Adiabatic theorem and anomalous commutators

V01ume 184, num6er 2,3

PHY51C5 LE77ER5 8

29 January 1987

AD1A8A71C 7 H E 0 R E M AND A N 0 M A L 0 U 5 C 0 M M U 7 A 7 0 R 5 5h1nj1 11DA a,6 and H1r05h1 KURA75UJ1 ° a Department 0fPhy51c5, Ky0t0 Un1ver51ty, Ky0t0 606, Japan 6 Re5earch Centerf0r Nuc1ear Phy51c5, 05aka Un1ver51ty, 05aka 567, Japan c Department 0fPhy51e5, R1t5ume1kan Un1ver51ty, Ky0t0 603, Japan

Rece1ved21 May 1986; rev15edmanu5cr1ptrece1ved 1 0ct06er 1986

we 5h0wthat the 9e0metr1ca1pha5e fact0r recent1yf0und 1n the 4uantum ad1a6at1cthe0rem m0d1f1e5the can0n1ca15tructure 0fthe pha5e 5pace.7h15 91ve5r15et0 the an0ma10u5term51n the 4uantum mechan1ca1c0mmutat10n re1at10n5.

7he 06ject w1th wh1ch we are c0ncerned 1n 4uantum mechan1c5 9eneraUy f0rm5 a 5y5tem c0mp05ed 0f a num6er 0f part5 under mutua11nteract10n. Apart fr0m a na1ve pertur6at10n the0ry, the meth0d dea11n9 w1th the 1nteract1n9 5y5tem515 0ften 6a5ed 0n the ••ad1a6at1c the0rem••, the c1a551ca1 examp1e 0fwh1ch date5 6ack t0 Ehrenfe5t 1n the day5 0f the 01d 4uantum mechan1c5 [ 1 ]. 7he 9enera1 f0rm 0f th15 the0rem 15 5tated a5 f0110w5 [2]: C0n51der tw0 1nteract1n9 5y5tem5 and, f0r the m0ment, free2e the m0t10n 0f 0ne 0f the 5y5tem5, that 15, the dynam1ca1 var1a61e5 0f the fr02en 5y5tem enter the ham11t0n1an f0r the rema1n1n9 5y5tem a5 externa1 parameter5. 7hen the rema1n1n9 ham11t0n1an 15 501ved 5uch that the 4uantum num6er5 6ec0me 1nvar1ant dur1n9 510w chan9e 0f the externa1 parameter5. 0 f c0ur5e, the ad1a6at1c the0rem d0e5 n0t h01d exact1y f0r the actua1 pr0ce55e5 0f f1n1te per10d5 0f m0t10n. H0wever, the c0ncept 0f ad1a6at1c chan9e 5erve5 a5 a p0werfu1 ••w0rk1n9 hyp0the515•• 1n treat1n9 a w1de c1a55 0f 4uantum pr061em5 1nc1ud1n9 even 4uantum f1e1d the0r1e5 [ 3 ], 51nce the ad1a6at1c 1nvar1ant 6ec0me5 a pert1nent 4uant1ty t0 c1a551fy the 4uantum 5tate5 f0r the 5y5tem under t1me-vary1n9 externa1 f1e1d5. Very recent1y 1t ha5 6een pr0ved that the c0nvent10na1 4uantum ad1a6at1c the0rem 15 n0t c0rrect f0r the 5pec1a1 51tuat10n that the chan9e 0f the externa1 parameter5 take5 p1ace a10n9 a c105ed 100p. F0r th15 ca5e the wave funct10n ac4u1re5 an extra pha5e fact0r 0f9106a1 nature depend1n9 0n the 100p c0nf19urat10n

242

1n the parameter 5pace [4-6]. 7he appearance 0fth15 5pec1f1c pha5e wh1ch we ca11 the 4uantum ad1a6at1c pha5e (a11a5 8erry•5 pha5e wh1ch we wr1te a5 F ) , ha5 6een 5u99e5ted ear1y 1n 4uantum chem15try [ 7,8 ]. 1n 0ur prev10u5 5tud1e5 [9] the dynam1ca1 ar9ument5 f0r the pha5e F have 6een 91ven u51n9 the path 1nte9ra1 f0rmu1at10n. 7he ma1n c0n5e4uence 06ta1ned there 15 that F appear5 a5 a t0p01091ca1 act10n added t0 the ad1a6at1c act10n funct10n 0f c0nvent10na1 f0rm f0r the externa1 5y5tem, wh1ch 1ead5 t0 the 5em1c1a551ca1 4uant12at10n 1nc1ud1n9 the effect 0 f F . Furtherm0re, the re1at10n 6etween the 4uant12ed Ha11 c0nductance and the ad1a6at1c pha5e, wh1ch wa5 f1r5t e5ta6115hed 6y 51m0n [ 5 ], ha5 6een extended t0 the 9enera1 ca5e 0f the many-part1c1e ham11t0n1an [ 10]. 1n th15 1etter, we addre55 the 4ue5t10n c0ncern1n9 the ad1a6at1c pha5e fr0m a new p01nt 0fv1ew, that 15, we c0n51der the pr061em 0f h0W the ad1a6at1c pha5e affect5 the 4uantum mechan1ca15tructure 1n an 0perat0r f0rm. 7he ma1n c0n5e4uence 15 that the ad1a6at1c pha5e 91ve5 r15e t0 an0ma10u5 term5 1n the 4uantum mechan1ca1 c0mmutat10n re1at10n5 a5 a re5u1t 0f the m0d1f1cat10n 0f the can0n1ca1 5tructure 0f the pha5e 5pace. 1n th15 c0nnect10n we ment10n 5evera1 recent w0rk5 dea11n9 w1th the an0ma10u5 c0mmutat10n re1at10n5 appear1n9 1n the 9enerat0r5 0f the 9au9e tran5f0rmat10n5 [ 11-13 ], a60ut wh1ch we w11191ve a 6r1ef acc0unt at the end 0fthe paper. We 5tart w1th a 6r1ef 5ummary 0f prev10u5 re5u1t5 [ 9 ]. C0n51der a dynam1ca1 5y5tem (wh1ch we ca11 an

V01ume 184, num6er 2,3

PHY51C5 LE77ER5 8

29 January 1987

externa1 5y5tem) 1nteract1n9 w1th a certa1n 1nterna1 5y5tem. 7he ham11t0n1an 0f th15 5y5tem 15 91ven 6y /~=/~0(X) +/~(4; X) where X1= (01 .... , an, e1, ..., Pn) and 4 den0te the can0n1ca1 var1a61e5 f0r the externa1 5y5tem and 1nterna1 var1a61e5 re5pect1ve1y. 7he ad1a6at1c pha5e appear5 a5 a c0n5e4uence 0f c0n5truct1n9 the effect1ve path 1nte9ra1 f0r the trace 0f the ev01ut10n 0perat0r K ( 7 ) = 7 r ( e x p [ - 1 f 1 7 / h ] ) 6y app1y1n9 the ad1a6at1c appr0x1mat10n t0 the 1nterna1 m0t10n ( we u5e the ad1a6at1c ener9y 1eve15 2n and the c0rre5p0nd1n9 e19en5tate 1n ) wh1ch 5at15f1e5 the 5nap5h0t e19enva1ue e4uat10n/~(4; X) 1n ) =,~n [n ) ):

0ther hand, the 1nte9rand 0f the pha5e F , that 15, 09 = 1(n 1d 1n ) m0d1f1e5 the 50-ca11ed ••can0n1ca1 term••: P.dQ (0r 1 h(21d1 ) ). 7he 1atter 5erve5 a5 a 10ca1 c0unterpart 0f the pha5e h010n0my 0f 9106a1 5tructure 0n wh1ch we f0cu5 0ur attent10n e5pec1a11y 1n the f0110w1n9. Let u5 den0te the ham11t0n1an 5y5tem a5 {M, £2, Had}, wh1ch 15 a550c1ated w1th the act10n 5eff, where M 15 a pha5e 5pace wh05e c00rd1nate5 are •91ven 6y X = {Q,P} and 92 15 a 5ymp1ect1c f0rm def1ned a5 12= d ( ~P~dQ~ + 1h ( n 1d 1n) ), wh1ch 15 reduced t0

K ( 7 ) = ~ J-eXp [ 1/h5~ff] 1-[d/2(Xt),

92~--2dP,~ ~ d Q ~ + h 2 A ( X j , X~) dX~ •dXj,

(1)

1

3(Xj, X1) = 1 ( n1 ( 0/0X~)( 0/0Xj)1n) - (1 ~ j ) .

where 5~ff = 5ad + hF , (C) w1th

(3)

5a =J 0--0- 0)at, F , ( C) =~< n[10/0X~ 1n> dX1.

(2)

C

Here 5ad 15 the ad1a6at1c act10n funct10n and the ad1a6at1c pha5e F repre5ent5 the pha5e h010n0my f0r an ad1a6at1c tran5p0rtat10n a10n9 the c105ed 100p C 1n the X-5pace. 1f we u5e the c0herent 5tate (0r h010m0rph1c) repre5entat10n 1n the a60ve f0rmu1a [ 14], the act10n funct10n f0r the externa1 5y5tem 15 91ven 6y 5 0 ( C ) = f ( 2 1 1 h 0 / 0 t - H 0 1 2 ) dt and the expre5510n 0f 7• may 6e 06ta1ned 6y rep1ac1n9 X 6y c0mp1ex var1a61e5 2 and 2* [ 2 = ( Q + 1 P ) / , 2 ~ ] 1n e4. (2). Acc0rd1n9 t0 the prev10u5 ar9ument5, F turn5 0ut t0 6e the ma9net1c f1ux a550c1ated w1th the ••effect1ve vect0r p0tent1a1•• (A1 = ( n [10/0X11 n ) ) Wh1Ch 15 pr0duced 6y effect1ve ••D1rac p01e5•• 10cated at the ••de9enerate p01nt5•• (name1y, the 1nter5ect10n p01nt5 0fthe ad1a6at1c ener9y 1eve15) 0ccurr1n9 1n the pha5e 5pace. 1n th15 way the t0p01091ca1 nature 0f F 15 1mp11ed fr0m the 065ervat10n that F take5 a va1ue even 1f the 100p C 1t5e1f d0e5 n0t 90 thr0u9h the de9enerate p01nt. N0w 5uch a 9106a1 feature 15 expected t0 take 0ver t0 the 10ca1 5tructure 0f the pha5e 5pace dynam1c5 f0r the externa1 5y5tem de5cr16ed 6y the can0n1ca1 var1a61e5 (Q,P). When C0n51der1n9 the 10ca1 5tructure 0fthe pha5e 5pace, we 5h0u1d pay attent10n t0 the fact that the 1nte9rand 0f the effect1ve act10n 15 d1v1ded 1nt0 tw0 part5 0f d1fferent nature: 0n the 0ne hand, 2, 91ve5 a 51mp1e m0d1f1cat10n f0r the ham11t0n1an term, and 0n the

N0te that the 5ec0nd term 0f e4. (3) m0d1f1e5 the u5ua1 5ymp1ect1c f0rm t2°=27dP~ • d Q , and 9enerate5 the n0ntr1v1a1 metr1c 0fthe pha5e 5pace 9 = J + hA ( J 15 the u5ua1 5ymp1ect1c matr1x). 7hu5 £2 15 c0mpact1y wr1tten a5 9-j= ~ 9 1 j ~ 1

A dXj .

(4)

1n 0rder t0 e5ta6115h the c0nnect10n 6etween the c1a551ca1 and 4uantum the0ry, we 1ntr0duce the P01550n 6racket ( P 8 ) . 8y u51n9 the 5ymp1ect1c f0rm, the P8 15 def1ned 6y [ 15-17] *1

0A 08 {A,8} =92(XA,X8) = 291J 0 X1 0Xj ~

(5)

wh1ch 15 rewr1tten a5

{A,8} = {A,8} 0 + h{A,8} 1 + ....

(6)

w1th

{A,8}0=• 0A j

08

L-~11 0 0xj~ 0A A 08 {A,8} 1 = E - ~ J0 (Xj, Xk)Jkt - - .

0x1

(7)

Here Xa den0te5 a vect0r f1e1d 9enerated 6y A (Q, P) *~A 51m11arf0rm 0f the 9enera112edP01550n 6racket ha5 prev10u51y 6een u5ed 1n the t1me-dependent Hartree-F0ck the0ry [ 15], where the canan0n1ca1term 1h(21 d 12) 15n0t wr1tten a5 a 51mp1ef0rm PdQ 6ut 91ven6y a m0re c0mp11catedf0rm due t0 the metr1c0fa curved pha5e 5pace. 243

V01ume 184, num6er 2,3

PHY51C5 LE77ER5 8

and 9a an 1nver5e 0f the metr1c 95j. Fr0m the a60ve c0n5truct10n we 5ee that the P8 f0r the externa1 5y5tem 15 a1tered fr0m the 0ne 0r191na11y def1ned. 7h15 feature 15 re9arded a5 a 10ca1 man1fe5tat10n 0f the pha5e F. Name1y, the 0r191na1 can0n1ca1 var1a61e5 (Q, P) are n0 10n9er can0n1ca1 1n the c0nvent10na1 mean1n9 a5 a c0n5e4uence 0f the e11m1nat10n 0f the de9ree5 0f freed0m 0fthe 1nterna1 5y5tem. E5pec1a11y, the e4uat10n5 0f m0t10n are m0d1f1ed a5

E91J dXj j

0H dt - 0 X 1 "

(8)

1fthe ham11t0n1an 5y5tem 15eventua11y 1nte9ra61e, th15 effect 91ve5 r15e t0 the m0d1f1cat10n 0fthe fre4uency 0f the m0t10n v and hence the act10n 1nte9ra1 J = fv-~ dE, wh1ch 1ead5 t0 the chan9e 0f the 5em1c1a551ca1 4uant12at10n ru1e [ 9 ]. N0w, we turn t0 the 4uantum mechan1ca1 c0rre5p0ndence 0fthe c1a551ca1 mechan1ca1 5tructure c0n5tructed a60ve. Acc0rd1n9 t0 the 5tandard pr0cedure, th15 15 acc0mp115hed 51mp1y 6y rep1ac1n9 the P8 6y the c0mmutat0r5. 7hu5, e4. (6) 1mp11e5 the f0110w1n9 c0mmutat10n re1at10n5:

[0., 0~1 = - 1 h : 2 ( ~ , ~ ) + ....

[/3,/5p] = -1h2A(~=, 0p) ~- ....

(9)

wh1ch f0rm a 6a515 0f the 4uantum mechan1c5 1nc1ud1n9 the effect 0f the ad1a6at1c pha5e F . Fr0m the a60ve expre5510n we 5ee an apparent dev1at10n fr0m the u5ua1 c0mmutat0r wh1ch 5ha116e ca11ed the "an0ma10u5 term5••. A5 an examp1e 0f the 9enera1 the0ry thu5 pre5ented, we c0n51der the f0110w1n9 m0de1 ham11t0n1an f0r wh1ch the 1nterna1 5y5tem c0n515t5 0f ju5t tw0 5tate5 (den0ted 6y [ 1 > and [2 > ), that 15

h=

(°

v~*

-~

29January1987

p = (Q2 + p 2 ) / 2 h • F• =f3~+ dQ~dP=~2(1 + d e ) , Ae =1[< + t ( 0 / 0 P ) ( 0 / 0 Q ) 1 + > , (Q*-*P)]

= ~ v2E/4e3h.

(11 c0nt•d)

Here we 1a6e1 the ad1a6at1c 1eve15 +. Due t0 the 5pec1a151tuat10n 0fthe pre5ent m0de1, A 6ec0me5 a funct10n 0 f p [ = (Q2+p2/2h] 0n1y, and furtherm0re the P8 15 m0d1f1ed fr0m the 0r191na1 0ne 6y the f0110w1n9 fact0r:

{A,8} = [ 1 +h3+ ( Q , P ) ] - 1 { A , 8 } ° . Hence we 06ta1n the c0mmutat0r f0r the 6050n

[~ ~ - ] = [1+~A+(0,P)]-~ = 1 - h 3 2 ( 0 , / 3 ) + ....

(12)

where the 5ec0nd and the rema1n1n9 term5 1n the expan510n may 6e re9arded a5 the ••an0ma10u5 term•• 1n the c0mmutat0r. N0w, we addre55 the 4ue5t10n 0f h0w we can under5tand the a60ve an0ma10u5 c0mmutat0r 1n the c0ntext 0f the c0nvent10na1 4uantum mechan1ca1 appr0ach. F1r5t, we n0t1ce that fr0m the expre5510n f0r the effect1ve path 1nte9ra1 [ 5ee e4. (1) ] the effect1ve ham11t0n1an 5y5tem a550c1ated w1th the act10n 5efr 15 d15cr1m1nated 6y the 4uantum num6er5 n. 7heref0re the re1evant 4uantum 5y5tem 5h0u1d n0t 6e c0n5tructed fr0m a fu11 H116ert 5pace 6ut fr0m a certa1n pr0jected 5pace 5pec1f1ed 6y n. 7h15 mean5 that the c0mmutat0r read5

[P.AP., Pn]~Pn] ,

(13)

where P, 15 the pr0ject10n 0perat0r 0nt0 a 5pace c0rrep0nd1n9 t0 the 1eve1 de519nated 6y n. 1n the pre5ent ca5e, n take5 0n + and we can ea511y 06ta1n the exact e19en5tate5 0f the ham11t0n1an a5

(10) 14/+(n)>

7he ad1a6at1c ener91e5 a5 we11 a5 the pha5e F are eva1uated a5 f0110w5:

E+ = + , , / ~ + v 2 p = + e(p),

244

(11)

=A•(n) 1 1 > × 1 n - 1 > + 8 + ( n ) 1 2 > × 1 n > ,

A-- (n) = vx/n1/x/2e(n)[e(n) ~e]

(14)

V01ume 184, num6er 2,3

PHY51C5 LE77ER5 8

8+• (n) = [ + e(n)-- E] /X/2e(n) [e(n)-7- e] , t

(14 c0nt•d) w1th the e19enva1ue5 E+~=+e(n)=+x//~+v2n, where 1n > are the e19en5tate5 0f the 6050n num6er 0perat0r. 7he pr0ject10n 0perat0r 0nt0 each ad1a6at1c 1eve1 + 15 def1ned a5 P•+ = ~1~u+ ( n ) ) <4/•+ (n) 1 •

(15)

n

1fwe are c0ncerned w1th the expectat10n va1ue 1n5tead 0f the c0mmutat10n re1at10n 1t5e1f, we 9et

( ~ +~(n)1[P+f1P+, P+~Ct+P+ ] 19t+• (m) > = {A2(n)A2(n+ 1) + (n-- 1)A2(n)[A2(n+ 1 ) --A2(n - 1 )] + 82( n )82( n +1) + n82(n)[82(n+ 1 ) --82(n - 1 )] +~ - - ~

29 January 1987

the0ry wh1ch 15 re9arded a5 an 1nf1n1te-d1men510na1 dynam1ca1 5y5tem. We 1mmed1ate1y 9ue55 that the f1e1d-the0ret1c ver510n 0f the c0mmutat0r (7) w0u1d 6e c105e1y re1ated t0 the 50-ca11ed 5chw1n9er term current1y rev1ved 1n 4uantum f1e1d the0ry. 1n c0nnect10n w1th th15 there have appeared 5evera1 w0rk5 dea11n9 w1th the an0ma10u5 c0mmutat0r5 a550c1ated w1th the ch1ra1 an0ma1y fr0m the v1ewp01nt 0f the ad1a6at1c pha5e: the appr0ach 6y Ne150n and A1vare26 a u m 6 6a5ed 0n the ham11t0n1an 1nterpretat10n [ 11 ], the appr0ach 6y N1em1 and 5emen0ff [ 12 ] u51n9 the pha5e h010n0my 0n the 9au9e f1e1d 5pace and the d1rect eva1uat10n 0f the ad1a6at1c pha5e 6y 50n0da [ 13]. 1n th15 w0rk, we have pre5ented the 9enera1 framew0rk wh1ch 5h0w5 that the ad1a6at1c pha5e 15 nece55ar11y c0nnected w1th the an0ma10u5 term5 1n the c0mmutat10n re1at10n thr0u9h the m0d1f1cat10n 0f the can0n1ca1 5tructure 0f the pha5e 5pace. 7heref0re the pre5ent the0ry w0u1d exp1a1n the an0ma10u5 c0mmutat0r5 a550c1ated w1th a ch1ra1 an0ma1y fr0m a 4u1te d1fferent p01nt 0f v1ew. H0wever, th15 expectat10n cann0t 6e fu11y exam1ned here and w1116e d15cu55ed 1n a f0rthc0m1n9 paper [ 18 ].

)2A(n)A(n+ 1 ) 8 ( n ) 8 ( n • 1) )2A(n-- 1)A( n)8( n)8(n-- 1)}6.m

= [ 1 + v2E/4e3(n) +...]~nm •

(16)

E4. (16) C0rre5p0nd5 t0 the an0ma10U5 C0mmutat0r e4. (12 ) 1n the 5en5e 0f expectat10n va1ue. 7he f0urth and f1fth 11ne5 0fe4. (16 ) y1e1d the expan510n 1n p0wer5 0fh Up t0 the f1r5t 0rder. 1fwe n0te that the 5em1C1a551ca1 4uant12at10n c0nd1t10n f0r th15 m0de1 91ve5 the re1at10n p = ( p 2 + Q 2)/2h = 1nte9er5 + 0 (h) and 1f We C0mpare e4. (12) W1th e4. (16), We Can 5ee that the m0d1f1ed C0mmUtat0r, Wh1Ch 15 06ta1ned 0n the 6a515 0f the 1nC1U510n 0f the ad1a6at1C pha5e, repr0duce5 the ••exact re5u1t•• up the appr0pr1ate 0rder. We 5ummar12e the pre5ent re5u1t: 7he e55ence 0f 0ur the0ry 15 that the ad1a6at1c pha5e m0d1f1e5 the can0n1ca1 5tructure 0f the dynam1ca1 5y5tem (wh1ch we ca11 the externa1 5y5tem). 7h15 91ve5 r15e t0 the an0ma10u5 term5 1n the 4uantum mechan1ca1 c0mmutat10n re1at10n5. 51nce F appear5 un1ver5a11y whenever 0ne dea15 w1th 1nteract1n9 5y5tem5, a5 wa5 prev10u51y 5u99e5ted [ 9 ], we expect that the pre5ent 1dea can 6e extended t0 the ca5e 0f the 9au9e f1e1d

7he auth0r w0u1d 11ke t0 thank Dr. 7. Hat5uda and Mr. 5. Y0r0 f0r the1r u5efu1 c0mment5. 0 n e 0f the auth0r5 (•1.) 15 1nde6ted t0 Japan 50c1ety f0r the Pr0m0t10n 0f 5c1ence f0r f1nanc1a1 5upp0rt.

N0te added. After we 5u6m1tted th15 paper, exper1menta1 065ervat10n5 0fthe ad1a6at1c pha5e have 6een made u51n9 the m01ecu1ar 5y5tem [ 19 ] and the 0pt1ca1 dev1ce [20]. Reference5 [ 1] 5. 70m0na9a, Quantum mechan1c5 M15u2u-5h060, 1964) [1n Japane5e]. [2] A. Me551ah, Quantum mechan1c5,V01. 2 (N0rth-H011and, Am5terdam, 1962). [3] 5ee e.9. J. 601d5t0ne and F. W11c2ek,Phy5. Rev. Lett. 47 (1981) 986. [4] M.V. 8erry, Pr0c. R, 50c. A 392 (1984) 45. [5] 8. 51m0n, Phy5. Rev. Lett, 51 (1983) 2167. [6] F. W11c2ekand A12ee, Phy5. Rev, Lett. 52 (1984) 2111. [7] H.C. L0n9ett-H1991n5,Pr0c. R. 50c. A 344 (1975) 147; A.J. 5t0ne, Pr0c. R. 50c. A 351 (1976) 141. [8] C.A. Mead and D. 7ruh1ar, J. Chem. Phy5. 70 (1979) 2284. [9] H. Kurat5uj1 and 5. 11da, Phy5. Lett. A 111 (1985) 220; Pr09r. 7he0r. Phy5. 74 (1985) 439; Phy5. Rev. Lett. 56 (1986) 1003. 245

V01ume 184, num6er 2,3

PHY51C5 LE77ER5 8

[10]H. Kurat5uj1, R1t5ume1kan prepr1nt PHAN70M 861 (1986). [ 11 ] P. Ne150n and L. A1vare2-6aum6, C0mmun. Math. Phy5. 99 (1985) 103. [12] A. N1em1 and 6 . w . 5emen0ff, Phy5. Rev. Lett. 55 (1985) 927; 56 (1986) 1019. [ 13] H. 50n0da, Nuc1. Phy5. 8 266 (1986) 410. [14] v. 8ar9mann, C0mmun. Pure App1. Math. 14 (1961) 187; 20 (1967) 1. [15] H. Kurat5uj1, Phy5. Lett. 8 103 (1981) 79; 8 108 (1982) 367.

246

29 January 1987

[ 16] F.A. 8ere21n, C0mmun. Math. Phy5. 40 (1975) 153. [17]8. K05tant, Lecture N0te5 1n Mathemat1c5, v01. 170 (5pr1n9er, 8er11n, 1970) 87. [ 18] H. Kurat5uj1 and 5.11da, t0 6e 5u6m1tted. [ 19 ] 6. De1acr6ta2, E.R. 6rant, R.L. whetten and L. W85te, Phy5. Rev. Lett. 56 (1986) 2598. [20] R.Y. Ch1a0 and Y.-5. wu, Phy5. Rev. Lett. 57 (1986) 933; A. 70m1ta and R.Y. Ch1a0, Phy5. Rev. Lett. 57 (1986) 937.