On a difficulty in constructing combined invariance theories

Volume 16, number 1

PHYSICS LETTERS

ON A D I F F I C U L T Y

IN C O N z S T R U C T I N G

1May1965

COMBINED

INVARIANCE

THEORIES

R. MIRMAN

Physics Department, Long Island University, Brooklyn, New York Received 22 March 1965

It has been shown by O ' R a i f e a r t a i g h [1] that t h e r e a r e four ways in which the L o r e n t z a l g e b r a can be imbedded in a l a r g e r algebra. Of these one involves the use of a (probably u n i n t e r e s t i n g ) solvable a l g e b r a and two do not allow m a s s s p l i t ting if only r e a l m a s s e s a r e c o n s i d e r e d . (However, s i n c e a l m o s t all p a r t i c l e s decay, and so have c o m p l e x m a s s e s , this l a s t objection is not n e c e s s a r i l y fatal). We wish h e r e to point out that the fourth case, in which the L o r e n t z a l g e b r a is imbedded in a s i m p l e a l g e b r a also leads to v e r y s e r i o u s difficulties. Theorem: If the L o r e n t z a l g e b r a , c o n s i s t i n g of the homogeneous o p e r a t o r s M 0 and the t r a n s l a tion o p e r a t o r s P~, is imbedded in a s e m i s i m p l e a l g e b r a with m u t u a l l y c o m m u t i n g o p e r a t o r s H i and step o p e r a t o r s E~, then at l e a s t one of the g Hi do not c o m m u t e with the P~, and with the m a s s squared operator. Proof: A s s u m e f i r s t that all of the P/~ a r e m e m b e r s of the set of H/; then

M0 : doTEs + dOig i

(1)

and by the well known c o m m u t a t i o n r u l e s for s e m i simple algebras

[% ,Mo] = R~O doyEy

(2)

which does not equal a Ptt as is r e q u i r e d by the c o m m u t a t i o n r e l a t i o n s for the inhomogeneous Lorentz algebra. Now let

% :

+ c.i

(3)

=RN(T)C~ C p v ( E T E a

+

EaET)

+

(5)

+~)~)C~>C~k(E>Hk + HkE~) which does not equal zero. Thus if the L o r e n t z a l g e b r a w e r e imbedded in a s e m i s i m p l e Mgebra t h e r e would be at l e a s t one c o m m u t i n g quantum n u m b e r o p e r a t o r which would not be t r a n s l a t i o n a l l y i n v a r i a n t , and f u r t h e r would label s t a t e s of different m a s s . Thus for c o n c r e t e n e s s only, taking this n o n - c o m m u t ing o p e r a t o r as the z component of isotopic spin the " n e u t r o n " and "proton" would have different m a s s e s , but while one o b s e r v e r would see a p a r t i c l e as a "proton" another displaced o b s e r v e r would see it as a m i x t u r e of a "proton" and " n e u tron". Thus imbedding the inhomogeneous L o r e n t z a l g e b r a in a s e m i s i m p l e a l g e b r a would seem to pose v e r y g r e a t i n t e r p r e t i v e p r o b l e m s , which would be w o r s e than those that r e s u l t from the t h e o r e m [2] that at l e a s t one of the H i cannote c o m m u t e with at l e a s t one L o r e n t z g e n e r a t o r . F o r in that case it was not c l e a r that the states having different values of that Hi would have to have different p h y s i c a l p r o p e r t i e s . Because of the t h e o r e m proved h e r e they must. F u r t h e r the p r e s e n t r e s u l t s u g g e s t s that c o n s t r u c t i n g a theory of this type which did not violate the law of c o n s e r v a t i o n of e n e r g y might be difficult.

So

[Hi' PU ] : Cg~Rie)EY

(4)

which does not equal z e r o for at l e a s t one i, so that at l e a s t one Pg does not c o m m u t e with at l e a s t one H i. C o m m u t i n g this p a r t i c u l a r Hj with the m a s s s q u a r e d o p e r a t o r we get

88

References 1. L.OWRaifeartaigh, Phys.Rev. Letters 14 (1964} 332. 2. F. Coester, M. Hammermesh and W. D. McGlinn, Phys.Rev. 135 (1964) B451; M.E.Mayer et al., Phys. Rev. 136 (1964) B888; A. Beskow and U.Ottoson, NuovoCimento 34 (1964) 248.