IN V E S T IG A C iÓ N R E V IS T A M E X IC A N A D E F iS IC A 4 -l (1 )7 3 -7 7 F E B R E R O 1 9 9 8
D iscretizin g th e d efo rm atio n
p ara m eter in th e
s u
q(2)q u an tu m
alg eb ra
B .E . P allad in o an d P . L eal F erreira
I n s titu to d e F ( s ic a T e ó r ic a , U n iv e r s id Q l/e E s ta d u a l P a u lis la R u a P a m p lo n a
145,
0 1 4 U 5 - 9 0 0 . S 1 w P a u lo . S .P ., B r a z ilR ecib id o el3 d e ab ril d e 1 9 9 7 ; acep tad o cl5 d e ju n io d e 1 9 9 7
In sp ircd in recen t w o rk s o f B icd cn h arn [1 , 2 ] o n th e rcalizatio n o flh e q -alg eb ra 8 1 1 '1 (2 ), w e sh o w in Ih is n o te th al th e co n d itio n [ 2 j
+
l]q=
N q (j)=
in teg er. im p lies th e d iscretizatio n o f th e d cfo rm atio n p aram eter ü , w h crc q:=
e<>. T h is d iscretizatio n rep laces th e co n tin u u m asso cialed lO oby
an in fin ite seq u en ce a l. 0 '2 . " '3... o b tain ed fo r Ih e v alu es o fj.
w h ich lab ellh e irrcp s o fs uq(2).
T h e alg eb raic p ro p ertieso f N q (j) are d iscu ssed in so rn e d etail. in clu d in g ils ro le as a trace. w h ich co n d u ciS lo th e C 1 eb sch .G o rd an series fo r th e d irect p ro d u ct 0 1 ' irrep s. T h e co n ~ eq u en ces o f th is p ro cess o f d iscretizatio n are d iseu ssed an d its p o ssib le ap p licatio n s are p o in ted o u t A lth o u g h n o t a n ecessary o n e, th e p resen t p rescrip tio n is v alu ab le d u e to its alg ch raic sim p licity esp ccially in th e reg im e o f ap p reciab le v alu es o f o .
K l'y " ,o r d s : s uq( 2 ) , q u an tu m alg eb ras. p aram eter d iscretizatio n
In sp irad o s p o r trab ajo s recien tes d e B ied en h am [1 ,2 ] so b re la realizacio n d e la q -alg eb ra s uq(2). m o stram o s en esta n o ta q u e la co n d ició n
1 2 j
+
l) q=
N q ( j)=
eO lero , im p lica la d iscretizació n d el p arám etro d e d efo rm ació n o . d o n d e q=
eO• E sta d iscretizació n su b stitu y e el
co n tin u o aso ciad o a o p o r u n a su cesió n o } , 0 2 . U 3 • ...o b ten id a p ara v alo res d e
j
q u e ro tu lan las irrep s d e s U q (2 ). L as p ro p ried ad es alg eb ráicas d eN
q(j)so n d iscu tid as co n alg ú n d etalle. in clu so su p ap el co m o u n trazo , q u e co n d u ce a la serie d e C leb sh .G o rd an p ara el p ro d u cto d irectod e irrep s. L as co n secu en cias d e este p ro cesso d e d iscrctizació n so n d iscu tid as y su s p o sib les ap licacio n es so n in d icad as. A u n q u e n o n ecesaria. la p resen te p rescrip ció n es d e in terés d eb id o a su sen cillez aIg eb ráica. esp ecialm en te en el rég im en d e v alo res ap reciab les d e o .
D e s c r ip to r e s : sUq (2 ). alg eb ras cu án ticas. d iscrelil< lci6 n d e p arám etro s
P A C S : 1 2 .1 5 F f,
IUON.
J. In tro d u c tio n
T h e b raek el in th e lefl-h an d si d e o f E q . (3 ) is Ih e q -n u m b er d cfin ed b y
Q u an lu m alg eb ras are b y n o w k n o w n to p lay a d islin g u isb ed ro le in sev eral field s o fth eo relieal p h y sies [1 ,
2J.
S o rn e y earsag o B ied en h arn [1 ) ex ten d ed Ih e Jo rd an -S eh w in g er p ro
ee-d u re lo th e 5uq
(2)
q u an tu m alg eb ra b y m ean s o f a p air o f m u-tu ally co m m u lin g q .h arm o n ic o scillato rs a i q an d a i q (i
=
1 , 2 ). H e eo n stru eled a b asisIj,
m ) q fo r ( 2 j+
1 ) d im en sio n alirrep s o f
5u
q(2),
D~j), fo r ev eryj
=
0 , 1 /2 , 1 , ... w ilh mru n n in g in th e ran g e -
j ~m ::;
j
b y in teg er step s.T h e aetio n o f lh e g en eralo rs o f
5u
q ( 2 ) o nIj,
m ) q o b ey , as is w ell k n o w n . th e fo llo w in g relatio n s:h lj,m ) q
=
([j:¡:m )q [j:l:m + IJq )I/2 Ij,m :l:1 )q ,(1)
J ,lj, m ) q
=
mlj,
r n ) q . ( 2 )It Is rem ark ab le lh at E q s. (1 ) an d (2 ) are sim ilar to Ih o se o f 811(2). ex cep t fo r th e ap p earcn ce o f th e q .n u m b er b rack els,
ch aracteristic 0 1 "th e q -d efo rm ed case. In th is n o te w e w ish lo slu d y a p ro b lem in th e fram ew o rk o f th e ab o v e 5 " q ( 2 ) alg
e-b ra. n am ely , w e w ish lo d iscu ss th e m ean in g an d alg eb raic
co n seq u en ccs o f th e co n d itio n
[ 2 j
+
IJo
=
N ,,( j)=
in leg er.(5)
(6)
(7)
jii)
N o ( j)=
¿
emo,m=-j
i) N ,,( j)
=
[ 2 j+
IJo,
i
i i)N o
(j)=
s_ il_ Il_ l(_(J_'
_+_1_/_2_),,_)
si!lh
(0/2)
F irslly , w e reeall th at N o ( j) can b e w rin en in th e fo llo w in g eq u iv alen t fo rro s:
w h cre th e p aram etcr Q is d efin ed b y q
=
ea, w ith q a realp o siliv e n u m b er. T h e eo n d ilio n (3 ) say s Ih at in th e rig h t-h an d
sid e N n ( j) is an in teg cr n u m b er. T h u s, it im p lies ad is c r e tiz a . lioll o f o , o b lain ed fo r Ih e v alu es o f
j
d efin in g irrep s o f th e s uq(2)
alg cb ra. In stead o f a co n tin u o u s Q w e h av e n o w a d is-crete set 0 1 'v alu es o l'Q:
0 '1 ,0 '2 ,0 '3 , .... In th e n ex t S ectio n sw e sh o w th e co n seq u en ces o f th is d iscrctizatio n an d so rn e o C
its u sefu l asp ects in p o ssib le ap p licatio n s. T h is p ap er is o
r-g an ized as fo llo w s: In S ee!. 2 w e d iseu ss !h e p ro p erties o f
N,,(j)
an d th e eo n seq u en ees o f lh e o d iseretizatio n . See-lio n 3 d iseu sses
N o
(j) as a trace (o r eh aracter), see E q . (3 0 ),an d d ev elo p s its eo n seq u en ees [see E q .
(32)],
an d th e use-fu i relatio n s, g iv en b y E q s.
(32)-(37),
w ith N o ( j) in teg er.F in ally , S ee!. 4 eo n tain s o u r eo n clu d in g rem ark s.
2 . P ro p e rtie s
oC
N o (j)
(4)
(3)
sin h ( x o /2 )
si!lh
(0/2) ,
e x o / 2 _ e - x n / 2
74
B E . P A L L A D IN O A N D P . L E A L F E R R E IR Aand so on.
S e v e ra l re la lio n s in v o lv in g N a (j) a ris e fro m E q s . (9 )-(1 4 ). F o r in s la n e e . o n e h a s
T h e e x p re s s io n fo r N a (j) g iv e n in E q . (7 ) w ill h e u s e fu l to
o b la in !h e e q u a Iio n s w h ie h fo llo w . w ilh a d e p e n d e n e e o n h y
-p e rb o lie fu n e tio n s . F ro m E q s . (5 ) a n d (6 ) o n e g e ls
E q u a tio n (5 ) is !h e d e fin in g e q u a Iio n fo r N a (j) a n d s h o w s Ih a l lh e d e fin e d p o s iliv e in le g e r n u m b e r N a e o r-re s p o n d s to w h a t is c a lle d q -d im e n s io n o f re p rc s c n ta tio n ,
n a m e ly
[2j
+ 1 J ,. E q u a lio n s (6 ) a n d (7 ) fo llo w re a d ily fro mE q . (3 ) in le n o s o f o . E q u a Iio n (6 ) fo llo w s fro m lh e fin ile
-s e rie -s -s u m fo r th e q -n u m b e r-s o f a n in tc g e r
n:
W c w ill c o n s id e r h e re th e p o s itiv e v a lu e s o f O . F ro m
E q s . (1 7 ) a n d (1 8 ) o n e u h la in s lh e re s u lIs d is p la y e d in T a -b le l.
T h c firs t 2 5 s o lu tio n s o r " ro o ts " O í w h ic h s a tis fy (h e c o n -d ilio n o f E q . (3 ) a re lis le d in T a b le l. F ro m lh a t lis l o n e
c a n n o te th a t s o rn e v a lu e s o f O í a re o f th e fo rm 2 o ). lik e
0 '6
=
2 0 2 ,0 1 3=
2 Q 3 . 0 '2 2=
2 0 4 • .... B e s id e s . th e s e v a l. u e s u fQ e o rre s p o n d lo th o s e in w h ie h N (I/2 ) is o f th e fo n ov m te g c r = in tc g c r. T h c s a m e ra e t w ill o c c u r fo r
j
=
3 /2 . 5 /2 , ... a l (h e s a m e " ro o ts " D . In s te a d , fo r j=
¡n te g e r. o n e h a s N (j) = in le g e r to o . In g e n e ra l. fo r Na(1 /2 ). w ilho o f lh e fo n n 2 0 ), o n e h a s
N
a;=2aj (1 /2 ) = J lO te g e r =J(j
+ 1 )2 =j
+ 1 . F o r in s ta n e e . 0 6 = 2 0 2 , e o rre s p o n d -in g ly . N a , (1 /2 ) =J 9
= 3 .A s a n o lh e r e x a m p le : w h e n d e a lin g w ilh lrip le ls (j = 1 ).
w e fo u n d
}' = ~ - [3 J 2 - 1 - V 2 (1 ) - 1 (2 0 )
n - {1
/2 1 ~ -
0 '= 0 ./ 2 - 1 0 .'= 0 ./ 2 .T h is q u a n tity . Yc •• is re la te d 1 0 th e tra c e o f m a s s m a tric e s fo r lh e
j
= 1 re p re s e n ta tio n s (s e e R e L 3 ). N O lie e th a l fo r a n y oo f th e fm m 2 0 :j o n e (h e n h a s }:l" a s a n in tc g c r n u m b e r, lo o . S o m c rc s u lts f U f Q i =2 0j a re g iv c n in T a b le
n.
In T a h le 1 Il w e s h o w lh e v a lu e s o f[2 j + 1 1 a = N a (j) fo r
th e firs l 6 " ro O IS " o . N O lie e ¡h a t w h e n [2 ]a = in le g e r. th e n
[4 1 a a n d [6 1 a a re in Ie g e rs 1 0 0 . a s w e a lre a d y m e n lio n e d .
W c n o te th a t
ir
w e c n u m c ra le th e d is c rc le v a )u e s o f Q inO U T T a h lc s h y m e a nSo f a n in d c x i(w ritin g o .¡), w c h a v e
[2 J " . =
N".
(1 /2 ) =JN
a.(1)
+ 1 = J i+ 3 , (2 1 )(9 )
(8)
n - 1
[n J,
=I :
q '/2l= - ( n - I)
[I]a
=
N a (O )=
l.
'lo .A n d u s in g E q s . (6 ) a n d (7 ). o n e h a s . fo r N a (j).
[2 )a = Na(1 /2 ) = 2 e o s h (0 /2 ). (1 0 )
[3 )a =
N
a(l)
= 2 e o s h o + l. (1 1 )[4 1 a = Na(3 /2 ) = 2 e o s h (3 0 /2 ) + 2 c o s h (0 /2 ). (1 2 )
[5 ]a = Na(2 ) = 2 c o s h (2 0 ) + 2 e o s h (0 ) + l. (1 3 )
[6 ]a = Na(5 /2 ) = 2 e o s h (5 0 /2 ) + 2 e o s h (3 0 /2 )
+ 2 e o s h (0 /2 ), (1 4 )
( 1 5 )
F u rth e n o o re . b y la k in g j = l. o n e g e ts . fro m E q . (1 \).
a s
[3 ]a .
=
N
a,(l)
= i+
2 (2 2 )1
e o s h 0 =
:¡[N
a(1)
-
1 ]. ( 16)If o n e w is h e s lo e X le n d T a b le 1 Il fo r h ig h e r v a lu e s o f j a n d
O í. th e fo llo w in g re la tio n s w ill a l5 0 b e u s e ru !:
In s p e c tio n o f th e a b o y e e q u a tio n s iriv o lv in g c o s h s h o w s th a t,
if [3 ]a =
N
a(1)
= in te g e r n u m b e r.lh e n fo r a ll in le g e r v al-u e s o f j o n e h a s [o d d n u m b e r]
=
in le g e r n u m b e r. w h ile if [2 J a = Na(1 /2 ) = in le g e r n u m b e r. lh e n o n e a ls o h a s [e v e nn u m b e r] =in le g e r n u m b e r. c o rre s p o n d in g lO h a lf-in le g e r v a l-u e s o fj.
T h u s , w c c a n Iis t th e v a lu e s o f O ' w h ic h s a tis fy c o n d itio n
(3 ) b y in v e rtin g E q . (1 6 ). w h ie h re a d s
0 = e o s h -1 [ ~ (N a (1 ) - 1 )] , (1 7 )
a n d b y ju s lla k in g lh e v a lu e s o f N a (1 ) w h ic h a re in le g e r n u m
-b e rs
2:
3 .E q u a lio n (1 7 ) g iv e s . fo r re a l o .
0 = :tln
(x
+ ~ ),
(1 8 )w ith x '"
(1/2)[N
a(1)
-
1 ] a n in le g e r o r h a lf-in te g e r n u m h e r 2:: l. N o tic e th a t. a s Q= e a a n e h a sf'ro m th e s e re la tio n s a I le s e c s th a t e v c ry lim e th c n u m b e r (i+ 3 ) is a p e rfe e t s q u a re .lh e n [2 J a . [4 )a . [6 J a , ... = in le g e r n u m b e rs . F u rlh e rm o re . th e n u m h e r Y o .d e fin e d in (2 0 ). c a n
b e w rittc n a s
[4 J a ;
=
N a .(3 /2 )=
(i+
1 ) J i + 3 ,(23)
[5 J a ; = N a .(2 ) = (i2 + 3 i + 1 ), (2 4 )
[6 ]a ;
=
N a .(5 /2 )=
(i+
2 ) i J i + 3 ( 2 5 )} 'a , = J i+ 3 ( J i+ 3 + 2 ) , (2 6 )
a n d th e n , in th e c a s e J i+ 3
=
in tc g e r. Y o ; is a n in te g e r 1 0 0 , w h ie h e x p la in s lh e re s u lIs fo r Y a in T a b len.
S e s id e s , w es h o u lJ m e n tio n th a t fo r th e 0 i
=
2 0 :iD n e h a si
=
( j+
1 )2 - 3 , w ilh lh e in d e x j = 1 , 2 . 3 .... (a s c a n b e e h e e k e d fro m th ev a lu e s in th e firs l c o lu m n o f T a b le I1 ) a n d th e n o n c fin d s , fo r
in s ta n c e
(2 7 )
Ya.~2",
=
( j +I)(J
+ 3 ) ,W ilh
j
= \.2 .3 ...(1 9 )
q = x + ~ o r ( x + ~ ) - 1
D I S C R E T l Z I N G T 1 l E D E F O R M A T l O N P A R A M E T E R I N T H E
SUQ(2)
Q U A N T U M A L G E B R A75
T A B L E 1 . L i S l o f s o l u t i o n s o f c o n d i t i o n ( 3 ) f o r r e a l o w i t h
N
u n t i l 2 5 . T h e s e s o l u t i o n s w e r c o b t a i n e d f o r i n t e g e r j . f r o m ( h e c a s e o f j = 1 . F o r h a l f .i n l e g e r j . t h e y s a t i s f yNo. =
( i n t e g e r ) 1 / 2 .N(I)
x
=~(N(I)
-
1 ) n iNa(~)
=JN(I)
+
13 1 0 1 = 0 1 4
4 3 / 2 o ,
=
0 . 9 6 2 4 2 3 6 V 55 2 " 3 = 1 . 3 1 6 9 5 7 9 / 6
6 5 / 2 '" = 1 . 5 6 6 7 9 9 2
../ 7
7 3 0 5 = 1 . 7 6 2 7 4 7 2 v '8
8 7 / 2 0 6 = 1 . 9 2 4 8 4 7 3 V 9
9 4 . " = 2 . 0 6 3 4 3 7 1 v T O
1 0 9 / 2 o , = 2 . 1 8 4 6 4 3 8 V i l
II
5 0 9 = 2 . 2 9 2 4 3 1 7 V 1 21 2 1 1 / 2 0 1 0 = 2 . 3 8 9 5 2 6 4 v T I
1 3 6 o " = 2 . 4 7 7 8 8 8 7 J I 4
1 4 1 3 / 2 o " = 2 . 5 5 8 9 7 9 0 V f 5
1 5 7 0 1 3
=
2 . 6 3 3 9 1 5 8 v 'i 61 6 1 5 / 2 " 1 4 = 2 . 7 0 3 5 7 5 8 v T 7
1 7 8 " 1 5 = 2 . 7 6 8 6 5 9 3
VT8
1 8 1 7 1 2 " 1 6 = 2 . 8 2 9 7 3 5 0 v 'I 9
1 9 9 0 1 7 = 2 . 8 8 7 2 7 0 9 v '2 O
2 0 1 9 / 2 0 '8 = 2 . 9 4 1 6 5 7 3 v '2 l
2 1 1 0 0 . 9 = 2 . 9 9 3 2 2 2 8
- m
2 2 2 1 / 2 0 '0 = 3 . 0 4 2 2 4 7 1 J 2 3
2 3 1 1 o " = 3 . 0 8 8 9 6 9 9 v '2 4
2 4 2 3 / 2 " " = 3 . 1 3 3 5 9 8 5 V 2 5
2 5 1 2 o " = 3 . 1 7 6 3 1 3 1 J 2 6
T A B L E J I . L i s t o f S o l u l i o n s o f E q . ( 3 ) f o r h a l f - i n t e g e r j . T h e y s a t i s f y Q = 2 0 / a n d a r e a l5 0 s o l U l i o n s f o r i n t e g e r j . N u m e r i c a l r e s u l l s f o r t h e f i r s (1 0o ' s o f ( h e f o r m O : i = 2 0 j a r e d i s p l a y e d .
O i
=
2 0 ) a l = 2 0 " 1 = 0 " 6 = 2 0 , = 1 . 9 2 4 8 4 7 3 0 1 3 = 2 0 3 = 2 . 6 3 3 9 1 5 8 0 , , = 2 0 . = 3 . 1 3 3 5 9 8 5 0 3 3 = 2 0 5 = 3 . 5 2 5 4 9 4 3 0 '6 = 2 0 8 = 3 . 8 4 9 6 9 4 6 " 6 1 = 2 " 7 = 4 . 1 2 6 8 7 4 1U 7 8 = 2 0 8 = 4 .3 6 9 2 8 7 6
0 9 7 = 2 " 9 = 4 . 5 8 4 8 6 3 3
O l l 8
=
2 0 1 0=
4 . 7 7 9 0 5 2 8[ 3 J a
=
N a ( l ) 38 1 5
24
35
4 8 6 3
80
99
120
[ 2 J a
=
N
a(I/2)
14=2
V 9 = 3 v 'i 6 = 4 V 2 5 = 5 v '3 6
=
6 1 4 9 = 7 / 6 4 = 8 v '8 l = 9V I O o = 1 O
V i 2 f
=
1 1Ya
8 1 5
24
3 5
48
6 3
80
99
120
1 4 3
O n th e o th e r s id e . w e c a n ta k e
j
=
1 a n d o n e g e n e r a lo r , s a yJ"
o f I h e 5 U ( 2 ) g r o u p , w h i c h i s r e p r e s e n l e d b y t h e 3 x 3 m a t r i x3.
N,,(j)
a s a t r a c eA f l e r a H l h e s e n u m e r i c a l r e m a r k s w e w i s h l o d i s c u s s I h e m e a n i n g o f
Na(j)
f r o m a n o l h e r p o i n l o f v i e w . A s d e s c r i h e d b y B i e d e n h a r n [ 2 ) , l h e q - d i m e n s i o n i s d e f i n e d b yL"p
qJ,.
a n d f o r
irreps
l a b e H e d b yj
i l i s g i v e n b y[2j
+
1 ) , . T h u s , a c c o r d i n g l o ( 3 ) .Na(j)
w a s i d e n l i t i e d w i t h I h e q - d i m e n s i o n .1 7 i
O 1 7 i
( 2 8 )
76
B .E . P A L L A D lN O A N I l P . L E A L F E R R E I R AT A B L E 1 1 1 . Y a l u c s o f
[2j
+ lj o f o r t h e f i r 5 t 6 " r o o t s " n i .o.
[ 2 ] 0=
N
o(I/2)
[ 3 ] 0=
N
o(l)
[ 4 ] "=
N" (3/2)
[ 5 ) 0=
N,,(2)
0 1= 0 V 4 3 2 V 4 5
0 2
=
0 .9 6 2 4 2 3 6 1 5 4 3 1 511
0 3
=
1 .3 1 6 9 5 7 9J6
54J6
19
a '
=
1 .5 6 6 7 9 9 2 - ./ 7 6 5 - ./ 729
O s = 1 . 7 6 2 7 4 7 2 v 's 7 6 v 's 4 1
0 6
=
1 .9 2 4 8 4 7 3 V 9 8 7 V 9 5 5[ 6 ] "
=
,\ ',,( 5 / 2 )3 V 4
8 1 5
15J6
2 4 - ./ 7
3 5 v 's
4 8 V 9
W e w i s h l O e x p o n e n t i a t e e oJ r . T o t h i s c o d . w c d i a g o
-n a l i z e J z a n d a p p ly lh e C a y le y - H a m illo n l h e o r e m
[4J.
O n eo b t a i n s
e
ol•=
1
+
( .i u h o ) J z+
( c o s h a - 1 ) J ; .A s T r J z
=
O , T r J ;=
2 , w e o b l a i n l h e l r a c eT r e n ) .., = 1 + 2 cOl'1h 0',
(29)
(30)
d i s p l a y e d i n l h e l a s l r o w . T h e n . i n E q s . ( 3 3 ) - ( 3 5 ) . o n e g e l s
No.(I/2)N,,¡;(1/2)
=
3
x3
=
9
=
1
+
8,
(36)
N",(1/2)N",,(1)
= : 1 x 8=
2 4=
3+
2 1 .(37)
N0 6( 1 ) ,\ ',,6 ( 1 )
=
8 x 8=
6 4=
1+
8+
5 5 . ( 3 8 )a n d s o o n .
0 1 1 t h e o t h e r h a n d , b y l a k i n g J I
=
h=
j,
o n e o b t a i n sN o ( j ) '
=
[ N o ( O )+
N,,(l)
+ ... +
N,,(2j)].
(39)a r e s u l l w h i c h c o i n c i d e s w i l h N o ( l) . A s i m i l a r r e s u l l i s v a l i d
f o r j
=
1 / 2 . O f c o u r s e , l h e s a m e h o l d s f m l h e o l b e r l w og e n e r a t o r s
J
y a n dJ
z• a s t h e i r t r a c e s a r e t h e s a m e . T h e n , [ r o m a s i m i l a r i t y t r a n s f o r m a t i o n a n e c a n r e w r i t e Lrep eo J za s T r
e
ol, w h i c h i s e q u a l l o N o(j),
a s w e h a v e s e e n f o r l h e c a s e j=
1 . T h e s a m e h a s 10b e v a l i d f m l h e o l h e r g e n e r a l m sa n d t h e r e s u l t , o f c o u r s e , c a n b e g e n c r a l i z e d . T h u s , w e w r i t c
A n d E q . ( 3 9 ) c a n b e r e w r i t t e n , b y l I l e a n s o f E 4 . ( 3 ) , a s
[2j
+
1 ] ~ ,=
( 1+
[ 3 ] "+ ... +
[4j
+
1 ] ,,) ,(40)
w h i c h g i v c r i s e l o a n u m b e r 0 1 ' i n l e r e s t i n g r e l a t i o n s . W e c a l l
a l t e n t i o n t o t h e r a e t t h a t m o s l o f t h c q - n u m b c r i d c n t i t i e s f o u n d
i n R e f . 2 c a n h e v c r i f i e d [ r o m I h e a b o y e c q u a t i o n s , w h c r e t h e
d i s c r e t i z a t i o n p l a y s i t s r o l e f o r i n t c g c r n u m h c r s
No.
(31 )
4 . C o n c l u d i n g r e m a r k s
T h e m a i n p o i n l i s l h a l h e r e
No(j)
i s a t r a c e ( n o l a d i-m e n s i o n ) a n d i s e x p r e s s e d i n f u n c t i o n o f t h e g e n e r a t o r s o f t h e
u s u a l S U ( 2 ) l h r o u g h a n a l o g o u s r e l a t i o n s o f l h o s e o f E q s . ( 2 9 )
a n d ( 3 0 ) .
T h e a b o y e r e m a r k h a s o b v i o u s c o n s e q u e n c c s . F o r c x a m
-p i e , i f w e m u l l i -p l y l w o N o ( j ) 's , c o r r e s p o n d i n g l o v a l u e s
j¡
a n d
izo
w e r e o b t a i n , f o r t h i s s p c c i f i c c a s e . t h e w e l l k n o w nC l e h s c h - G o r d a n s u m m a t i o n f o r t h e
direct product
il+h
No(j,)N,,(j,)
=
L
No(j).
(32)
] = I j ¡ - h l
a r e s u l l l h a l c a n b e e a s i l y v e r i f i e d b y u s i n g E q s .
(9)-(
1 4 ) . F mi n s t a n c e , o n e h a s
No(1/2)No(1/2)
=
N o ( O )+
N o ( l) ,(33)
N,,(1/2)No(1)
=
No( 1 / 2 )+
N,,(3/2),
(34)
N
o(l)N
o(1)
=
N o ( O )+
N o ( l )+
N o ( 2 ) , ( 3 5 )e t c ....
C o n s i d e r , f o r e x a m p l e , t h e c a s e o f o
=
0 6 . T h c v a l u e so f
N,,(j)
a r e g i v e n i n T a b l e 1 I l . F m " 6 w e h a v eN(O)
=
1 ,N(I/2)
=
3 ,N(l)
=
8 ,N(3/2)
=
2 1 ,N(2)
=
5 5 , a s\ V c w o u l d l i k c l o a d d r e s s n o \ \ ' l o s o r n e c o n c l u d i n g c o n s i d e r
-a t i o n s . T h r o u g h o u t t h i s n o t c w c h a v e s h o w n s e v e r a l u s c f u l
r e l a t i o n s f o r q - n u m h e r s w h i c h a r e c o n s c q u c n c c 0 1 ' t h e c o n d i
-l i o n I h a t [ 2 ) + l ] u
=
i n l e g e r . T h i s c o n d i t i o n l e d u s t o a d i s-c r -c t i z a t i o n a l ' t h e s e t o f \ ' a l u e s 0 1 ' o . V e r y p r a c l i c a l r e l a t i o n s .
l i k e E q s . ( 2 1
)-(25)
1 m e x a m p l e , a l l o w a q u i e k e a l c u l a l i o n o f[ n ] n f o r a n y O : i o r l h e e n u m e r a l e d l i s t o f o ' s . \ V c h a v e a p p l i e d
t h e p r c s e n t s c h e m e i n a c a l c u l a t i o n o f f u n d a m e n l a l f e r m i o n
l 1 l a s s e s u s i n g l h e a l g e b r a o f t h e
SU
q(2)
g r o u p a s a s p e c l r u mg c n e r a t i n g d c f o r r n e d a l g e b r a 1 3 ] . T h e s i m p l i f i c a t i o n s o c c u r
-i n g h y l h e u s e o f t h e d i s c r e l i z a l i o n w e r e g r a t i f y i n g . W c r c
-m a r k , h u w e v c r , l h a l l h e v a l u c s 0 1 'O : f o r t h e p r o h l e m l r e a t c d
i n R c f . 3 w c r c t y p i c a l l y r a t h e r l a r g c s , 0 1 ' l h e o r d e r o = 2 . 6
-4 .8 . N o t i c e t h a t f o r v a l u c s 0 1 '( t i i n T a b l e l - o b l a i n e d f r o m
d i s c r e l i z a t i o n - w e h a v e 0 . D 6 < ( } i < 3 .1 8 . W e r e c a l l l h a t
f o r i n s t a n c e , r m l h e p r o h l c m 0 1 ' n u c l e a r r o ta l io n a l b a n d s 1 5 1
l r c a l e d w i l h d e f o r m a l i o n , v a l u c s o f t h e o r d e r 1 0 '1
=
0 . 0 3 0w e r e u b t a i n c d . T h c r e i s a d i f f c r e n c c 0 1 ' S O I 1 l Co r d c r s 0 1 ' m a g n i
-t u d e f u r l h e v a l u c s o l 't i i n t h c l w o p r o b l e m s . \ V c c ) ; p l a i n l h i s
a p p a r e n t d i s c r c p a n c y b y t h e v c r y d i f f e r e n t c o n t e x t o f e a c h
t r c a l m e n t . \ V h i l c h e r c , a n d i n t h e w o r k o f R e f . 3 , w h i c h i s d i
-r c c t l y h a s e d o n l h e
su
q(2)
a l g c h r a . t h e e n e c t 0 1 ' d c f o r m a t i o ni s l a r g c , i n o l h c r 1 1 l 0 d e l s i n h a d r o n i c , n u c l e a r a n d m o l e c u l a r
p h y s i c s l h e d c f o r m c d a l g e h r a s a r e m e a n l l o d e s c r i b e s m a l l
D IS C R E T IZ IN G T H E D E F O R M A T lO N P A R A M E T E R IN T H E SUQ(2) Q U A N T U M A L G E B R A
77
d c v ia tio n s fro m a n a lm o st c x a c t sy rn m c try . im p ly in g l1luch sm a Ile r v a lu c s o f th e d e fo rm a tio n p a ra m e te r, n o l n e c e ssa rily h e lo n g in g loOUT"d isc rc tiz e d " 5Ct.
W e p o in l o U llh a l fo r O
=
ilol.
Ih a l is, fo rq
a p u re p h a se fa c to r(q
=
ei1ul), th e sa m e d isc re tiz a tio n p ro c e ss c o u ld h e c a rric d o u t le a d in g , h o w c v e r, to a d iffe rc n l re g im e , c h a ra c-te riz e d b y c a ,h o -; c o sh
ilol
=
c a s101,
F in a lly , w e w ish 10e m p h a siz e th a t th e p re se n t p re sc rip tio n o f in te g e r v a lu c s0 1 ' th e q -d irn c n sio n . in g e n e ra l a p p lic a tio n s is n o l a " e c e s . s a r y a lle . N e v c rth e le ss. d u c lo its a lg e b ra ic sim p lic ity , it is a n u sc fu l p ro c c d u rc e sp e c ia lly in d c a lin g w ith p ro b le m s in v o lv -in g la rg e v a lu c s o f Q . T h is. o f c o u rse , m e a o s th a t w e d o 1101
e x c lu d e th e c a se o f o r d in a r y q .d im e n sio n in o th e r p h y sic a l a p p lic a tio n s. W e fu rth e r re m a rk Ih a l o n e o f lh e u se fu i e o n
-1 . L .e . B ie d e n h a m . J. P h y s . A : M a lh G e n . 2 2 (1 9 8 9 ) L 8 7 3 ; A J. M a c fa rla n e , J P h y s . A : M a tlt. G e n . 2 2 (1 9 8 9 ) 4 5 8 1 . 2 . L .e . B ie d e n h a m . in th e X V III I n te r n a tio n a J C o //o q u iw 1 J C H !
G r o u p T h e o r e tic a l M e th o d s in P h y s ic s , M o sc o w , U .S .S .R .• (S p rin g e r-V e rla g , B e rlin . 1 9 9 0 ), p . 1 4 7 ; L .e . B ie d e n h a rn a n d r..t.A . L o h e , Q u a n lu m G r o u p S y m m e tr y a n d q - T e n s o r A lg e h r c u , (W o rld S c ie n tific . S in g a p o re . 1 9 9 5 ).
3 . S .E . P a lla d in o a n d P . L e a l F c rre ira , F u n d a m e n ta l F e r m ir m
se q u e n c c s o f th e d isc re tiz a tio n is th e p o ssib ility o f o b ta in in g
re la liu n s Jik e E q s.
(32)-(40).
w ilh N o (j) in le g e r, in a sim p le w a y .A c k n o w le d g m e n l~
W e a re g ra le fu l 1 0 P ru f. l .A .C a stilh o A le a rá s fo r h e lp fu l d is-c u ssio n s a n d fo r a is-c a rd u l re a d in g o f th e m a n u sc rip t. T h a n k s a re a lso d u c to L . T o m io fo r d isc u ssio n s a n d fo r h e lp o n c 1 o
s-in g Ih e m a n u sc rip \. W e a re g ra le fu l lo F A P E S P , S a o P a u lo , fo r p e rip h e ric a l su p p o r\. O n e o f u s, P .L .F ., is g ra te fu l lo th e C o n se lh o N a c io n a l d e D e se n v o lv im e n to C ie n tífic o e T e c
-n o ló g ic o (C N P q ), B ra z iJ. fo r a re se a rc h g ra n \.
M a s u s fm m D e fo r m e d S U q ( 2 ) T r ip le ls ,I F f- P .0 3 8 1 9 6 ;
JJ
M w v o C u n . Iln A (1 9 9 7 ) 3 0 3 .4 . P c tc r L a n c a ste r, T J lt'o r y o f M a tr ic e s , (A c a d e m ic P re ss. N e w Y o rk a n d L o n d o n . 1 9 6 9 ), p . 6 6 .
5 . D . B o n a tso s, P .P . R a y c h e v . R .P . R o u sse v . a n d Y u F . S m im o v , C /u 'm . P h ) 's . L e n . 1 7 5 (1 9 9 0 ) 3 0 0 ; D . B o n a tso s,
