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HAL Id: jpa-00224238

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Submitted on 1 Jan 1984

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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�

(2)

I n s t i t u t fiir Theoretische Physik der Universittit,

0-7400

Tiibingen, 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

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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)

E

S O ( ~ ) / S O ( ~ - 3 )

I

C 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

to

Q - -

=

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

2

x2 -> 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 , V

n 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

(4)

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

'T

g = 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,

(5)

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 -

3

c 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

1

j

11

'

S

3 3

L X . . = X s

S

x . . :

=

1

1

J g = l v = g + l 1 (oig(~)Ojv(~)+oiv(~)ojg(~))- gv,

l j

j 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

(6)

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 '

E

GL+(3, R) and Ow($)

E

SO(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

E

3. 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

0

41. 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 ' )

E

SO ( 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

(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 '

T

where 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 ; ~

XIS

1= 1

where

and

(8)

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

cp

r 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

P

3

Lo ,

c p

g < v

=

1 , 2 , 3

w i t h

TI

c 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

P '

bCv9

The l a s t l i n e i n itp:~ll'; i n c l u d i n g L ~ l k , g v (B) related t o the matrix block L 0,, ( 8 ) wh~ch

&pehds

on t h e i n t r i n s i c c o o r d i n a t e s 6) s p o i l s t h e s e p a r a - t i o n . I f we would p o s t u l a t e t h a t L o ~ K ( B )

=

0 , what i s e q u i v a l e n t t o t h e c o n s t r a i n t s L Y , , ( R ) ~ o / 7 / we would o b t a i n t h e s e p a r a t i o n . I f t h i s c o n d i t i o n i s compared w l t h r e f . / 8 / , one f i n d s t h a t t h e r e l a t i o n ( 2 . 9 ) v o s t u l a t e d t h e r e mav be w r i t t e n a s

b , v = 1 , 2 , 3 . A c t u a l l y i n t h i s r e f e r e n c e o" i s some 3xn m a t r i x which i s n o t i d e n t i f i e d a s o"

E

S O ( n ) / ( S 0 ( 3 ) x S O ( n - 3 ) ) , a n d t h e r e l a t i o n ( 2 . 9 ) means t h r e e nonholomonic c o n s t r a i n t s from a l t o g e t h e r 9 c o n s t r a i n t s on ogs t h e r e d e n o t e d by x & . n

=

1 , ..., N (NrA-1) and a

=

1 , 2 , 3 . These c o n s t r a i n t s d e f i n e 3n-9 I n t r i n s i c c o o r d i n a t e s P7 d e n o t e d t h e r e - b y - E n . I t s h o u l d be s t r e s s e d t h a t t h e c o n s t r a i n t s L ~ a , [ ~ )

=

0 do n o t followfrom t h e t h e o r y o f t h e L i e groups i n v o l v e d . S i m i l a r c o n c l u s i o n f o l l o w s i f one s t u d i e s t h e p r o b l e m o f t h e s e p a r a t i o n o f t h e G L + ( 3 , I R ) c o l l e c t i v e k i n e t i c e n e r g y / 7 / , which i s a l s o p o s s i b l e u n d e r t h e same c o n d i t i o n s . R e f e r e n c e s

/ 1 / LIPKIN H . J . , DE SHALIT A. and TALMI I . , Nuovo C i m . (1955) 773.

/ 2 / VILLARS F . , N u c l . P h y s . 3 (1957) 240.

/ 3 / VILLARS F.M.H. and COOPER G . , Ann.Phys. 56 (1970) 224.

/ 4 / ZICKENDRAHT W . , J.Math.Phys. 12 (1971) 1 x 3 .

/ 5 / DZYUBLIK A.Ya., OVCHARENKO V.T, STESHENKO A . I . and FILIPPOV G.F., S o v . J . N u c l . P h y s . 15 (1972) 487.

/ 6 / ROWE D . J . and R O S ~ S T E E L G . , Ann.Phys. (N.Y.) 126 (1980) 198.

/ 7 / PAPADOPOLOS Z . , D i s s e r t a t i o n , U n i v e r s i t a t Tubingen, 1983 / 8 / GULSHANI P . and ROWE D . H . , Canad.J.Phys. 54 (1976) 970.

/ 9 / BIEDENHARN L . C . and BRUSSAARD P . J . , 'hulomb E x c i t a t i o n , C l a r e n - don P r e s s , Oxford 1965.

/ l o / WEAVER O . L . , CUSSON R . Y . and BIEDENHARN L . C . , Ann.Phys. (N.Y.) 102 (1976) 493.

/ 1 1 / WEAVER L . , BIEDENHARN L . C . and CUSSON R . Y . , Ann.Phys. 77 (1973) 250.

/ 1 2 / ROSENSTEEL G . and ROWE D . J . , Ann-Phys. 123 (1979) 36.

/ 1 3 / ROWE D . J . , i n "The M i c r o s c o p i c R e a l i z a t i o n o f N u c l e a r C o l l e c t i v e Models", Group Theory a n d I t s A p p l i c a t i o n s i n P h y s i c s , e d . by T.H. Seligman (AIP Conf. P r o c . 71, New York 1981)

/ 1 4 / BUCK B . , BIEDENHARN L . C . a n d CUSSON R . Y . , N u c l . P h y s . A317 (1979)

205.

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