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THE NUCLEAR VELOCITY FIELD WITH AND WITHOUT THE VORTEX SPIN
Z. Papadopolos
To cite this version:
Z. Papadopolos. THE NUCLEAR VELOCITY FIELD WITH AND WITHOUT THE VORTEX SPIN. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-297-C6-303. �10.1051/jphyscol:1984636�.
�jpa-00224238�
I n s t i t u t fiir Theoretische Physik der Universittit,
0-7400Tiibingen, F.R. G.
RCsum6 - L'existence du champ de vitesse d6fini dans le cadre d'lme theorie collective GL,(3, W) est 6tudiCe.
A b s t r a c t - The e x i s t e n c e o f t h e c o l l e c t i v e v e l o c i t y f i e l d w . r . t . GL+(3, Ill) c o l l e c t i v i t y i s s t u d i e d .
We a r e i n t e r e s t e d i n t h e e x t r a c t i o n o f t h e c o l l e c t i v e q u a n t i t i e s , s u c h a s t h e c o l l e c t i v e v e l o c i t y f i e l d , o u t o f t h e many p a r t i c l e t h e o r y t h r o u g h some c o o r d i n a t e t r a n s f o r m a t i o n f o r t h e s y s t e m o f A-1 : n J a c o b i v e c t o r s s u c h t h a t t h e c o l l e c t i v e c o o r d i n a t e s b e l o n g t o t h e s e t o f t h e new 3n c o o r d i n a t e s . For t h i s i d e a s s e e r e f s . / 1 , 2 , 3 / .
F i r s t we d e f i n e a s t h e c o l l e c t i v e c o o r d i n a t e s t h r e e r o t a t i o n a l c o o r d i - n a t e s , a n g l e s
y,, )I =1 , 2 , 3 and t h r e e v i b r a t i o n a l c o o r d i n a t e s al
1
=1 , 2 , 3 which o n e o b t a i n s t h r o u g h t h e d i a g o n a l i z a t i o n o f t h e c o o r d i - n a t e q u a d r u p o l e t e n s o r ,
n
w i t h i,j= 1 , 2 , 3 t h e s p a c e i n d e x a n d s
=1 , . . , n t h e J a c o b i v e c t o r i n d e x . Such a c o o r d i n a t e t r a n s f o r m a t i o n h a d b e e n done i n r e f s . /4,5/ i n o r d e r t o s t u d y t h e problem o f s e p a r a t i o n of t h e c o l l e c t i v e k i n e t i c e n e r g y . We d i d t h e c o o r d i n a t e t r a n s f o r m a t i o n u s i n g t h e t e c h n i q u e o f an o r b i t a n a l y s i s w . r . t . t h e SO(3)xSO(n) group / 6 , 7 / . The t y p e o f c o l l e c t i v i t y r e l a t e d t o t h i s o r b i t a n a l y s i s we c a l l SO(n) c o l l e c t i v i t y . The group SO(3)xSO(n) a c t on t h e 3 n - d i m e n s i o n a l E u c l i d e a n m a n i f o l d p a r a m e t r i z e d by t h e C a r t e s i a n c o o r d i n a t e s x i s s u c h t h a t t h e " c o l l e c t i v e " group SO(3) a c t s on t h e i n d e x i and t h e " i n t r i n s i c " group SO(n) on t h e i n d e x s . Through t h e o r b i t a n a l y s i s one g e t s r i d o f t h e r e d u n d a n t c o o r d i n a t e s w i t h o u t i n t r o d u c i n g c o n s t r a i n t s / 6 , 7 / , and a s t h e r e s u l t one w r i t e s t h e p o i n t o f t h e c o n f i g r u a t i o n s p a c e a s a f u n c t i o n o f t h e new c o o r d i n a t e s i n t h e u n i q u e f a s h i o n
( t h e r e s t k i c t i o n from SO(3) t o SO(3)/M i s n o t i A p o r t a n t i n f u r t h e r
* Work s u p p o r t e d by Deutsche F o r s c h u n g s g e m e i n s c h a f t
** Permanent a d d r e s s : I n s t i t u t e o f P h y s i c s , B e l g r a d e , J u g o s l a v i a
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984636
C6-298 JOURNAL DE PHYSIQUE
a n a l y s i s b e c a u s e M i s a d i s c r e t e g r o u p ) ;
o'(B)
ES O ( ~ ) / S O ( ~ - 3 )
IC R t h e r i g h t c o s e t s p a c e .
Both v 1 s and B's a r e e x p o n e n t i a l p a r a m e t e r s . The v i b r a t i o n a l variables gl
1
=1 , 2 , 3 one o b t a i n s t h r o u g h t h e d i a g o n a l i z a t i o n o f Q . . 1 3 0 2
toQ - -
=1 o i l ( v l x l
l J
1 l j (TI -
Out o f t h i s r e l a t i o n oge s e e s t h a t t h e x ' s must be chosen
LJ-a3 an ordered s e t , f o r example xi
2x2 -> g3. We r e s t r i c t o u r a n a l y s i s t o x l >
?i2> g3x,
r e l a t e d w i t h t h e group M. Through t h i s r e s t r i c t i o n , u n i m p o r t a n t f o r t h e quantum s y s t e m , f o r t h e c l a s s i c a l s y s t e m we a r e bound t o one t y p e of o r b i t s / 7 / . The same r e s t r i c t i o n was u s e d i n r e f . / 8 / .
The c a n o n i c a l l y c o n j u g a t e momenta r v v
=1 , ..., 3n t o t h e new c o o r d i - n a t e s
a r e r e l a t e d t o t h e C a r t e s i a n momenta pis by t h e i n v e r s e J a c o b i m a t r i x J - 1 ,
3n P i s
=1 J T '
v= 1
l S , Vn v ' where
- 1 - - - ' i s , v
a x i s
The J a c o b i m a t r i x J i s g i v e n by i t s m a t r i x e l e m e n t s
where
R ? ~ l p )and L?(p) a r e e s s e n t i a l l y g i v e n by t h e e x p r e s s i o n s
f o r any S O ( j ) g r o u p , compare r e f . / 7 / f o r t h e d e t a i l s . We s t a t e t h e f o l l o w i n g p r o p o s i t i o n w i t h o u t p r o o f :
1 P r o p o s i t i o n : The i n v e r s e J a c o b i m a t r i x J - I i s g i v e n by i t s m a t r i x
e l e m e n t s
c p b
P
=1 , 2 , 3 t h e group p a r a m e t e r s up t o t h e d i s c r e t e group M) a s
L L L
I L r 5 T , ( 6 ) o f R
=o,,, (Rl Y T I T (Bl = 6 G T
T T 1
The e x p r e s s i o n
one c a n u s e b o t h t o c a l c u l a t e t h e k i n e t i c e n e r g y / 7 / a n d t h e v e l o c i t y f i e l d . F o r t h e v e l o c i t y f i e l d we p r o c e e d a s i n r e f . / 8 / . The C a r t e s i a n momenta become
where t h e new c l a s s i c a l momenta a r e d e f i n e d a s f o l l o w s :
311-6 I,
I : = - I Og,v7
'Tg = 1 , 2 , 3 v
=4 , ..., n a n d
gv T=l -
rw a r e t h e momenta c a n o n i c a l l y c o n j u g a t e t o t h e c o o r d i n a t e gW,
C6-300 JOURNAL DE PHYSIQUE
w = 1 , 2 , 3 . The aomenta Lgv a r e a n g u l a r momenta " r e f e r r e d " t o t h e body i s t h e v o r t e x s p i n / l o / .
: ~ / t ~ ~ d c ~ f ' ) ; e c t i v e p a r t i n C a r t e s i a n momenta we mean t h e p a r t t h a t c a n be e x p r e s s e d a s -
3c o l l
=P i s j = 1 1 X i j x j s
w i t h c o e f f i c i e n t s X d e p e n d i n g o n l y on c o l l e c t i v e c o o r d i n a t e s and new momenta r e l a t e d t o t h e c o l l e c t i v e m a n i f o l d . F o r l o c a l h a m i l t o n i a n s we have
m v .
=1 s P i s We e x t r a c t vE:l1 a s
7
where
3 0 - l T
xVtb
11 : =1 o j w w ,, X v i b
=X y j b .
W =
1
1j
11'
S
3 3
L X . . = X s
Sx . . :
=1
1
J g = l v = g + l 1 (oig(~)Ojv(~)+oiv(~)ojg(~))- gv,
l jj i *
v g
As t h e a u t h o r s i n r e f . / 8 / we d e f i n e t h e v e l o c i t y f i e l d a t t h e p o i n t + x t o be t h e c o l l e c t i v e component o f t h e v e l o c i t y 3 s f o r t h e r e l a t i v e v e c t o r s a t 2, 3
I t i s c l e a r t h a t t h e SO(n) ( c o l l e c t i v e ) v e l o c i t y f i e l d s e p a r a t e s com- p l e t e l y from t h e i n t r i n s i c p a r t and i s i r r o t a t i o n a l , r o t 8
=0 .
I n o r d e r t o s t u d y more r e a l i s t i c t y p e o f c o l l e c t i v i t y i n which f o r example t h e v e l o c i t y f i e l d i s s u c h t h a t r o t f 0 , Weaver e t a 1 /11,10/
and Rowe e t a 1 / 8 , 1 2 / i n t r o d u c e d t h e p h e n o m e n o l o g i c a l C M ( 3 ) o r MQC
model (Mass Quadrupol C o l l e c t i v e Model) b a s e d on t h e i d e a o f s p e c t r u m
g e n e r a t i n g a l g e b r a s . These a u t h o r s i n t r o d u c e d 9 c o l l e c t i v e d e g r e e s o f
freedom: t h e s i x o n e s a l r e a d y mentioned f o r t h e v i b r a t i o n s and t h e r o -
t a t i o n s and t h r e e SO(3) d e g r e e s o f freedom which b e l o n g t o t h e v o r t e x
s p i n and s h o u l d y i e l d t h e c o l l e c t i v e f l o w which i s n o t i r r o t a t i o n a l .
The c o l l e c t i v e c o o r d i n a t e s i n t h e MQC model a r e t h e GL+(3, IR) parame-
t e r s . T h i s t y p e o f c o l l e c t i v i t y we c a l l t h e G t + ( 3 , IR) c o l l e c t i v i t y .
Note t h a t t h e v o r t e x s p i n we a l r e a d y i n t r o d u c e d i n SO(S)xSO(n) o r b i t
a n a l y s i s , b u t w . r . t . t h e SO(n) c o l l e c t i v i t y i t was a n i n t r i n s i c quan-
t i t y . I n r e f . / 8 / t h e a u t h o r s t r i e d t o i n t r o d u c e t h e MQC model t h r o u g h
t h e c o o r d i n a t e t r a n s f o r m a t i o n which was s i m i l a r t o t h e one o f V i l l a r s
/ 3 / ( r e d u n d a n t c o o r d i n a t e s ) , i n t r o d u c i n g 9 G L + ( 3 , IR) c o l l e c t i v e co-
o r d i n a t e s a n d n i n e c o n s t r a i n t s from which 6 were h o l o n o m i c , b u t t h r e e
nonholonomic which we a r e g o i n g t o d i s c u s s l a t e r on. Due t o t h e s e 9
c o n s t r a i n t s t h e y e x t r a c t e d t h e c o l l e c t i v e G L + ( 3 , IR) v e l o c i t y f i e l d
which was no l o n g e r i r r o t a t i o n a l and a l s o o b t a i n e d t h e t o t a l s e p a r a -
t i o n o f t h e k i n e t i c e n e r g y i n t o a GL+(3, IR) c o l l e c t i v e p a r t and t h e
i n t r i n s i c o n e . O t h e r a t t e m p t s were done i n o r d e r t o o b t a i n t h e m i c r o s -
c o p i c a l l y b a s e d MQC model: Rowe and R o s e n s t e e l / 6 , 1 3 / i n t r o d u c e d t h e
c o o r d i n a t e t r a n s f o r m a t i o n b a s e d on t h e o r b i t a n a l y s i s which h a s t o b e
s e e n a s t h e t r a n s f o r m a t i o n o f momenta r a t h e r t h a n c o o r d i n a t e s . Buck,
B i e d e n h a r n and Cusson / 1 4 / d i d s i m i l a r work. A l l t h e s e a u t h o r s o b t a i n e d
from t h e p o i n t o f view o f t h e GL+(3, W) c o l l e c t i v i t y . They d i s c u s s e d p a r t i c u l a r l y t h e problem of t h e s e p a r a t i o n o f t h e GL+(3, IR) c o l l e c t i v e k i n e t i c e n e r g y from t h e i n t r i n s i c o 6 t us t u r n back t o o u r e x p r e s - s i o n f o r p i , g i v e n above. The f i r s t t h r e e l i n e s of p i s a r e t h e " c a n d i - d a t e s " f o r t h e ( c o l l e c t i v e ) v e l o c i t y f i e l d w. r . t . t h e GL+ ( 3 , IR) c o l l e c - t i v i t y : t h e momenta L g V can s t i l l be s e e n a s r e l a t e d t o t h e SO(3) sub- m a n i f o l d o f t h e r i g h t c o s e t s p a c e SO(n)/SO(n-3) - cR, b u t t h e p a r t p r o p o r t i o n a l t o I t s i s r e l a t e d t o t h e whole CR and c a n n o t be d e f i n e d a s p u r e l y i n t r i n s i c w.r.t.GL+(S, IR) c o l l e c t i v i t y .
S i m i l a r problems a p p e a r i f one s t u d i e s t h e problem of t h e s e p a r a t i o n o f t h e c o l l e c t i v e k i n e t i c e n e r g y / 7 / .
We c l a i m t h a t t h e c o o r d i n a t e t r a n s f o r m a t i o n i n t r o d u c e d f o r t h e SO(n) c o l l e c t i v i t y i s n o t t h e a d e q u a t e one f o r GL+(3, IR) c o l l e c t i v i t y , and f o r GL+(3, IR) c o l l e c t i v i t y we i n t r o d u c e t h e c o o r d i n a t e t r a n s f o r - mation ( c . t . ) b a s e d on t h e o r b i t a n a l y s i s w . r . t . t h e GL+(3, IR)xSO(n) group / 7 / . Through t h e o r b i t a n a l y s i s w . r . t . t h e GL+(3, IR)xSO(n) group one c o n c l u d e s t h a t t h e whole 3n-dimensional E u c l i d e a n s p a c e i s g i v e n by t h e o n l y o r b i t p a r a m t r i z e d a s
1
X .
1 s
=i x i j o ~ , ( B ) i
=1 , 2 , 3 s
=I , . . . , n
j
=I
where x '
EGL+(3, R) and Ow($)
ESO(n)/[S0(3)xSO(n-3) 1 , r i g h t c o s e t s p a c e . T h i s d e c o m p o s i t i o n i s u n i q u e . The c o o r d i n a t e s x:- a r e t h e 9 c o l l e c t i v e c o o r d i n a t e s f o r t h e GL+(3, IR) c o l l e c t i v i t y . +he c . t . 41 we d e f i n e by
The c . t . 41 i s o n l y a k i r s t s t e p : we would l i k e t o o b t a i n some c o l l e c - t i v e c o o r d i n a t e s which we can i n t e r p r e t a s r o t a t i o n a l c o o r d i n a t e s , co- o r d i n a t e s f o r v i b r a t i o n s and d e f o r m a t i o n s and c o o r d i n a t e s f o r v o r t i - c i t i e s . These c o o r d i n a t e s c a n be e a s i l y i n t r o d u c e d f o r m a l y u s i n g t h e r e s u l t o f t h e c o o r d i n a t e t r a n s f o r m a t i o n r e l a t e d t o t h e SO(n) c o l l e c - t i v i t y o n l y f o r n
E3. So we i n t r o d u c e c . t . 4 2 , and 4 which i s de- f i n e d a s 4 :
=412
041. The c . t . $2 i s
i n t r o d u c e d through t h e unique d e c o m p o s i t i o n o f x ' -
where ~ ( c p ) sSO(3) / M ?he l e f t c o s e t - s p a c e , 0' (cp ' )
ESO ( 3 ) .
We r e s g r i c t a g a i n a s f o r SO(n) c o l l e c t i v i t y t o t h e o r b i t s o f t h e t y p e 2 1 > 2 2 > ~ 3 > 0 r e l a t e d t o t h e group M / 6 , 7 / .
F i n a l l y t h e c . t . 4 becomes
@
0( x i s l i = 1 , 2 , 3 s = l , ..., n)
+(qu,xi,cp~,l,6T~p,p',i=1,2,3
T =I , ..., 3n-9)
and i s r e l a t e d t o t h e f o l l o w i n g unique d e c o m p o s i t i o n o f x
7 7
C6-302 JOURNAL DE PHYSIQUE
The C a r t e s i a n momenta a r e e x p r e s s e d t h r o u g h t h e momenta c a n o n i c a l l y c o n j u g a t e t o t h e new c o o r d i n a t e s a f t e r t h e c . t . a s
p = J;' ~ 1 '
Twhere t h e J-' and J-' a r e i n v e r s e J a c o b i m a t r i c e s f o r t h e c o o r d i n a t e
transformations 2nd @ 2 r e s p e c t i v e l y . The m a t r i x J j l we a l r e a d y de- t e r m i n e d i n p r o p o s i t i o n 1 and we s t a t e w i t h o u t p r o o f / 7 / :
2 P r o p o s i t i o n : The i n v e r s e J a c o b i m a t r i x J i l f o r t h e c o o r d i n a t e t r a n s - f o r m a t i o n $ 1 i s g i v e n by i t s m a t r i x e l e m e n t s
The m a t r i x b l o c k (8) f o r m=l , 2 , 3 k=4,. . , n and
i.1,. . ,3n-S i s o f t h e t y p e L@,,(B) i n p r o p o s i t i o n 1 and t h e b l o c k t k l ,mk(B) f o r m t = 1 , 2 , 3 k 1 = 4 , ... , n a n d m , k = 1 , 2 , 3 i s o f t h e t y p e Lrn O,,(B) / 7 / .
Now we c a n t r y t o d e f i n e t h e c o l l e c t i v e v e l o c i t y f i e l d . Making use o f J - l
=J j l 121 we o b t a i n t h e c a n d i d a t e f o r p ' f g l l
3
I T
P i s ~011"
=1 . x ; ~
XIS1= 1
where
and
3
L = I R o (cp)vP
18'v
=, 2 2 3
gv u = l gv,u
w i t h vP c a n o n i c a l l y c o n j u g a t e t o
cpr e p r e s e n t t h e a n g u l a r momenta r e f e r r e d t o t h e body and
P3
Lo ,
c pg < v
=1 , 2 , 3
w i t h
TIc a n o n i c a l l y c o n j u g a t e t o
c p 'i s t h e v o r t e x s p i n .
lJt