Relativistic model of the anyon

v01ume 248, num6er 1,2

PHY51c5 LE77ER5 8

27 5eptem6er 1990

Re1at1v15t1c m0de1 0f the any0n M.5. P1yu5hchay

1n5t1tutef0r H19hEner9yPhy51c5,5u-142284 Pr0tv1n0,M05c0wRe910n, u55R Rece1ved 29 May 1990

7he m0de10f the re1at1v15t1c5p1nn1n9part1c1e15c0n51dered1n (2 + 1) d1men510n5.1t155h0wnthat there are tw0 6050n1c0r tw0 ferm10n1c 5tate5 0f 5p1n55= + 0t 1n the 4uantum 5pectrum 0f the m0de1 when the parameter 0f the m0de1, a, 15, re5pect1ve1y, 1nte9er 0r ha1f-1nte9er, and at 0ther va1ue5 0f a there 150n1y 0ne any0n1c 5tate 1n the phy51ca15pectrum. 7he c0nnect10n 0f the m0de1 w1th re1at1v15t1cpart1c1ew1tht0r510n, wh05e act10nappeared 1n the 805e-Ferm1 tran5mutat10n mechan15m,15p01nted 0ut.

51nce the an9u1ar m 0 m e n t u m a19e6ra 1n tw0 d1men510n515 a tr1v1a1 c0mmutat1ve a19e6ra, the ex0t1c 5p1n and 5tat15t1c5 can take p1ace 1n ( 2 + 1 )-d1men510na1 4 u a n t u m the0r1e5 [ 1 ]. 7 h e 4 u a n t u m 06ject, wh1ch ha5 unu5ua15tat15t1c5, wa5 ca11ed the any0n [ 2 ]. After that, 5evera1 paper5 0n fract10na1 5p1n and 5tat15t1c5 appeared [ 3 ] and 1ntere5t1n9 phy51c5 1n (2 + 1 ) d1men510n5 wa5 p01nted 0ut [ 4 ]. At pre5ent, there 15 9reat 1ntere5t 1n the fract10na15tat15t1c51n c0nnect10n w1th h19h temperature 5uperc0nduct1v1ty [ 5 ]. 1n ref. [6] new m0de15 0f ma551ve and ma551e55 5p1nn1n9 part1c1e5 1n (3 + 1 )-d1men510na1 5pace-t1me were 5u99e5ted. 7he5e m0de15 were c0n5tructed 6y ana109y w1th p5eud0c1a551ca1 mechan1c5 [ 7 ] and there the c0mmut1n9 vect0r and 5ca1ar var1a61e5 were u5ed 0n the c1a551ca1 1eve1 f0r the de5cr1pt10n 0f the 5p1n de9ree5 0f freed0m. 1n the pre5ent paper we 5ha115h0w that the m0de1 0 f a ma551ve part1c1e [ 6 ] 6e1n9 carr1ed 1nt0 the 5pace-t1me 0f (2 + 1 ) d1men510n5, can 6e c0n51dered a5 the re1at1v15t1c m0de1 0f the any0n. F1r5t we 5ha11 de5cr16e the c1a551ca1m0de1. We 5ha11 5h0w that 0ne 0f the ham11t0n1an c0n5tra1nt5 0f the 5y5tem 15 reduced t0 the c0nd1t10n 5 2 - a 2 = 0, where 5 15 the c1a551ca1 5p1n and a 15 the parameter 0f the m0de1. A5 a c0n5e4uence, 0n the c1a551ca1 1eve1, the m0de1 de5cr16e5 the 5tate5 0f 5p1n5 5 = + a. Part1a11y f1x1n9 the 9au9e freed0m c0nta1ned 1n the 5y5tem, we 5ha11 perf0rm the 4uant12at10n 0f the m0de1 0n the reduced pha5e 5pace. 1t appear5 that 1n the ca5e when 0t = k 0r a = k + •, where k 15 an ar61trary 1nte9er, the m0de1 de5cr16e5 0n the 4 u a n t u m 1eve1 6050n1c 0r ferE15ev1er5c1encePu6115her58.V. (N0rth-H011and)

m10n1C 5tate5 0f 5p1n5 5= •+ a, re5pect1ve1y. At the 5ame t1me, 1f a ~ •k, then there 15 0n1y the 5tate 0f 5p1n 5 = a 1n the phy51Ca1 5peCtrum (0r 5 = - - a , depend1n9 0n the ch01ce 0f rea112at10n 0f the 5p1n 0perat0r). Under 2n r0tat10n 0f the wave funct10n 0f 5uch a 5tate 1n a 5pace, a pha5e fact0r d1fferent fr0m •+ 1 appear5, 1.e., 5uch a 5tate 15 any0n1c. 51nce the m0de1 15 de5cr16ed 6y a 5et 0f f1r5t-c1a55 c0n5tra1nt5, 1t can 6e 4uant12ed a150 1n an exp11c1t1yc0var1ant way, w1th0ut 1mp051n9 9au9e c0nd1t10n5. We 5ha11 5h0w that c0var1ant 4uant12at10n 91ve5 the 5ame re5u1t5 a5 4uant12at10n w1th f1x1n9 9au9e5. 1n c0nc1u510n, we p01nt 0ut the c0nnect10n 0f the 5u99e5ted m0de1 w1th a re1at1v15t1c part1c1e w1th t0r510n, wh05e act10n appeared 1n the 805e-Ferm1 tran5mutat10n mechan15m

[8]. 7 h e m0de1 0f a re1at1v15t1c ma551ve 5p1nn1n9 part1c1e 1n (3 + 1 ) d1men510n5 15 91ven 6y the f0110w1n9 1a9ran91an [ 6 ]: 1 . 2 L = ~ee(x~,--v4u) - ~1e m 2 - - m v 4 5 - a x / ~ ,

(1)

where x u are 5pace-t1me C00rd1nate5 0f the part1C1e, e(2) and v(2) are La9ran9e mU1t1p11er5, m 15 a parameter w1th the d1men510n 0f ma55, wh11e 0t 15 a d1men510n1e55 parameter (1n natura1 un1t5), 4~(2) 15 a tran51at10na11y-1nvar1ant vect0r, 45 (2) 15 a L0rent2 5ca1ar, f 1 M = ~ M / x / ~ , ~M= (4~, 45) and we 5upp05e that [6] 4 2 = 4 M 4 N 9 M N > 0 , H2~0, 9 M N = ( -- 1, 1. . . . . 1 ); and we den0te the der1vat1ve 0ver the ev01ut10n parameter 2 6y a d0t. 7 h e vect0r 4u 5erve5 t0 de5cr16e 107

V01ume 248, num6er 1,2

PHY51C5 LE77ER5 8

the 5p1n de9ree5 0f freed0m and 0w1n9 t0 the pre5ence 0fthe 5ca1ar 45, the ham11t0n1an c0n5tra1nt5 f0110w1n9 fr0m (1) are f1r5t-c1a55 c0n5tra1nt5 [6] (4u and 45 are 5u65t1tute5 here f0r the 6ra55mann var1a61e5 ~u and ~5 pre5ent 1n p5eud0c1a551ca1 mechan1c5 [ 7 ] ). Let u5 tran5fer n0w th15 m0de1 1nt0 the 5pacet1me 0f ( 2 + 1 ) d1men510n5 putt1n9/1=0, 1, 2, and pa55 t0 the ham11t0n1an de5cr1pt10n 0fthe 5y5tem. F0r th15 we 1ntr0duce the can0n1ca1 m0menta PA =- (Pu, ~M, Pe, P v ) = 0 L / 0 Q a, C0njU9ated re5pect1Ve1y t0 Q~---- (X u, 71~, e, V), and the P01550n 6raCket5 {0~, P~} =~f3,

{0A, Q~}={PA, P ~ } = 0 .

(2)

7here 15 the f0110W1n9 5et 0f the ham11t0n1an C0n5tra1nt51n the 5y5tem [ 6 ]: 0~ = ~ - ~ 0 ,

02=~(~2~2-a2) ~0,

0~=p~,~¢0,

04=p~,~0,

0~=p2+m2~,0,

(3)

06=p4+m45~0,

07 = p ~ 2 + m n 5 ~ 0 ,

(5)

(6)

15 the C0n5erved an9u1ar m0mentum. 7he C0n5tra1nt 01 can 6e rewr1tten 1n the e4u1va1ent f0rm ~2 = ~ ( 52--0t2) ~ 0

(7)

0n the 5urface 0fC0n5tra1nt5 0~ ~ 0, 6 = 1, 5, 6, 7, where

1

5 = ~--~--/-/-/-/-/-/~-2~m~ppuMvp 2x/--p~

(8)

15 the 5p1n 0f the part1c1e. Fr0m re1at10n (7) 1t f0110w5 that 0n the c1a551ca1 1eve1 the m0de1 under c0n51derat10n de5cr16e5 the part1c1e w1th 5p1n5 5 = + a. 1n 0rder t0 pa55 t0 the 4uant12at10n 0f the 5y5tem 0n the reduced pha5e 5pace, we f1x part1a11y the 9au9e 108

2/=02-1~0,

1

2 2 = e - - -m~ 0 ,

23=v~0,

24=P 7~-m~5"~0 , X5 = P 4 - m 4 5 ~ 0 •

(9) (10)

6au9e5 (9), (10) and c0n5tra1nt5 0c, c = 1, 3, 4, 6, 7, f0rm a 5et 0f 5ec0nd-c1a55 c0n5tra1nt5. 7he re4u1rement 0f c0n5ervat10n 0f the 9au9e c0nd1t10n5, d26/ d2~ 0, 6 = 1, ..., 5, effect5 part1a1 f1x1n9 0f the mu1t1p11er51n the ham11t0n1an (5); w~ = w3= w4= 0. Let u5 pa55 n0w fr0m the P01550n 6racket5 t0 the D1rac 6racket5 (D8•5) [9] u51n9 the a60ve 5et 0f c0n5tra1nt5 0c and 9au9e5 X6. After that the5e c0n5tra1nt5 and 9au9e5 can 6e c0n51dered e4ua1 t0 2er0 1n the 5tr0n9 5en5e. 7he c0n5tra1nt5 0c90 and the 9au9e5 26 ~ 0 51n91e 0ut a 5urface f~ 1n the pha5e 5pace, wh1ch can 6e parametr12ed 1n the f0110w1n9 way:

4u=etu1~n1(~0), nu=e(u1)m~1)(~0)~5,

p1ay5 the r01e 0f the t0ta1 ham11t0n1an [9 ], where wa=wa(r), a = 1, ..., 4, are ar61trary funct10n5. 7he can0n1ca1 m 0 m e n t u m pu 15 at the 5ame t1me the c0n5erved ener9y-m0mentum vect0r 0fthe part1c1e, and the 4uant1ty Muv = x u p v~x~,pu + 4un~,~ 4,, nu

ar61trar1ne55 c0nta1ned 1n the 5y5tem. F0r th15 we 1mp05e the f0110w1n9 5et 0f 9au9e c0nd1t10n5:

(4)

am0n9 wh1ch the c0n5tra1nt5 ( 3 ) are pr1mary and (4) are 5ec0ndary c0n5tra1nt5. A11the c0n5tra1nt5 f0rm the a19e6ra 0f the f1r5t-c1a55 c0n5tra1nt5 and the 4uant1ty [6]

H=~e0~ +V06 +wa0a

27 5eptem6er 1990

45=~25=v=pe=pv=0,

1

e=--, m

(11)

where n(4~) = (c05 ~0, 51n ~0), m(~0) = ( - 51n ~0, c05 ~0), 0~<~<2~2, - 0 ~ < 5 < c ~ , and we 1ntr0duce the n0rma15 e
{(p, 5 } * = 1 .

(12)

7he rema1n1n9 c0n5tra1nt5 05 ~ 0 and 02 ~ 0, the 1atter 6e1n9 rewr1tten n0w 1n f0rm (7), p1ay the r01e 0f the re5tr1ct10n5 0n the 1n1t1a1 data 0n the reduced pha5e 5pace 0f the 5y5tem, and the 4uant1ty /~= (1/ 2m) 05 + w2ff2 9enerate5 0n 1t the ev01ut10n 6y mean5 0fthe D8•5. Let u5 4uant12e the m0de1. 1n the repre5entat10n where the 0perat0r5/)u are d1a90na1, we rea112e ~u 1n the f0rm ~ u= 10/0p u. 7ak1n91nt0 acc0unt that the an-

V01ume 248, n u m 6 e r 1,2

PHY51C5 LE77ER5 8

9u1ar var1a61e ~0var1e5 fr0m 0 t0 2rt, we 5ha11w0rk 0n the 5pace 0f 27r-per10d1c funct10n5 0f the f0rm ~u(p,~0)= ~ ~2(p)exp(1/~0), t

~2.

(13)

51nce the 0perat0r ~ take5 0ut funct10n5 0f f0rm ( 13 ) fr0m the 5pace 0f 2n-per10d1c funct10n5, we u5e the 0perat0r5 51n ~ and c05 (01n5tead. 7hen 1n acc0rdance w1th (12), 1t rema1n5 t0 rea112e the 0perat0r ~, 5at15fy1n9 the permutat10n re1at10n5 [51n ~, 5] =1 c05 ~, [c05 ~, ~] = - 1 51n ~. 1n the 9enera1 ca5e 5uch an 0perat0r can he rea112ed 1n the f0rm ~=-1~

+c,

(14)

where c 15 an ar61trary rea1 c0n5tant, and then the wave funct10n5 4/(p) exp (11~0) are 1t5 e19enfunct10n5 w1th e19enva1ue51+ c. 7he c0n5tra1nt5 ~5 and (7) turn n0w 1nt0 e4uat10n5 f0r the wave funct10n5 0f phy51ca1 5tate5 (/~2+m2) ~v=0,

(~2-a2) ~=0.

27 5 e p t e m 6 e r 1990

t10n5 0f the 0perat0r (18) w1th e19enva1ue5 •k and •k, re5pect1ve1y. At the 5ame t1me we f1nd that when 0t# •k, 0n1y the wave funct10n ~ ( p , ~0) fr0m (19) 15the 501ut10n 0fe4. (16), wh1ch 15 the e19enfunct10n 0 f ~ w1th e19enva1ue 5 = a. C0n5truct n0w w1th the he1p 0f re1at10n5 (6), ( 11 ) the an9u1ar m0mentum 0perat0r -

~ ~ ~ 1)~(2) Mu,=2[up,1+~(e1u e,] +p(u~
(20)

where at u6~j =au6~-a~6 u. Ca1cu1at1n9 the c0mmutat0r5 [M u~, M p°] and [M u~, ~P], we 9et c0nv1nced that the 0perat0r5 h7/u" and pu f0rm the P01ncar6 a19e6ra 1n d = (2 + 1 ). 7heref0re the c0n5tructed 4uantum the0ry 15 re1at1v15t1c1nvar1ant, and w1th the he1p 0f (8) and (20) we f1nd that the 0perat0r (18) 15 the 5p1n 0perat0r 0f the part1c1e. 1t 15 06v10u5 that 1f 1n (17) we ch005e the 519n ••-••, 1.e., rea112e ~ 1n the f0rm ,~= - 1 ~

-a,

(21)

(15)

Let u5 c0n51der n0w the ca5e when a 15 an 1nte9er 0r ha1f-1nte9er num6er, a = •k, where k 15 an ar61trary 1nte9er. A5 a re5u1t we 06ta1n the wave funct10n5

1n5tead 0f (18), then we f1nd that when 0t= •k, the funct10n5 (19) w111a9a1n de5cr16e the phy51ca15tate5, and 1n th15 ca5e the f1r5t and the 5ec0nd 5tate5 w111 have the 5p1n5 5= - •k and 5= •k, re5pect1ve1y, wh11e at 0t # • k the f1r5t wave funct10n fr0m (19) w111de5cr16e the phy51ca1 5tate w1th 5p1n 5 = - 0 t . 1t 15 a150 06v10u5 that when a = •k, repre5entat10n (21) can 6e 06ta1ned fr0m repre5entat10n (18) 6y mean5 0f the un1tary tran5f0rmat10n 0,~(~0)~(,~)09~(~0)= 5 ( ~ ) , where 0~(~0)=exp(21a~0). 8ut when 0t# •k, the act10n 0f the 0perat0r 0~ (~0) 15n0t def1ned 0n the 5pace 0f funct10n5 (13), and theref0re 1n th15 ca5e the 4uantum repre5entat10n5 (18) and (21 ) are 1ndependent. Let u5 5h0w n0w that at a # • k the m0de1 rea11yde5cr16e5 the any0n. F0r th15 we pa551nt0 the re5t frame 0f the part1c1e, p = 0, and r0tate the part1c1e at an91e A~0. 7h15 r0tat10n 15 de5cr16ed 6y the 0perat0r /~ (A~0) = exp (1A~0~). Act1n9 6y 1t 0n the wave funct10n 0f the phy51ca1 5tate ~u,~(p, ~0), we 06ta1n the e4ua11ty

~U~(p, 49)=6(p2+m2)~°(p) ,

R(A~0) ~ (p, ~0) =exp( •+ 1aA~0) ~v~(p, ~0)

~ - ~ (p, ~0) =6 (p2 + m 2) 4/- 2~(p )exp ( - 21a(0),

1n wh1ch the 519n • +•• c0rre5p0nd5 t0 repre5entat10n (18 ), and the 519n ••-•• c0rre5p0nd5 t0 repre5entat10n (21 ). Hence, under r0tat10n at the an91e A~0=2n, the pha5e fact0r 15d1fferent fr0m + 1 and we dea1 w1th

7he f1r5t c0nd1t10n fr0m (15) ha5 the wave funct10n5 (13) w1th ( / ( p ) = 6 ( p 2 + m 2 ) ~ 1 ( p ) a5 the 501ut10n5, and the 5ec0nd c0nd1t10n 15 rewr1tten 1n the e4u1va1ent f0rm

ffd(p) [ (1+c)2~0t2]exp(11~0) = 0 .

(16)

1

7h15 e4uat10n ha5 501ut10n5 d1fferent fr0m 1dent1ca1 2er0 0n1y when

c=+(a-[a1+n),

(17)

where [ a ] 15 an 1nte9er part 0f a, and n 15 50me ar61trary 1nte9er. W1th0ut 1055 0f 9enera11ty 1et u5 put n = [ 0t ], and, 6e51de5, ch005e the 519n • +•• 1n (17), 1.e., rea112e ~ 1n the f0rm 0

~ = - 1 ~ +,~.

(18)

(19) a5 the 501ut10n5 0f e4. (16), wh1ch are the e19enfunc-

109

the any0n. At the 5ame t1me 1n the ca5e 0t= •k, the wave funct10n5 0f phy51ca1 5tate5 (19) under 2n-r0tat10n 15 mu1t1p11ed 6y 4- 1 and - 1 at even and 0dd k, re5pect1ve1y. Let u5 c0n5truct n0w the exp11c1t1y c0var1ant 5cheme 0f 4uant12at10n 0f the 5y5tem, w1th0ut 1mp051n9 9au9e c0nd1t10n5. 1n th15 ca5e the 4uantum ana1095 0f c0n5tra1nt5 (3), (4) turn 1nt0 e4uat10n5 f0r the wave funct10n5 0f the phy51ca1 5tate5: ( p 2 + m 2 ) hv=0,

/)~u=0,

1(~+~)~u=0,

(23) (24)

where ~,MN= ~M~N~ ~N~M. Here we have wr1tten the 4uantum ana109 0fthe c0n5tra1nt 4~ 1n herm1t1an f0rm and, 6e51de5, we have taken the 4uantum ana109 0f the c0n5tra1nt ~2 1n a 5pec1f1c f0rm 1n 0rder t0 06ta1n c0rre5p0ndence w1th the 4uantum 5cheme 0n the reduced pha5e 5pace. 7he 4uantum ana1095 0f c0n5tra1nt5 (3), (4) f0rm the a19e6ra 0f f1r5t-c1a55 c0n5tra1nt5 and theref0re, the 5et 0f 4uantum c0nd1t10n5 ( 2 2 ) - (24) 15 c0n515tent. 1n the repre5entat10n where the 0perat0r5/) u, ~M, E, and 6 are d1a90na1, we rea112e the 0perat0r5 h5,/~e a n d / ~ 1n the f0rm h5 = - 10/045, /)~= - 1 0 / 0 e , / ~ = -10/0v, and f0r ~u and hu we take the f0110w1n9 rea112at10n:

(25)

" - 0- +cr-2(4(1)e(2)u-4(2)e (1)u) ,

(26)

where c 15 an ar61trary rea1 c0n5tant, 4 ~1)= e(u1)4u and r = [ (4t ~) ) 2 + (4 (2)) 2 ] ~/2. 8y a d1rect ca1cu1at10n 0ne 9et5 c0nv1nced that the 0perat0r5 PA and Q~ 5at15fy the can0n1ca1 permutat10n re1at10n5, wh1ch are the 4uantum ana1095 0f (2). Repre5entat10n5 (25 ), (26) can 6e c0nnected w1th the ••5tandard•• repre5entat10n, ~ =10/0p u, h~ = -10/04 u, thr0u9h the re1at10n 0¢(~0)00tJ~ -~ (~0) =0¢, where 00 15 ~ 0r ~t~, and 0¢ 15 the c0rre5p0nd1n9 0perat0r (25) 0r (26); 0c(p) =exp(1c(0) and here exp(1~0) =r-~(4~)+14 ~2) ). 1n the ca5e c = k the 0perat0r 0~ 15 un1tary, 6ut when c ~ k, the re1at10n 5h0u1d 6e c0n51dered a5 f0rma151nce 110

F(p, ~) =J(p2+m2)J(p4+m45)(~)-~/2,

(27)

are the 501ut10n5 0f e45. (22), (23). the act10n 0fthe 5p1n 0perat0r, wh1ch 15 the 4uantum ana109 0f (8), 0n the wave funct10n5 (27) ha5 the f0rm .

0

(28) and e4. (24) 15 n0w reduced t0 the e4uat10n

0..~ c) 2- a2](/(p, exp (1~0))=0.

(29) 51nce we w0rk 1n the c1a55 0f 51n91e-va1ued funct10n5, the funct10n ~/(p, exp(1~a)) can 6e dec0mp05ed 1n a 5er1e5 0f f0rm (13). 7hen fr0m re1at10n5 (28) and (29) 1t f0110w5 that we have reduced the pr061em 0f c0var1ant 4uant12at10n 0f the 5y5tem t0 the a60ve 5cheme 0f 4uant12at10n. Hence, putt1n9 c=a 0r c = - a 1n (25), (26), we repr0duce the re5u1t5 c0rre5p0nd1n9 t0 repre5entat10n (18) 0r (21 ), and here the wave funct10n5

~a(p, ~) =F(p, ~)~,0(p), ~-~(p, ~)=F(p, ~)4/-2~(p)exp ( - 21a~0),

Ycu = 10p0u +cr~24~(4(2)0ue~)~40)0ue~2)),

- 104 u

~F(p, ~)=F(p, ~)(/(p, exp(1~0)),

(22)

( ~,~,MN,~,MN- 2 - - a 2 ) ~v=0,

~u=

1n th15 ca5e Uc 15 n0t a 51n91e-va1ued 0perat0r. 7he wave funct10n5 0f the f0rm

~

(/~c~+m~5) hu= 0 ,

( / ~ + m~5) ~ = 0 , /)~=0,

27 5eptem6er 1990

PHY51C5 LE77ER5 8

V01ume248, num6er 1,2

c0rre5p0nd t0 wave funct10n5 (19). 7hu5 we have 5h0wn that the m0de1 0f a 5p1nn1n9 part1c1e 1n (2 + 1 ) d1men510n5, 91ven 6y 1a9ran91an ( 1 ), can 6e c0n51dered a5 the re1at1v15t1c m0de10fthe any0n. A 5ca1ar pr0duct 1n 60th 4uantum 5cheme5 (w1th f1x1n9 9au9e5 0r w1th0ut 1t) can 6e c0n5tructed acc0rd1n9 t0 the 9enera1 pr0cedure [ 10 ] f0r 5y5tem5 w1th f1r5t-c1a55 c0n5tra1nt5 (5ee a150 ref. [ 11 ] ). 7he c1a551ca11a9ran91an ( 1 ) ha5 1nvar1ance under t1me rever5a1, 7: x ° ~ - - x °, 40~ •40, and par1ty, P: X 1 ~ X 1, 41•+•41, X2~X 2, 42~42. Each 0f the5e 0perat10n5 1ead5 t0 a chan9e 0f 519n 0f the 5p1n (8): 5-~ - 5. 0 n the 4uantum 1eve1, a5 ha5 6een 5h0wn, at a # • k the phy51ca1 5pectrum c0nta1n5 0n1y 0ne 5tate, e1ther w1th 5p1n 5=a 0r 5= - a . Fr0m here we c0nc1ude that 0n the 4uantum 1eve1 1n the ca5e when 0t ~ • k the P and 7 1nvar1ance5 are v101ated.

V01ume 248, num6er 1,2

PHY51C5 LE77ER5 8

7 h e rea50n 0 f the a p p e a r a n c e 0 f fract10na1 5p1n 1n the m0de1 15 c0nd1t10ned 6y the f0110w1n9 c1rcum5tance. A5 we have 5een, after 1mp051n9 9au9e c0nd1t10n5 ( 9 ) , ( 1 0 ) , the pha5e 5pace c00rd1nate5 are d1v1ded 1nt0 the ••externa1•• c00rd1nate5 pu a n d 2 u a n d ••1nterna1•• c00rd1nate5 5 a n d ~0. 7 h e c00rd1nate5 2 u are the ana109ue5 0 f the N e w t 0 n - W 1 9 n e r c00rd1nate5 [ 12 ], wh11e the ••1nterna1•• pha5e 5pace 15, 06v10u51y, the pha5e 5pace 0 f the tw0-d1men510na1 r0tat0r. A n d 51nce the c0nf19urat10n 5pace 0 f the r0tat0r, wh1ch 15 de5cr16ed 6y the an9u1ar var1a61e ~0, 15 mu1t1c0nnected, n 1 ( 5 0 ( 2 ) ) = 2 , mu1t1-va1ued repre5entat10n5 are a110wed [ 13 ]. 7 h e ••r0tat0r•• nature 0 f the m0de1 can 6e 5een 0n the 1a9ran91an 1eve1 t00. 1ndeed, 1n the 1n1t1a1 1a9ran91an ( 1 ) the 9au9e 1nvar1ance 9enerated 6y the c0n5tra1nt 07 Can 6e f1xed 1f putt1n9 4 ° + 4 5 = 0 1n 1t. 7 h e n f0r the 1a5t t e r m 1n ( 1 ) we 06ta1n the expre5510n AL= -a1~61,

(30)

where the an91e ~05et5 the un1t 5pace vect0r 4/] 4 ] w1th re5pect t0 50me f1xed reference frame. U p t0 the m0du1u5 519n, expre5510n ( 3 0 ) c01nc1de5 w1th the t e r m 0 f ••5tat15t1ca1 1nteract10n•• 1n the 1a9ran91an 0f the 51mp1e5t n0nre1at1v15t1c m0de1 0f the any0n (5ee, e.9., ref. [ 14 ] ), AL = - a(0, where the re1at1ve vect0r, x, - x 2 , 0f tw0 part1c1e5 w1th c00rd1nate5 x1 a n d x2, act5 1n the capac1ty 0f4. 7 h e pre5ent m0de1 15 c0nnected w1th the re1at1v15t1c part1c1e w1th t0r510n, wh05e act10n appeared 1n ref. [ 8 ] wh1ch wa5 d e v 0 t e d t0 the 805e-Ferm1 tran5mutat10n5, 1n the f0110w1n9 way. 1f 1n5tead 0f9au9e c0nd1t10n5 ( 1 0 ) we 1mp05e the 9au9e c0nd1t10n5 24 = ~5 ~ 0 a n d 2 = 45 ~ 0, wh1ch them5e1ve5 f0rm a 5et 0 f 5ec0nd-c1a55 c0n5tra1nt5, a n d redef1ne the 6racket5 w1th the he1p 0f the 5et 0 f c0nd1t10n5 ¢~3~ 0, ~4~ 0, •2 ~ 0, X3 ~ 0, 24 ~ 0, 25 ~ 0, then we exc1ude the var1a61e5 45, n5, e, Pe, v a n d Pv, a n d the 6racket5 f0r the re5t 0f the var1a61e5 d0 n0t chan9e. A5 a re5u1t, we 5ha11 have 1n the 5y5tem the c0n5tra1nt5 4 n ~ 0 , (42~2--0~2) ~0, ~5~0, P 4 ~ 0 , p n . ~ 0 a n d the 9au9e 4 2 1 ~ 0. 7h15 wh01e 5et 0fc0nd1t10n5 appear5 a5 the f1na15et 0 f t h e c0n5tra1nt51n the m 0 d e 1 0 f t h e part1c1e w1th t0r510n 1n the 11m1t x ~ 0 , where x 15 the curvature 0 f the w0r1d traject0ry, a n d theref0re, the pre5-

27 5eptem6er 1990

ent m0de1 de5Cr16e5, 1n fact, the 5tate5 0f the part1C1e w1th t0r510n w1th the max1mUm Va1Ue 0 f the ma55: M = m (5ee ref. [15] ~1). 1n C0nC1U510n we n0te that 1t w0U1d 6e 1ntere5t1n9 t0 C0n5truCt a C0rre5p0nd1n9 f1e1d 1a9ran91an, wh1Ch w0U1d 91Ve 4UantUm C0nd1t10n5 ( 2 2 ) - ( 2 4 ) a5 the f1e1d e4Uat10n5 0 f m0t10n 1n 0rder t0 perf0rm 5eC0nd 4Uant12at10n 0f the m0de1.

~1 1n ref. [ 15] the 4uant12at10n 0fthe pan1c1e w1th t0r510n wa5 c0n51dered 1n the 11m1tx~0 0n1yf0r the rea112at10n0fthe 5p1n 0perat0r 1n the f0rm (14) w1th c=0.

Reference5 [1] J.M. Le1naa5 and J. Myrhe1m, Nu0v0 C1ment0 8 37 (1977)1; F. W11c2ek,Phy5. Rev. Lett. 48 (1982) 1144. [2] E W11c2ek,Phy5. Rev. Lett. 49 (1982) 957. [3] F. W11c2ekand A. 2ee, Phy5. Rev. Lett. 51 (1983 ) 2250; Y.-5. Wu and A. 2ee, Phy5. Lett. 8 147 (1984) 325; D.P. Ar0va5, R. 5chr1effer, F. W11c2ekand A. 2ee, Nuc1. Phy5. 8 251 (1985) 117. [4] R.8. Lau9h11n, Phy5. Rev. 8 23 (1983) 3383; 8.1. Ha1per1n, Phy5. Rev. Lett. 52 (1583; D.P. Ar0va5, R. 5chr1efferand F. W11c2ek,Phy5. Rev. Lett. 53 (1984) 722. [ 5 ] V. Ka1meyerand R.8. Lau9h11n, Phy5. Rev. Len. 59 ( 1987 ) 2095; R.8. Lau9h11n, 5c1ence 242 (1988) 525; Phy5. Rev. Lett. 60 (1988) 2677; 1. D2ya105h1n5k11,A. P01yak0v and P.8. W1e9mann, Phy5. Len. A 127 (1988) 112. [6] M.5. P1yu5hchay, Phy5. Lett. 8 236 (1990) 291. [7] F.A. 8ere21n and M.5. Mar1n0v, 50v. Phy5. JE7P Lett. 21 ( 1975 ) 321; Ann. Phy5. (NY) 104 ( 1977 ) 336; R. Ca5a16u0n1, Nu0v0 C1ment0 A 33 (1976) 369; L. 8r1nk, 5. De5er, 8.2um1n0, P.D1 Vecch1aand P. H0we, Phy5. Lett. 8 64 (1976) 435; L. 8r1nk, P.D1Vecch1aand P. H0we, Phy5. Lett. 8 65 (1976) 471. [8] A.M. P01yak0v, M0d. Phy5. Lett. A 3 (1988) 325; 5ee a150C.-H. 72e, 1ntern. J. M0d. Phy5. A 3 (1988) 1959. [9] P.A.M. D1rac, Lecture5 0n 4uantum mechan1c5 (8e1fer 6raduate 5ch0010f 5c1ence, Ye5h1vaUn1ver51ty,New Y0rk, 1964). [ 10 ] V.6.8udan0v, A.V. Ra2um0v and A.Yu. 7aran0v, Pr0c. 1V 5em1nar 0n Pr061em5 0f h19h ener9y phy51c5and 4uantum f1e1dthe0ry, V01.2 (Pr0tv1n0, 1981 ) p. 273.

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PHY51C5 LE77ER5 8

[ 11 ] M.5. P1yu5hchay, Ma551e55part1c1e w1th r191d1ty a5 a m0de1 f0r de5cr1pt10n 0f 6050n5 and ferm10n5, Pr0tv1n0 prepr1nt 1HEP 90-32 (1990). [ 12] 7.D. Newt0n and E.P. W19ner, Rev. M0d. Phy5. 21 (1949) 400.

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[ 13 ] L.5. 5chu1man, Phy5. Rev. 176 ( 1968 ) 1558; 7echn14ue5 and app11cat10n5 0f path 1nte9rat10n (W11ey, New Y0rk, 1981). [ 14] R. Macken21e and F. W11c2ek, 1ntern. J. M0d. Phy5. A 3 (1988) 2827. [ 15] M.5. P1yu5hchay, Phy5. Lett. 8 235 (1990) 47.