Group-theoretical method for internal symmetries of elementary particles from space-time geometry determinatio

Zmuidzinas, J. S.

English

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RICE - $- aOMETRICAL ORIGIN OF INTZRNAL SYMMETRIES* 
Jonas S t a s p  Vpuidzinas 
I I .  
Jet & o p d :  zoo w n o d A 1 1 7 1 1 V l  , 
Cal i forn ia  I n s t i t u t e  of ~echnology,  
Pasadena, Ca l i fo rn ia  
rd copy (HC) 
crofiche (M F) 
~ l y  65 
In  t h i s  note we wish t o  ou t l ine  a group- theore t ica l  method 
 of obtaining i n t e r n a l  symmetries of elementary p a r t i c l e s  from the 
geometry of space-time. 
b 
1) The bas i s  of our theory i s  t h e  group T 
having as t'ne generators  of i t s  Lie a lgebra  3 ' t h e  i d e n t i t y ,  momentum; 
2 )  <;rid pos i t i on  operators  I, Pp, and X JL , JL = 0, 1, 2, 3, s a t i s f y i n g  
(gpv)  = diag  (1, -1, -1, -1) . 
These equat ions represent  t he  s implest  covar ian t  gene ra l i za t ion  of t h e  
n o n - r e l a t i v i s t i c  canonical  commutation r e l a t i o n s  [p i' qj]  = -isij. 
c i b l e  un i t a ry  representa t ions  of T are labe led  by the eigenvalue Q 
( r e s t r i c t e d  t o  be > 0 f o r  physical  reasons)  of I; f o r  t he  maxim1 
abe l i an  subalgebra of 
t hus  lap>, t o  be phys ica l ly  in t e rp re t ed  as t h e  momentum e igens t a t e s  
.;f matter i n  which a l l  spin,  isospin,  e t c . ,  dependence i s  'Lashed out"  
or ignored. 
be regenerated by s t a r t i n g  with the bas i c  s t a t e s  only. 
I r redu-  
we choose the  f o u r  P P .3) The bas i c  states are 
We aim t o  show how these a d d i t i o n a l  quantum numbers can 
*This paper r ep resen t s  t h e  results of 
ou t  by tne  J e t  Propulsion Laboratory, 
under Zontract NAS7-loo, sponsored by 
one 
Cal 
t h e  
phase of research  c a r r i e d  
i f o r n i a  I n s t i t u t e  of TechriGl 
National Aeronautics and 
Space Administration. 
https://ntrs.nasa.gov/search.jsp?R=19650019808 2020-03-17T02:00:52+00:00Z
. I  
.. * 
-2 - 
Tke allowed operat ions on t h e  states a r e  the  usual  ones i n  
quantum mechanics: superposit ion,  o r  formation of l i n e a r  combinations 
of states wi th  complex coe f f i c i en t s ,  and composition, or formation of 
t enso r  product states. Every “physical”  state i s  t o  be obtained by 
a repeated app l i ca t ion  of t hese  two opera t ions  t o  our bas ic  states. 
The new s t a t e s  so  obtained, i f  they are chosen t o  be the  e igens t a t e s  
of momentum, may again  be regarded as the  bas i c  states if one sup- 
p re s ses  t h e i r  newly generated quantum numbers. 
By passing t o  the  enveioping a lgebra  of y, we m y  introduce 
t h e  angular  momentum opera tors  M = x P /I = (X P - x P )/I = L ~ ~ .  
PV CP V I  P V  V P  
One f i n d s  the usual  commutation r e l a t i o n s  between the  operators  
X and M namely, IMP,, P ] = i P  M ]=iM g pd P J  PV’ P IKCO 9 3 V l ”  
e t c .  Thus M i f  adjoined t o  y, y i e l d s  a new Lie a lgebra  which i s  
PV’ 
t he  Lie  a lgeb ra  of what we c a l l  t h e  augmented Poincar; group P. 4) 
The t o t a l  angular  momentum M i s  equal t o  t h e  o r b i t a l  p a r t  L; t h i s  i s  
t r u e  only f o r  t h i s  simple case of t he  bas i c  representa t ions  of T. 
To i l l u s t r a t e  t he  s t a t e -bu i ld ing  process  t o  the  lowest order  e+ 
of complexity, l e t  us consider the  t enso r  product s t a t e s  1 ulpl) 631 ci2s2> 
diagonal iz ing  the  operators  I(i) and P (i), i = 1, 2, of t h e  Lie a lgeb ra  
y( 1) 633 T(2). Define t h e  ex te rna l  Lie a lgebra  
by e x h i b i t i n g  i t s  b a s i s  elements: I = I(1) + I(2), P = P (1) + 
P ( 2 )  , x = x (1) + ~ ~ ( 2 ) .  
p a r t i c l e  system” i s  by d e f i n i t i o n  j u s t  t he  sutn of the  ind iv idua l  anguls r  
P 
(isomorphic t o  T )  e x t  
P P 
a ~ i x  t o t a l  angular  monentm of t h i s  “two- 
P P P 
moment a : 
. \  
L .  
. 
where 
L = x  P / I ,  
PV rcl VI 
- -  
and x form the  i n t e r n a l  
pP) F 
and s i m i l a r l y  f o r  . 5 ,  The operators  I, 
P 
L i e  a lgebra  Tint isomorphic and orthogonal t o  Text, i . e . ,  [A, B ]  = 0 
f o r  all A i n  Text, B i n  Tint. We may now d iagonal ize  I, P , I, and 
P and s o  obta in  the  states l a  p; u p > charac te r ized  by exterr-a1 and 
i n t e r n a l  quantum numbers ( r e l a t i v e  t o  t h e  group T).  
- 
P - -  - 
,U 
Equation (l) shows 
tha t  w e  no longer have M = L The sp in  p a r t  S of t h e  t o t a l  
PV PV' PV 
angular  momentum arises as a consequence of t h e  ex is tence  of t ke  
i n t e r n a l  opera tors  and x . The ( i n t r i n s i c )  s p i n  t enso r  opera tor  
S 
P P 
i s  r e l a t e d  t o  the  usual p a r t i c l e  sp in  through the  equat ions W2 = 
U V  
SVPPa.6) S obviously commutes w i t h  I, 1 = - E in2 s ( s  + 1) and W 
P 2 PvPa WV 
and X and hence w i t h  L as it should. Ins tead  of t he  states 
- P I-I v' - -  pP , 
I cr p; 0 p >, one could, e.g. ,  t ake  \ a  p s s 3 j  CI f K >, diagonal iz ing  
I, PPI E2, S3, I, $ - S2, and a c e r t a i n  combination of the  opera tors  
? and z2. 
-.J rv 
The var ious  s t a t e s  so obtained are r e l a t e d  t o  each o ther  
by t h e  Clebsch-Gordan c o e f f i c i e n t s  appropr ia te  t o  T or P, In  any 
case,  it should now be c l e a r  how "complex" states may be generated 
from the  bas i c  ones. If one reduces an  n-fold t enso r  product of t h e  
- -  - -  
bas i c  s-cates, one obta ins  the  s t a t e s  1 u p; u lp l*- -  0 p > corres -  
ponding t o  one e x t e r n a l  and n-1 i n t e r n a l  Lie a lgebras ,  a l l  mutually 
n-1 n-1 
orthogonal and isomorphic t o  y. 7 )  
-4- 
We proceed now t o  show how t o  obta in  i n t e r n a l  symnetry g , r o q s  
of p a r t i c l e s  w i t h  t h e i r  t r a d i t i o n a l l y  %on-geometric” quantun n.mbers 
such as isospin,  hypercharge, e tc .  Consider the above reducea n-fold 
t enso r  product states. The t o t a l  angular  momentum M = En M (k)  
P V  k=l  Pt 
may again  be decomposed i n t o  i t s  o r b i t a l  and sp in  pa r t s .  The sp in  
tensor  S now rece ives  cont r ibu t ions  from the  n-1 i n t e r n a l  Lie 
PV 
algebras;  it i s  unique although its decomposition S =xn x (k)P (k)/y(k) PV k = l  CP V I  
depends on the choice of i n t e r n a l  Lie a lgebras .  If V i s  any of the 
CI 
i n t e r n a l  four-vector .operators,  then it obviously commutes w i t h  I, P , c: 
v ] = ( s  v ] = ~ v  g The oniy 
PV, P 11.1 VIP. 
and X j u t  not  w i t h  M 
P 
combinations of i n t e r n a l  operators  commuting w i t h  the generators  of t he  
e x t e r n a l  L i e  a lgebra  of the augaented PoincarL group (and i n  p a r t i c u l a r  
w i t h  the elements of the  Poincar4 group i t s e l f )  are the 1’s and the  
Lorentz s c a l a r s  or  inva r i an t s  formed from the 2(n-1) i n t e r n a i  fou r -  
vec tors .  To cons t ruc t  a l l  the inva r i an t s  it i s  usefu l  t o  introduce t h e  
c omb i n a t  i ons 
+ 1 
AL(k) = [.lo (k)  2 i x (k) /Jo]  [ 2y(k)]-z , k = 1, 2,. . ., n - 1, 
CI P 
where ,l ( t h e  fundarenta l  length!) has the dimensions of length or 
0 
inverse  mass and serves  t o  make the A’s dimensionless. Define 
i + 
3 P 
5 .  = A (i) A-’(j) = ( & ) *  , 
si’ = A i ( i )  A+’”(j) = 5 j i  = ( E ~ ~ ) *  . 
By convention the upper and lower ind ices  r e spec t ive ly  correspond t o  
the A ‘ s  and A-’s ;  thus ,  e.g. ,  
-k 
= A;(i)  A-’(j). We f i n d  the  ‘ij 
following commutation r e l a t ions :  
c 
-5 - 
The first equat ion i n  conjunction wi th  (ii)* = 6' shows t h e t  tze f' &re  
J i J 
t h e  genera tors  of the  uni ta ry  group U U1 x Sun. The r e m i s i n g  equa- n .  
t i o n s  r e v e a l  t h a t  U 
which i s  e a s i l y  i d e n t i f i e d  as the symplectic group C '). 
i n t e r n a l  'symmetry group exis ts  for t h i s  case of n-fold t enso r  product 
i s  a subgroup of a l a r g e r  i n t e r n a l  symmetry group n 
No higher  n 
s t a t e s  because the  only o the r  poss ib le  i n t e r n a l  i nva r i an t s  a r e  of t h e  
+ + 
form E "'' At(i) Av(j) , A i ( , )  At(Q), and they f a i l  t o  form a f i n i t e -  
dimensional Lie a lgebra.  For f ixed  n, one may l a b e l  t h e  s t a t e s  by The 
quantum numbers a s soc ia t ed  wi th  U 1  and the  ,sequence of compact groups 
.'I Which subgroup SU, of SU should be i d e n t i f i e d  n su, c su3 c - c sun 
wi th  the  phys ica l  i sosp in  group can only be decided once a d e f i n i t e  
choice of i n t e r n a l  L i e  a lgebras  i s  made and dynamical quest ions i n v e s t i -  
gated.  The f a c t  t h a t  t he  SU a re  subgroups of C means t h a t  symmetry 
schemes based on any of them cannot be exac t  but  are necessa r i ly  braker, 
i n  a d e f i n i t e  manner. 
k n 
Further  development of o u r  theory,  including i t s  d e t a i l e d  phys ica l  
i n t e r p r e t a t i o n  and predic t ions ;  dynamical ca l cu la t ions ;  r e l a t i o n s  bs- 
tween i n t e r n a l  and 3x te rna l  quantum numbers, e t c . ,  w i l l  be presented 
i n  a comprehensive paper cow being prepared. 
I wish t o  acknowledge the encouragement and t h e  s t i m d a t i o n  
rece ived  from the  members of the t h e o r e t i c a l ' p h y s i c s  group of t h i s  
Laboratory, e s p e c i a l l y  from Drs. P. B u r t  and M. M. Saffren.  
'. I 
I .  
r 
0 I4 a 
0 0  1 I ? 
-6 - 
FOOmOTES 
The motivation f o r  introducing T and t h e  d e t a i l e d  d iscuss ion  of 
it w i l l  be given elsewhere (J. S. Zmuidzinas, t o  be publ ished) .  
We merely quote t h e  following 6 x 6 matrix r e a l i z a t i o n  of t h i s  
group: T i s  t h e  set of a l l  matr ices  of t he  form 
The phys ica l  i n t e r p r e t a t i o n  of t hese  operators  i s  the  fol lowing,  
The P are the  usual momentum opera tors  o r  generators  of space- 
time t r a n s l a t i o n s .  The pos i t i on  opera tors  X generate  t r a n s -  
l a t i o n s  i n  the  momentum space; they  are i n  a sense the genera tors  
CL 
CL 
of dynamics, as one can e a s i l y  convince himself i f  he th inks  of 
t h e i r  e f f e c t  i n  mixing masses. The i d e n t i t y  opera tor  I i s  the  
gecera tor  of phases of quantum mechanical states [cf .  t he  case 
of t he  G a l i l e i  group: Jean-Marc kvy-Leblond, Jour.  Math. 
Phys. - 4, 776 (1963)l. 
This choice i s  obviously ne i the r  unique nor necessary;. however, 
it i s  convenient for physical  i n t e r p r e t a t i o n .  Another convenient 
choice i s  that of the  f o u r  X . In  f a c t ,  t he  whole theory  i s  
inva r i an t  under P - X , I - -I. 
The group of i nva r i an t  autoaorphisms of t he  Lie a lgeb ra  ( w i t h  t h e  
CL 
P P 
i d e n t i t y  element I) of the  r e s t r i c t e d  Poinear; group P: i s  ve ry  
. '  
b -7 - 
c 
c l c s e l y  r e l a t e d  t o  the  augmented Poincard group P whose group 
l a w  i s  
(a ' ,  v', a ' ,  i')(a, v, a, i) 
= (a' + + v'. a '  a, v '  + i' v, a' + 4 '  a, 1 ' ~ ) ;  
We bel ieve  t h a t  t h e  aug- T and P: are of course subgroups of P. 
mented Poincarg group i s  of fundamental s ign i f icance  i n  elementary 
p a r t i c l e  physics.  
5 )  
6) 
A l l  barred opera tors  w i l l  henceforth be i d e n t i f i e d  as i n t e r n a l .  
The i n t r i n s i c  sp in  operators  S = (S, T )  form a Lie a lgeb ra  whLch 
pv - -4 
i s  t h a t  of t he  (proper homogeneous) Lorentz group; i t s  i r r e d u c i b l e  
un i t a ry  r ep resen ta t ions  are charac te r ized  by the  values  of i t s  two 
inva r i an t s  
wherev t akes  a continuous range of values  and k = 0, 1/2, 1, 3/2, ... 
The eigenvalues of S2 = s (  s + l), s = k, k + 1, . . . , r ep resen t  t h e  
S SPv = f = 1 + v2 - k2 and 1 SPVSPd = g = &v, - 2 -  pv F E p v p r  
N 
sp in  of a p a r t i c l e  def ined i n  i t s  rest frame. The occurrecce of 
h a l f - i n t e g r a l  sp ins  should not be mysterious i f  one remembers t h a t  
S i s  def ined i n  terms of i n t e r n a l  va r i ab le s  and t h a t  t h e r e  i s  
no reason t o  i n s i s t  on the  single-valuedness of any func t ion  of 
PV 
t hese  unobservable var iab les .  
7) The reduct ion  process i s  c l e a r l y  not unique: t he re  a r e  nany poss ib l e  
sets of n-1 mutually orthogonal i n t e r n a l  Lie a lgebras .  The i n t e r e s t -  
ing  p o s s i b i l i t y  of choosing h i e ra rch ie s  of i n t e r n a l  Lie a lgebras  
wi th  progress ive ly  diminishing commutators [ p, x] w i l l  be discussed 
elsewhere. The ex is tence  of such h i e ra rch ie s  a l lows us t o  approxi- 
m t e  states by neglec t ing  higher order  i n t e r n a l  quantum numbers 
and thus  a p o s t e r i o r i  j u s t i f i e s  o m  i n i t i a l  cons idera t ion  of t he  - 
bas ic  s t a t e s  1 0  p > . 
-a - 
8)  It i s  impor’iant t o  note t h a t  t h e  group generated by the  j ’ s ,  
although isomorphic t o  t h e  complex group C i s  not  compact. 
This i s  most e a s i l y  seen by examining C1 which is contained i n  
every C * n’ 
i(i1’ - ~ ~ ; ) / 4 ,  and K3 = E1/2 generate t h e  Lie a lgebra  of the 
non-compact 3-dimensional Lorentz group. 
A d i scuss ion  of such hierarchy of un i t a ry  groups has r e c e n t l y  
been given by Nevi l le  [D.  E. Nevi l le ,  phys. Rev. Let ters13 
118 (1964)j. 
Phys. Rev. 135 
Phys. Rev. Letters13, - 173 (1964). 
n’ 
the  hermit ian operators  Kl = ( E l 1  + t11)/4, = 
1 
9) 
-’ 
See a l s o  C. R. Hagen and A. J. Macfarlane, 
B434 (1964); and F. Gksey  and L. A. Radlcat i ,  -’ 


Geometrical Origin of Internal Symmetries

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