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The BRST treatment of collective rotational coordinates in axially-symmetric systems
V01ume 248, num6er 1,2
PHY51C5 LE77ER5 8
27 5eptem6er 1990
7he 8 R 5 7 treatment 0f c011ect1ve r0tat10na1 c00rd1nate5 1n ax1a11y-5ymmetr1c 5y5tem5 D.R. 8e5 1, R. De LUCa a n d J. KUrChan Phy51c5Department CNEA, Av. de1L16ertad0r8250, 14298uen05A1re5,Ar9ent1na Rece1ved 19 Apr111990; rev15edmanu5cr1pt rece1ved 17 Ju1y 1990
7he 8R57 treatment 0f c011ect1vec00rd1nate5 15extended t0 ca5e5 1n wh1ch there 15a part1a16reakd0wn 0f 5ymmetr1e51n the mean-f1e1d appr0x1mat10n. 7he re5u1tant 5ymmetr1e50f the wave funct10n5 are d15cu55ed, a5 we11a5 the f0rm 0f the re51dua1 1nteract10n5and, 1n part1cu1ar,0fthe u5ua1C0r10115term. An app11cat10nt0 the 80hr c011ect1veham11t0n1an15made a5 an 111u5trat10n.
7 h e 8 e c c h 1 - R 0 u e t - 5 t 0 r a - 7 y u t 1 n 1nvar1ance [ 1 ] ha5 6ec0me a fundamenta1 t001 f0r the 4uant1f1Cat10n 0f 9au9e the0r1e5. 7 h e treatment 0f c011ect1ve C00rd1nate5 a5 an app11cat10n 0f the 8 R 5 7 1nvar1ance 15 carr1ed 0ut 1n ref. [2 ]. 7 h e 1dea 15 t0 c0n51der the parameter5 0f a 9r0up 0f tran5f0rmat10n5 (f0r 1n5tance, the 0r1entat10n an91e5 0v) a5 rea1 c011ect1ve var1a61e5 and t0 1nc1ude extra de9ree5 0f freed0m (La9ran9e mu1t1p11er5, 9h05t5) 50 that the 501ut10n 0f the t0ta1 5pur10u5 5ect0r 6ec0me5 a re1at1ve1y 51mp1e pr0cedure. App11cat10n5 t0 a 51mp1e (a6e11an) tran5f0rmat10n and t0 (n0na6e11an) r0tat10n5 w1th h19h an9u1ar m 0 m e n t u m are made 1n ref. [ 3 ] and ref. [ 4 ], re5pect1ve1y. 5ee a150 ref5. [ 5,6 ] f0r a c0mp1ete d15cu5510n and ref. [ 7 ] f0r a 9enera112at10n t0 f1e1d the0ry. 7 h e 0r191na1 ham11t0n1an H 15 rep1aced 6y the 8 R 5 7 ham11t0n1an H8R5-r=H+ [p, Q1 + , where the (n11p0tent) 8 R 5 7 ••Char9e••, h11ate5 phy51ca1 wave funct10n5, 15
( 1) wh1Ch ann1-
51te 519n); (92v, Pv) pa1r5 0f c0nju9ate var1a61e5 a550c1ated w1th La9ran9e mu1t1p11er5 and (4v, nv), (0v, 9v) pa1r5 0f c0nju9ate 9h05t var1a61e5. 7 h e p 0perat0r 15 ar61trary. 1n ref. [2 ] we have taken
A
(3)
A, 8 are ar61trary C0n5tant5 and 5h0U1d n0t appear 1n any f1na1 re5U1t. 7 h e phy51Ca1mean1n9 0fthe 9au9ef1x1n9 funct10n5 6v 15 that the van15h1n9 0f the1r 9r0und 5tate expectat10n va1ue def1ne5 the 1ntr1n51c 5y5tem. 7he re5u1tant ham11t0n1an 15 91ven 1n e4. (17) 0f ref. [ 2 ]. Here we app1y th15 f0rma115m t0 the pr061em 0f c011ect1ve r0tat10n5 1n rea1 5pace. Hence Jv (v = 1, 2, 3 ) are the an9u1ar m 0 m e n t u m 0perat0r5 and 1v the c0rre5p0nd1n9 c011ect1ve term5. 1n the 501ut10n 0fH8957 pre5ented 1n ref. [ 3 ], the 6.5at15fy, t0 1ead1n9 0rder [6~, J,0] =16,,0 1
[H, 6 , 1 = - - ~ J v
(4) (5)
Q = P ~ v - - ( J v - - 1 , ) 4 v + ~~1~w0¢4vrh0;9~ (V, C0, ~= 1, ..., n) .
(2)
Here Jv are the 9enerat0r5 0f the tran5f0rmat10n W1th 9r0Up 5truCtUre C0n5tant5 ev,0~;1v the1r C011eCt1Vever510n (5at15fy1n9 the 5ame a19e6ra 6ut w1th an 0pp0Fe110w0fthe C0N1CE7, 8uen05 A1re5,Ar9ent1na.
F0r a many-60dy 5y5tem, the5e e4uat10n5 can 6e 501ved ~1 w1th1n the RPA [8 ] and y1e1d, 1n add1t10n, the 7h0u1e55-Va1at1n m 0 m e n t 0f 1nert1a Jv. H0wever, th15 501ut10n cann0t 6e app11ed t0 ax1a11y 5ymat A c0rre5p0nd1n9 (11near12ed) appr0x1mat10n a150 ex15t5 f0r def0rmed 6050n1c5y5tem5.
0370-2693/90/$ 03.50 • 1990 - E15ev1er5c1encePu6115her58.V. ( N0rth-H011and )
1
V01Ume248, num6er 1,2
PHY51C5 LE77ER5 8
metr1c 5y5tem5, 6ecau5e e45. (4) and (5) cann0t 6e fu1f111eda10n9 the 5ymmetry d1rect10n (p = 3 ). 1n an ax1a11y5ymmetr1c ca5e 1t 15n0t p05161e t0 treat c011ect1ve1y the r0tat10n ar0und the 5ymmetry ax15 6ecau5e the fre4uency 0f th15 r0tat10n w0u1d 6e the 5ame a5 the 0ne a550c1ated w1th the c0rre5p0nd1n9 v16rat10n. Hence we 5h0u1d n0t f1x the 0r1entat10n 0f the m0v1n9 frame w1th re5pect t0 r0tat10n5 ar0und th15 ax15. We try a 51m11arexpre510n t0 (3) 6ut w1th the 5ummat10n re5tr1cted t0 5, t = 1, 2. 7he re5u1t1n9 8 R 5 7 ham11t0n1an
H8R57 = H-- Q5( J5 --1~) +11t5~5 + 1 ( 6 5 P 5 - - 2~ P2 +43f15[65, J3]+r1fh[6t, J5])
+1~5t3925(7ttt13 - -
7~3~t)
,
(6)
d0e5 n0t c0nta1n ne1ther the 0perat0r5 t23,/•3, 93 and t73, wh1ch hence unc0up1e fr0m the rema1n1n9 de9ree5 0f freed0m, n0r the 9au9e f1x1n9 0perat0r 6 3. H0wever, the 9h05t 0perat0r5 43, n3 5t111appear and n0 k1net1c term 15 a550c1ated w1th them. 7hu5 the 9h05t 5pace may 6e d1v1ded 1nt0 tw0 (de9enerate) 5u65pace5 c0rre5p0nd1n9 t0 the tw0 d1fferent 5tate5 0f the 43 9h05t. 7h15 de9eneracy d0e5 n0t affect the pertur6at1ve ca1cu1at10n5, 51nce the 9h05t 0perat0r5 appear pa1rw15e 1n (6) and thu5 1ntermed1ate 5tate5 c0nta1n1n9 9h05t have at 1ea5t the (f1n1te) exc1tat10n ener9y 0f the 5-9h05t wh1ch ha5 6een 51mu1tane0u51y created. 7he ham11t0n1an (6) adm1t5 a5 6a51c 5et 0f 5tate5 the un1f1ed m0de1 wave funct10n5 [ 9 ] 1[~11MKK~n ~-
2//2/2/2/2/2/2/2/2N 21+ 1 1 4 - ~ 2 DUr(0~)2X~,(41, 92~, r1p,~p) , (7)
1n Wh1Ch the 1ntr1n51C 5eCt0r may 6e 1a6e11ed 6y the e19enVa1Ue K• 0f the 0perat0r J3 (p1U5 add1t10na1 4UantUm num6er5 n). 7he pertur6at10n the0ry 15 6a5ed, a5 U5Ua1,1n 5tate5 that are ann1h11ated 6y Q (t0 1ead1n9 0rder). HenCe We c0n51der UnpertUr6ed 5tate5 5UCh that K=K• (t0 1ead1n9 0rder). H0WeVer, 51nCe (/3--J3) 15 n0t a C0n5tant 0f the m0t10n, 5tate5 W1th K 5 K• d0 n0t UnC0Up1e. NeVerthe1e5, there rema1n5 a C0n5erved 4Uant1ty. Let U5 def1ne the 9enerat0r5
N 3 = ~35tff25Pt, 73 = --1E35t~151rt, f3 =--1e3~4~9~.
(8)
27 5eptem6er 1990
We c0n51der the 0perat0r L3 --=-13-13 +N3 +r3 +f3
= [Q, (1~35tY1592t"~/~3) ]
,
(9)
wh1ch c0mmute5 w1th Haa5r and Q. 7hu5, the phy51ca1 wave funct10n5 can 6e c1a51f1ed6y the e19enva1ue5 E and 2 0f H8a5a- and L3, re5pect1ve1y. 1f2~ 0, 1
1
1~,2)= ~L31¢,2) = ~ 01~0),
(10)
f0r 50me 1~0). Due t0 the n11p0tency 0f Q, the 5tate 1E, 2) ha5 2er0 n0rm ( 1 f 2 ~ 0 ) [6]. 7hu5 we 0n1y need t0 c0n51der 5tate5 wh1ch are 1nvar1ant w1th re5pect t0 ar61trary 51mu1tane0u5 r0tat10n5 (9enerated 6y L 3) 1n the1r 1ntr1n51c and c011ect1ve 5ect0r5. 51m11ar ar9ument5 can 6e app11ed t0 the 0perat0r that r0tate5 the c011ect1ve, 1ntr1n51c and 9h05t wavefunct10n5 6y 180 ° ar0und an ax15 perpend1cu1ar t0 the 5ymmetry ax15. 7h15 15 a 9enera112at10n 0f the d15crete 1ntr1n51c-c011ect1ve5ymmetr1e5 de5cr16ed 1n ref. [91. 51nce
J ~ = 1 2 - [ (J~ +1~)n~, 0 ] + ,
(11)
and thu5
j2 ~1MKx,n=1( 1 + 1 ) ~11MKK,n + Q1 ~0),
( 12 )
the 5tate (7) 15an an9u1ar m0mentum pr0jected 5tate, Up t0 a 2er0 n0rm 5tate. 7here are n0 2er0 m0de5, 51nCe J~ d0e5 n0t c0mmute w1th//8a57 and thu5 pertur6at10n the0ry 6ec0me5 fea5161e. F1na11y, the 11near r0tat10na1-1ntr1n51c c0up11n9 term, t2J5,1n the ham11t0n1an (6) may 6e e11m1nated thr0u9h the tran5f0rmat10n 7 = eXp[115 ( - ~ - 6 , ) ] , Wh1Ch y1e1d5the effect1Ve ham11t0n1an
(13)
V01ume248, num6er 1,2
PHY51C5 LE77ER5 8
Heft = 7HnR57 71
=HaR57+ ~---~12--15(~25+1j5--1[H, 65]
1
+ ~ 9 ~ t + -5 ~5~3(n3 t1, - n~t13 ) +...
)
1 1~r[-9 55+[6,,(:[9,6A+1J,)1 +~-j
--~.2t3(-~ --6t)2+...] --113[~5t3(~ --6t)~5 +...~
+
3 2F[Pt ~ 6 ~ 2 + . . . 1 , 23Lt,7 ) J
(14)
where 95t ( = - 1[65, Jt ] - 85t) 15 a55umed t0 6e a 5ma11 4uant1ty w1th re5pect t0 un1ty. 7he f1r5t term pr0p0rt10na1 t0 15 cance15 the r0tat10na1-1ntr1n51c c0up11n9 ~2~/51n H8P,57,a5 expected. 7he next c0ntr16ut10n 15 (1/J)J5-1[H, 65]. 1t5 1ead1n9 0rder term van15he5 1f the c0nd1t10n (5) 15 5at15f1ed. 7he term - ( 1 / J ) 1 J 5 15 the C0r10115 1nteract10n. U5ua11y the 1ead1n9 (RPA) term5 0f th15 1nteract10n are ne91ected 0n the 9r0und5 that they 91ve r15e t0 the m0ment 0f 1nert1a tr0u9h the 1n9115 pre5cr1pt10n and 5h0u1d n0t 6e c0unted tw1ce (they c0nnect, f0r 1n5tance, the 9r0und 5tate w1th tw0-4ua51 part1c1e 5tate5). 7h15 ne91ect 15 re4u1red 6y e4. (5), 51nce th05e 11near term5 are cance11ed 6y the c0rre5p0nd1n9 term5 1n [H, 6~]. 7he C0r10115 1nteract10n c0n515t5 0f the rema1n1n9 (n0n 11near) term5 1n the 0perat0r J~ (f0r 1n5tance, 0f the term5 c0n5erv1n9 the num6er 0f 4ua51-part1c1e5). A 5ke1et0n 1n the nuc1ear phy51c5 c105et ha5 6een the (50 far unexp1a1ned) attenuat10n 0f the C0r10115 term. 7he ham11t0n1an (14) may 501ve th15 r1dd1e, 51nce 1t 1nc1ude5 a150 n0n-11near term5 fr0m 1[H, 6~]. 1n fact, 1n the E1110t 5U (3) m0de1 [ 10 ], the Ca51m1r 0perat0r 15 pr0p0rt10na1 t0 H - ( 1 / 2 J ) J ~ . 7hu5 the C0r10115 1nteract10n 15 exact1y cance11ed 1f 0ne u5e5 the n0n-d1a90na1 c0mp0nent5 0fthe 4uadrup01e m0ment (wh1ch are 9enerat0r5 0f the 5U (3) 9r0up) a5 9au9e-f1x1n9 0perat0r5. 0 n the 0ther hand, 1n the mean f1e1d appr0x1mat10n f0r the ham11t0n1an H, the c0mmutat10n w1th 6 y1e1d5 0n1y 11near-RPA term5 (1.e., term5 creat1n9 tw0 4ua51-part1c1e5), and thu5 the n0n-11near term5 1n the C0r10115 1nteract10n w0u1d rema1n una1tered. Expre5510n (14) 5h0u1d 6e a6e1 t0
27 5eptem6er 1990
5y5temat1ca11y treat the C0r10115 effect5 1n 1ntermed1ate rea115t1c 51tuat10n5. 51nce the 5c0pe 0f the f0rma115m 15 n0t re5tr1cted t0 pure m1cr05c0p1c ham11t0n1an5, we u5e the 80hr c011ect1ve ham11t0n1an a5 an 111u5trat10n. We a55ume a p0tent1a1 ener9y 5urface 1nc1ud1n9 4uadrat1c, cu61c and 4uart1c term5 1n the 4uadrup01e c00rd1nate5 4u, P - u 6e1n9 the1r c0nju9ate m0ment ~2. 7he 5u61nd1ce5 #, ( ( = 0, + 1, •+ 2) den0te 5pher1ca1 c0mp0nent5, wh11e the carte51an v 0r 5 have the 5ame mean1n9 a5 6ef0re. 7he three free parameter5 are ch005en t0 6e the e4u1116r1um def0rmat10n f1 and the fre4uenc1e5 0f the 6eta and 9amma v16rat10n5 0~0, t09 (we a55ume a 5ta61e def0rmed 5y5tem, 1.e., c06,9f12>> 1 ). 1n part1cu1ar, the an9u1ar m0mentum and the 9au9e-f1x1n9 funct10n5 read
Ja - - 7 ( 1 )
4"1(2)
JCu~)=1t:~/2pu, J~u2)=x/~ [4P11 1
6u=6~u ~)- j1/21L4u
(/t=0, • 1 ) ,
(#=+1).
(15)
where 40 fr0m n0w 0n mea5ure5 the d15tance fr0m p, and the the u5ua1 1rr0tat10na1 m0ment 0f 1nert1a J (=3f12) 15 u5ed (1t can a150 6e 06ta1ned fr0m e4. (5) ). U51n9 the tran5f0rmat10n5 (3.13 ) and (3.20) 0fref. [ 3 ], the 4uadrat1c part 0fthe ham11t0n1an (6) can 6e ca5ted 1nt0 n0rma1 m0de5
H28 R 5 7 =
1n2 1 (/)2 ~ 2 .~ (/)2 ~ ~/~ 0 -[- p + 2 p - - 2 "~[-~ 6 t / 0 9 t 1 + 2 t,/--2
{(J(1))2 A 1 ~ --~5J~ 1 ) - 28 2 P ~ + - ~ 6 5 P ~ 1 ~545) E
• (F~+ F~+~)
,u= + 2
+C0 ~ /~=--+1
+ r~1 - F ,~0r,~0 + ( F ~, + a,,a,, + 9~6 ~ ) , (16)
#2 1n th15ca5ethere 15an apparent am619u1tyc0ncern1n9the u5e 0f the adject1vec011ect1ve,51ncethe f1ve4uadrup01evar1a61e5 can 6e 1nterpreteda5 c011ect1venuc1earc00rd1nate5. H0wever, 1n the pre5ent c0ntext they are the 0r191na1 (n0n-c011ect1ve) 6050n var1a61e50fthe pr061em,0ut 0fwh1ch, f0r 1n5tance,the three 9enerat0r5 J~ ( 15) are c0n5tructed. 7he5e are d1fferent fr0m the three c011ect1ve0perat0r5 1~ act1n9 0n the D1x funct10n5.
v01ume 248, num6er 1,2
PHY51C5 LE77ER5 8
where [F~,1, F~11 = [F~,+0,F,01 = t a , , a•1 + = t6., 6¢1 + = ~,¢, t 0 = 8 -1/2,
( 17 )
A=8/J.
(18)
N0te (1) the 1ndef1n1te metr1c a550c1ated w1th the ph0n0n5 (#0) and (11) the 5uper5ymmetry 1n the 5pur10u5 5ect0r 0f the 4uadrat1c ham11t0n1an. 7 h e 5pur10u5 5ect0r 0f the unpertur6ed phy51ca1 5tate5 15 ann1h11ated 6y the 0perat0r5 Fj,1, F ~ , a,, 6~, (/2= + 1 ) and, c0n5e4uent1y, 6y the 1ead1n9 0rder term51n the char9e Q. 7 h e c0rrect10n5 t0 the unpertur6ed v16rat10na1 and r0tat10na1 ener91e5 can 6e w0rked 0ut fr0m (14) taken 1n acc0unt the rema1n1n9 part5 0f Heff (wh1ch 1nc1ude the cu61c and 4uart1c term5 0f H ) . F0r 1n5tance, f0r the 0ne-6eta-ph0n0n 5tate 9 ( 204, AE(1)=-~-~1+~3--~ 1 t02
+
t0-4 +
1 t03
t09 +~
1 t0:~
t0-7 +
3 9 309, +~-~P22 ~9 + - - + - - - +
2t0
1t02~
(19)
7 h e f1r5t term repre5ent5 a c0rrect10n t0 the ener9y 0f the 0ne-ph0n0n 6 a n d head and the 5ec0nd 0ne a ren0rma112at10n 0 f t h e m 0 m e n t 0f1nert1a 0 f t h e 6eta6and. 7 h e y are pr0p0rt10na1 t0 the (5ma11) parameter 1/Jt06,9. 7he5e c0rrect10n5 a9ree w1th the va1ue der1ved fr0m the 0r191na1 8 0 h r ham11t0n1an and are, 1n part1cu1ar, 1ndependent 0f the ar61trary parameter t0. N0te h0wever that f0r a 9enera1 def0rmed 5y5tem the5e pertur6at1ve c0rrect10n5 cann0t 6e 06ta1ned w1th0ut the 8 R 5 7 treatment (0r any 0ther treatment
4
27 5eptem6er 1990
that may a150 0verc0me the pre5ence 0f 1nfrared 51n9u1ar1t1e5). We have 5h0wn that ca5e5 1n wh1ch there 15 a part1a1 6reakd0wn 0f 5ymmetr1e5 can 6e treated w1th1n the 8 R 5 7 thr0u9h a 5119ht m0d1f1cat10n 0f the prev10u51y de5cr16ed pr0cedure5. 7 h e re5u1tant 5ymmetr1e5 0f the wave funct10n5 and the re51dua1 1nteract10n d15p1ay5 51m11ar1t1e5 6ut a150 50me d1fference5 fr0m th05e 0 f t h e un1f1ed m0de1 [9]. D15cu5510n5 w1th V. A1e55andr1n1 are 9ratefu11y ackn0w1ed9ed.
Reference5 [ 1] C. 8ecch1, A. R0uet and 8. 5t0ra, Phy5. Lett. 8 52 (1974) 344; Ann. Phy5. (NY) 98 (1976) 278; 1.v. 7yut1n, Le6edev Rep0rt N0. F1AN 39 (1979), unpu6115hed; 8.L. v0r0n0v and 1.v. 7yut1n, 7e0r. Mat. F12. 50 (1982) 218. [2] J. Kurchan, D.R. 8e5 and 5. Cru2 8arr105, Phy5. Rev. D 38 (1988) 3309. [ 3 ] D.R. 8e5, 5. Cru2 8arr105 and J. Kurchan, Ann. Phy5. (NY) 194 (1989) 227. [4] J. Kurchan, D.R. 8e5 and 5. Cru2 8arr105, Nuc1. Phy5. A 509 (1990) 306. [5] M. Henneaux, 1n: Quantum mechan1c5 0f fundamenta1 5y5tem5 1, ed. C. 7e1te1601m (P1enum, New Y0rk, 1988); Ann. Phy5. (NY) 194 (1989) 281. [6] D.R. 8e5 and J. Kurchan, 7he treatment 0f c011ect1ve c00rd1nate51nmany-60dy5y5tem5,LectureN0te51n Phy51c5, N0. 34 (w0r1d 5c1ent1f1c,51n9ap0re), t0 6e pu6115hed. [7] J. A1far0 and P.H. Dam9aard, Ann. Phy5. (NY), t0 6e pu6115hed. [ 8 ] E.R. Mar5ha1ekand J. wene5er, Phy5. Rev. C 2 (1970) 1982. [9] A. 80hr and 8.R. M0tte150n, Nuc1ear 5tructure, v01. 11 (8enjam1n, Read1n9, MA, 1975). [ 10] M. Harvey, Advan. Nuc1. Phy5. 1 (1968) 67.
PHY51C5 LE77ER5 8
27 5eptem6er 1990
7he 8 R 5 7 treatment 0f c011ect1ve r0tat10na1 c00rd1nate5 1n ax1a11y-5ymmetr1c 5y5tem5 D.R. 8e5 1, R. De LUCa a n d J. KUrChan Phy51c5Department CNEA, Av. de1L16ertad0r8250, 14298uen05A1re5,Ar9ent1na Rece1ved 19 Apr111990; rev15edmanu5cr1pt rece1ved 17 Ju1y 1990
7he 8R57 treatment 0f c011ect1vec00rd1nate5 15extended t0 ca5e5 1n wh1ch there 15a part1a16reakd0wn 0f 5ymmetr1e51n the mean-f1e1d appr0x1mat10n. 7he re5u1tant 5ymmetr1e50f the wave funct10n5 are d15cu55ed, a5 we11a5 the f0rm 0f the re51dua1 1nteract10n5and, 1n part1cu1ar,0fthe u5ua1C0r10115term. An app11cat10nt0 the 80hr c011ect1veham11t0n1an15made a5 an 111u5trat10n.
7 h e 8 e c c h 1 - R 0 u e t - 5 t 0 r a - 7 y u t 1 n 1nvar1ance [ 1 ] ha5 6ec0me a fundamenta1 t001 f0r the 4uant1f1Cat10n 0f 9au9e the0r1e5. 7 h e treatment 0f c011ect1ve C00rd1nate5 a5 an app11cat10n 0f the 8 R 5 7 1nvar1ance 15 carr1ed 0ut 1n ref. [2 ]. 7 h e 1dea 15 t0 c0n51der the parameter5 0f a 9r0up 0f tran5f0rmat10n5 (f0r 1n5tance, the 0r1entat10n an91e5 0v) a5 rea1 c011ect1ve var1a61e5 and t0 1nc1ude extra de9ree5 0f freed0m (La9ran9e mu1t1p11er5, 9h05t5) 50 that the 501ut10n 0f the t0ta1 5pur10u5 5ect0r 6ec0me5 a re1at1ve1y 51mp1e pr0cedure. App11cat10n5 t0 a 51mp1e (a6e11an) tran5f0rmat10n and t0 (n0na6e11an) r0tat10n5 w1th h19h an9u1ar m 0 m e n t u m are made 1n ref. [ 3 ] and ref. [ 4 ], re5pect1ve1y. 5ee a150 ref5. [ 5,6 ] f0r a c0mp1ete d15cu5510n and ref. [ 7 ] f0r a 9enera112at10n t0 f1e1d the0ry. 7 h e 0r191na1 ham11t0n1an H 15 rep1aced 6y the 8 R 5 7 ham11t0n1an H8R5-r=H+ [p, Q1 + , where the (n11p0tent) 8 R 5 7 ••Char9e••, h11ate5 phy51ca1 wave funct10n5, 15
( 1) wh1Ch ann1-
51te 519n); (92v, Pv) pa1r5 0f c0nju9ate var1a61e5 a550c1ated w1th La9ran9e mu1t1p11er5 and (4v, nv), (0v, 9v) pa1r5 0f c0nju9ate 9h05t var1a61e5. 7 h e p 0perat0r 15 ar61trary. 1n ref. [2 ] we have taken
A
(3)
A, 8 are ar61trary C0n5tant5 and 5h0U1d n0t appear 1n any f1na1 re5U1t. 7 h e phy51Ca1mean1n9 0fthe 9au9ef1x1n9 funct10n5 6v 15 that the van15h1n9 0f the1r 9r0und 5tate expectat10n va1ue def1ne5 the 1ntr1n51c 5y5tem. 7he re5u1tant ham11t0n1an 15 91ven 1n e4. (17) 0f ref. [ 2 ]. Here we app1y th15 f0rma115m t0 the pr061em 0f c011ect1ve r0tat10n5 1n rea1 5pace. Hence Jv (v = 1, 2, 3 ) are the an9u1ar m 0 m e n t u m 0perat0r5 and 1v the c0rre5p0nd1n9 c011ect1ve term5. 1n the 501ut10n 0fH8957 pre5ented 1n ref. [ 3 ], the 6.5at15fy, t0 1ead1n9 0rder [6~, J,0] =16,,0 1
[H, 6 , 1 = - - ~ J v
(4) (5)
Q = P ~ v - - ( J v - - 1 , ) 4 v + ~~1~w0¢4vrh0;9~ (V, C0, ~= 1, ..., n) .
(2)
Here Jv are the 9enerat0r5 0f the tran5f0rmat10n W1th 9r0Up 5truCtUre C0n5tant5 ev,0~;1v the1r C011eCt1Vever510n (5at15fy1n9 the 5ame a19e6ra 6ut w1th an 0pp0Fe110w0fthe C0N1CE7, 8uen05 A1re5,Ar9ent1na.
F0r a many-60dy 5y5tem, the5e e4uat10n5 can 6e 501ved ~1 w1th1n the RPA [8 ] and y1e1d, 1n add1t10n, the 7h0u1e55-Va1at1n m 0 m e n t 0f 1nert1a Jv. H0wever, th15 501ut10n cann0t 6e app11ed t0 ax1a11y 5ymat A c0rre5p0nd1n9 (11near12ed) appr0x1mat10n a150 ex15t5 f0r def0rmed 6050n1c5y5tem5.
0370-2693/90/$ 03.50 • 1990 - E15ev1er5c1encePu6115her58.V. ( N0rth-H011and )
1
V01Ume248, num6er 1,2
PHY51C5 LE77ER5 8
metr1c 5y5tem5, 6ecau5e e45. (4) and (5) cann0t 6e fu1f111eda10n9 the 5ymmetry d1rect10n (p = 3 ). 1n an ax1a11y5ymmetr1c ca5e 1t 15n0t p05161e t0 treat c011ect1ve1y the r0tat10n ar0und the 5ymmetry ax15 6ecau5e the fre4uency 0f th15 r0tat10n w0u1d 6e the 5ame a5 the 0ne a550c1ated w1th the c0rre5p0nd1n9 v16rat10n. Hence we 5h0u1d n0t f1x the 0r1entat10n 0f the m0v1n9 frame w1th re5pect t0 r0tat10n5 ar0und th15 ax15. We try a 51m11arexpre510n t0 (3) 6ut w1th the 5ummat10n re5tr1cted t0 5, t = 1, 2. 7he re5u1t1n9 8 R 5 7 ham11t0n1an
H8R57 = H-- Q5( J5 --1~) +11t5~5 + 1 ( 6 5 P 5 - - 2~ P2 +43f15[65, J3]+r1fh[6t, J5])
+1~5t3925(7ttt13 - -
7~3~t)
,
(6)
d0e5 n0t c0nta1n ne1ther the 0perat0r5 t23,/•3, 93 and t73, wh1ch hence unc0up1e fr0m the rema1n1n9 de9ree5 0f freed0m, n0r the 9au9e f1x1n9 0perat0r 6 3. H0wever, the 9h05t 0perat0r5 43, n3 5t111appear and n0 k1net1c term 15 a550c1ated w1th them. 7hu5 the 9h05t 5pace may 6e d1v1ded 1nt0 tw0 (de9enerate) 5u65pace5 c0rre5p0nd1n9 t0 the tw0 d1fferent 5tate5 0f the 43 9h05t. 7h15 de9eneracy d0e5 n0t affect the pertur6at1ve ca1cu1at10n5, 51nce the 9h05t 0perat0r5 appear pa1rw15e 1n (6) and thu5 1ntermed1ate 5tate5 c0nta1n1n9 9h05t have at 1ea5t the (f1n1te) exc1tat10n ener9y 0f the 5-9h05t wh1ch ha5 6een 51mu1tane0u51y created. 7he ham11t0n1an (6) adm1t5 a5 6a51c 5et 0f 5tate5 the un1f1ed m0de1 wave funct10n5 [ 9 ] 1[~11MKK~n ~-
2//2/2/2/2/2/2/2/2N 21+ 1 1 4 - ~ 2 DUr(0~)2X~,(41, 92~, r1p,~p) , (7)
1n Wh1Ch the 1ntr1n51C 5eCt0r may 6e 1a6e11ed 6y the e19enVa1Ue K• 0f the 0perat0r J3 (p1U5 add1t10na1 4UantUm num6er5 n). 7he pertur6at10n the0ry 15 6a5ed, a5 U5Ua1,1n 5tate5 that are ann1h11ated 6y Q (t0 1ead1n9 0rder). HenCe We c0n51der UnpertUr6ed 5tate5 5UCh that K=K• (t0 1ead1n9 0rder). H0WeVer, 51nCe (/3--J3) 15 n0t a C0n5tant 0f the m0t10n, 5tate5 W1th K 5 K• d0 n0t UnC0Up1e. NeVerthe1e5, there rema1n5 a C0n5erved 4Uant1ty. Let U5 def1ne the 9enerat0r5
N 3 = ~35tff25Pt, 73 = --1E35t~151rt, f3 =--1e3~4~9~.
(8)
27 5eptem6er 1990
We c0n51der the 0perat0r L3 --=-13-13 +N3 +r3 +f3
= [Q, (1~35tY1592t"~/~3) ]
,
(9)
wh1ch c0mmute5 w1th Haa5r and Q. 7hu5, the phy51ca1 wave funct10n5 can 6e c1a51f1ed6y the e19enva1ue5 E and 2 0f H8a5a- and L3, re5pect1ve1y. 1f2~ 0, 1
1
1~,2)= ~L31¢,2) = ~ 01~0),
(10)
f0r 50me 1~0). Due t0 the n11p0tency 0f Q, the 5tate 1E, 2) ha5 2er0 n0rm ( 1 f 2 ~ 0 ) [6]. 7hu5 we 0n1y need t0 c0n51der 5tate5 wh1ch are 1nvar1ant w1th re5pect t0 ar61trary 51mu1tane0u5 r0tat10n5 (9enerated 6y L 3) 1n the1r 1ntr1n51c and c011ect1ve 5ect0r5. 51m11ar ar9ument5 can 6e app11ed t0 the 0perat0r that r0tate5 the c011ect1ve, 1ntr1n51c and 9h05t wavefunct10n5 6y 180 ° ar0und an ax15 perpend1cu1ar t0 the 5ymmetry ax15. 7h15 15 a 9enera112at10n 0f the d15crete 1ntr1n51c-c011ect1ve5ymmetr1e5 de5cr16ed 1n ref. [91. 51nce
J ~ = 1 2 - [ (J~ +1~)n~, 0 ] + ,
(11)
and thu5
j2 ~1MKx,n=1( 1 + 1 ) ~11MKK,n + Q1 ~0),
( 12 )
the 5tate (7) 15an an9u1ar m0mentum pr0jected 5tate, Up t0 a 2er0 n0rm 5tate. 7here are n0 2er0 m0de5, 51nCe J~ d0e5 n0t c0mmute w1th//8a57 and thu5 pertur6at10n the0ry 6ec0me5 fea5161e. F1na11y, the 11near r0tat10na1-1ntr1n51c c0up11n9 term, t2J5,1n the ham11t0n1an (6) may 6e e11m1nated thr0u9h the tran5f0rmat10n 7 = eXp[115 ( - ~ - 6 , ) ] , Wh1Ch y1e1d5the effect1Ve ham11t0n1an
(13)
V01ume248, num6er 1,2
PHY51C5 LE77ER5 8
Heft = 7HnR57 71
=HaR57+ ~---~12--15(~25+1j5--1[H, 65]
1
+ ~ 9 ~ t + -5 ~5~3(n3 t1, - n~t13 ) +...
)
1 1~r[-9 55+[6,,(:[9,6A+1J,)1 +~-j
--~.2t3(-~ --6t)2+...] --113[~5t3(~ --6t)~5 +...~
+
3 2F[Pt ~ 6 ~ 2 + . . . 1 , 23Lt,7 ) J
(14)
where 95t ( = - 1[65, Jt ] - 85t) 15 a55umed t0 6e a 5ma11 4uant1ty w1th re5pect t0 un1ty. 7he f1r5t term pr0p0rt10na1 t0 15 cance15 the r0tat10na1-1ntr1n51c c0up11n9 ~2~/51n H8P,57,a5 expected. 7he next c0ntr16ut10n 15 (1/J)J5-1[H, 65]. 1t5 1ead1n9 0rder term van15he5 1f the c0nd1t10n (5) 15 5at15f1ed. 7he term - ( 1 / J ) 1 J 5 15 the C0r10115 1nteract10n. U5ua11y the 1ead1n9 (RPA) term5 0f th15 1nteract10n are ne91ected 0n the 9r0und5 that they 91ve r15e t0 the m0ment 0f 1nert1a tr0u9h the 1n9115 pre5cr1pt10n and 5h0u1d n0t 6e c0unted tw1ce (they c0nnect, f0r 1n5tance, the 9r0und 5tate w1th tw0-4ua51 part1c1e 5tate5). 7h15 ne91ect 15 re4u1red 6y e4. (5), 51nce th05e 11near term5 are cance11ed 6y the c0rre5p0nd1n9 term5 1n [H, 6~]. 7he C0r10115 1nteract10n c0n515t5 0f the rema1n1n9 (n0n 11near) term5 1n the 0perat0r J~ (f0r 1n5tance, 0f the term5 c0n5erv1n9 the num6er 0f 4ua51-part1c1e5). A 5ke1et0n 1n the nuc1ear phy51c5 c105et ha5 6een the (50 far unexp1a1ned) attenuat10n 0f the C0r10115 term. 7he ham11t0n1an (14) may 501ve th15 r1dd1e, 51nce 1t 1nc1ude5 a150 n0n-11near term5 fr0m 1[H, 6~]. 1n fact, 1n the E1110t 5U (3) m0de1 [ 10 ], the Ca51m1r 0perat0r 15 pr0p0rt10na1 t0 H - ( 1 / 2 J ) J ~ . 7hu5 the C0r10115 1nteract10n 15 exact1y cance11ed 1f 0ne u5e5 the n0n-d1a90na1 c0mp0nent5 0fthe 4uadrup01e m0ment (wh1ch are 9enerat0r5 0f the 5U (3) 9r0up) a5 9au9e-f1x1n9 0perat0r5. 0 n the 0ther hand, 1n the mean f1e1d appr0x1mat10n f0r the ham11t0n1an H, the c0mmutat10n w1th 6 y1e1d5 0n1y 11near-RPA term5 (1.e., term5 creat1n9 tw0 4ua51-part1c1e5), and thu5 the n0n-11near term5 1n the C0r10115 1nteract10n w0u1d rema1n una1tered. Expre5510n (14) 5h0u1d 6e a6e1 t0
27 5eptem6er 1990
5y5temat1ca11y treat the C0r10115 effect5 1n 1ntermed1ate rea115t1c 51tuat10n5. 51nce the 5c0pe 0f the f0rma115m 15 n0t re5tr1cted t0 pure m1cr05c0p1c ham11t0n1an5, we u5e the 80hr c011ect1ve ham11t0n1an a5 an 111u5trat10n. We a55ume a p0tent1a1 ener9y 5urface 1nc1ud1n9 4uadrat1c, cu61c and 4uart1c term5 1n the 4uadrup01e c00rd1nate5 4u, P - u 6e1n9 the1r c0nju9ate m0ment ~2. 7he 5u61nd1ce5 #, ( ( = 0, + 1, •+ 2) den0te 5pher1ca1 c0mp0nent5, wh11e the carte51an v 0r 5 have the 5ame mean1n9 a5 6ef0re. 7he three free parameter5 are ch005en t0 6e the e4u1116r1um def0rmat10n f1 and the fre4uenc1e5 0f the 6eta and 9amma v16rat10n5 0~0, t09 (we a55ume a 5ta61e def0rmed 5y5tem, 1.e., c06,9f12>> 1 ). 1n part1cu1ar, the an9u1ar m0mentum and the 9au9e-f1x1n9 funct10n5 read
Ja - - 7 ( 1 )
4"1(2)
JCu~)=1t:~/2pu, J~u2)=x/~ [4P11 1
6u=6~u ~)- j1/21L4u
(/t=0, • 1 ) ,
(#=+1).
(15)
where 40 fr0m n0w 0n mea5ure5 the d15tance fr0m p, and the the u5ua1 1rr0tat10na1 m0ment 0f 1nert1a J (=3f12) 15 u5ed (1t can a150 6e 06ta1ned fr0m e4. (5) ). U51n9 the tran5f0rmat10n5 (3.13 ) and (3.20) 0fref. [ 3 ], the 4uadrat1c part 0fthe ham11t0n1an (6) can 6e ca5ted 1nt0 n0rma1 m0de5
H28 R 5 7 =
1n2 1 (/)2 ~ 2 .~ (/)2 ~ ~/~ 0 -[- p + 2 p - - 2 "~[-~ 6 t / 0 9 t 1 + 2 t,/--2
{(J(1))2 A 1 ~ --~5J~ 1 ) - 28 2 P ~ + - ~ 6 5 P ~ 1 ~545) E
• (F~+ F~+~)
,u= + 2
+C0 ~ /~=--+1
+ r~1 - F ,~0r,~0 + ( F ~, + a,,a,, + 9~6 ~ ) , (16)
#2 1n th15ca5ethere 15an apparent am619u1tyc0ncern1n9the u5e 0f the adject1vec011ect1ve,51ncethe f1ve4uadrup01evar1a61e5 can 6e 1nterpreteda5 c011ect1venuc1earc00rd1nate5. H0wever, 1n the pre5ent c0ntext they are the 0r191na1 (n0n-c011ect1ve) 6050n var1a61e50fthe pr061em,0ut 0fwh1ch, f0r 1n5tance,the three 9enerat0r5 J~ ( 15) are c0n5tructed. 7he5e are d1fferent fr0m the three c011ect1ve0perat0r5 1~ act1n9 0n the D1x funct10n5.
v01ume 248, num6er 1,2
PHY51C5 LE77ER5 8
where [F~,1, F~11 = [F~,+0,F,01 = t a , , a•1 + = t6., 6¢1 + = ~,¢, t 0 = 8 -1/2,
( 17 )
A=8/J.
(18)
N0te (1) the 1ndef1n1te metr1c a550c1ated w1th the ph0n0n5 (#0) and (11) the 5uper5ymmetry 1n the 5pur10u5 5ect0r 0f the 4uadrat1c ham11t0n1an. 7 h e 5pur10u5 5ect0r 0f the unpertur6ed phy51ca1 5tate5 15 ann1h11ated 6y the 0perat0r5 Fj,1, F ~ , a,, 6~, (/2= + 1 ) and, c0n5e4uent1y, 6y the 1ead1n9 0rder term51n the char9e Q. 7 h e c0rrect10n5 t0 the unpertur6ed v16rat10na1 and r0tat10na1 ener91e5 can 6e w0rked 0ut fr0m (14) taken 1n acc0unt the rema1n1n9 part5 0f Heff (wh1ch 1nc1ude the cu61c and 4uart1c term5 0f H ) . F0r 1n5tance, f0r the 0ne-6eta-ph0n0n 5tate 9 ( 204, AE(1)=-~-~1+~3--~ 1 t02
+
t0-4 +
1 t03
t09 +~
1 t0:~
t0-7 +
3 9 309, +~-~P22 ~9 + - - + - - - +
2t0
1t02~
(19)
7 h e f1r5t term repre5ent5 a c0rrect10n t0 the ener9y 0f the 0ne-ph0n0n 6 a n d head and the 5ec0nd 0ne a ren0rma112at10n 0 f t h e m 0 m e n t 0f1nert1a 0 f t h e 6eta6and. 7 h e y are pr0p0rt10na1 t0 the (5ma11) parameter 1/Jt06,9. 7he5e c0rrect10n5 a9ree w1th the va1ue der1ved fr0m the 0r191na1 8 0 h r ham11t0n1an and are, 1n part1cu1ar, 1ndependent 0f the ar61trary parameter t0. N0te h0wever that f0r a 9enera1 def0rmed 5y5tem the5e pertur6at1ve c0rrect10n5 cann0t 6e 06ta1ned w1th0ut the 8 R 5 7 treatment (0r any 0ther treatment
4
27 5eptem6er 1990
that may a150 0verc0me the pre5ence 0f 1nfrared 51n9u1ar1t1e5). We have 5h0wn that ca5e5 1n wh1ch there 15 a part1a1 6reakd0wn 0f 5ymmetr1e5 can 6e treated w1th1n the 8 R 5 7 thr0u9h a 5119ht m0d1f1cat10n 0f the prev10u51y de5cr16ed pr0cedure5. 7 h e re5u1tant 5ymmetr1e5 0f the wave funct10n5 and the re51dua1 1nteract10n d15p1ay5 51m11ar1t1e5 6ut a150 50me d1fference5 fr0m th05e 0 f t h e un1f1ed m0de1 [9]. D15cu5510n5 w1th V. A1e55andr1n1 are 9ratefu11y ackn0w1ed9ed.
Reference5 [ 1] C. 8ecch1, A. R0uet and 8. 5t0ra, Phy5. Lett. 8 52 (1974) 344; Ann. Phy5. (NY) 98 (1976) 278; 1.v. 7yut1n, Le6edev Rep0rt N0. F1AN 39 (1979), unpu6115hed; 8.L. v0r0n0v and 1.v. 7yut1n, 7e0r. Mat. F12. 50 (1982) 218. [2] J. Kurchan, D.R. 8e5 and 5. Cru2 8arr105, Phy5. Rev. D 38 (1988) 3309. [ 3 ] D.R. 8e5, 5. Cru2 8arr105 and J. Kurchan, Ann. Phy5. (NY) 194 (1989) 227. [4] J. Kurchan, D.R. 8e5 and 5. Cru2 8arr105, Nuc1. Phy5. A 509 (1990) 306. [5] M. Henneaux, 1n: Quantum mechan1c5 0f fundamenta1 5y5tem5 1, ed. C. 7e1te1601m (P1enum, New Y0rk, 1988); Ann. Phy5. (NY) 194 (1989) 281. [6] D.R. 8e5 and J. Kurchan, 7he treatment 0f c011ect1ve c00rd1nate51nmany-60dy5y5tem5,LectureN0te51n Phy51c5, N0. 34 (w0r1d 5c1ent1f1c,51n9ap0re), t0 6e pu6115hed. [7] J. A1far0 and P.H. Dam9aard, Ann. Phy5. (NY), t0 6e pu6115hed. [ 8 ] E.R. Mar5ha1ekand J. wene5er, Phy5. Rev. C 2 (1970) 1982. [9] A. 80hr and 8.R. M0tte150n, Nuc1ear 5tructure, v01. 11 (8enjam1n, Read1n9, MA, 1975). [ 10] M. Harvey, Advan. Nuc1. Phy5. 1 (1968) 67.