A. 2he1tukh1n, Khark0v prepr1nt KhF71 88-59 ( 1988 ).

[ 6 ] J.A. De A2carra9a and J. Luk1er5k1, Phy5. LetL 8 113 ( 1982 ) 170.

[7] W. 51e9e1, Phy5. Len, 8 128 (1983) 397. [ 8 ] D.P. 50r0k1n, v.1.7kach and D.v. V01k0v, Khark0v prepr1nt

KhF71 88-31 (1988). [9] E. 50katchev, Phy5. Lett. 8 169 (1986) 209.

[ 10] E. 50katchev, C1a55. Quant. 6rav. 4 (1987) 237. [ 11 ] E.R. N1551m0v and 5.L. Pacheva, Phy5. Lett. 8 198 (1987)

57. [ 12] E.R. N1551m0v, 5.L. Pacheva and 5. 5010m0n, Nuc1. Phy5.

8296 (1988) 462. [ 13] E.R. N1551m0v, 5.L. Pacheva and 5. 5010m0n, Nuc1. Phy5.

8297 (1988) 349. [ 14 ] R.E. Ka1105h and M.A. Rahman0v, Phy5. Lett. 8 209 ( 1988 )

233. [ 15 ] E.A. 1van0v and V.1. 091evet5ky, 7e0r. Mat. F12.25 ( 1975 )

169. [ 16] 7. 5h1rafuj1, Pr09. 7he0r. Phy5. 70 (1983) 18. [ 17] E. w1tten, Nuc1. Phy5. 8 266 (1986) 245.

306