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Supersymmetry and local gauge symmetry

L. Castell Max-Planck-Institut zur Erforschung der Lebensbedingungen der wissenschaftlich-technischen Welt,

0-813 Starnberg, Riemerschmidstr. 7, Germany (Received 17 June 1975)

We show that the representation of the Lie algebra of the 15-dimensional local relativistic gauge group by contracted para-Bose operators contains the supersymmetry algebra as a subalgebra.

The supersymmetry algebra proposed by Wess and Zumino' can be interpreted in the framework of a larger group than the Poincar6 group. In a ser ies of papers Roman and co-workers have explained how local gauge symmetry leads in the relativistic case to the 15-dimensional relati- vistic Galilei group and its central extension.' The Lie algebra of this group can be obtained as a contraction from the Lie algebra of the con- formal group.3 In Ref. 3 all contractions of the Lie algebra I of S00(4, 2 ) which leave the Poincar6 subalgebra invariant have been given. The con- traction from I to I1 is given by replacing the gen- erators of the translations P , and of the special conformal transformations K, by RP, and RK, , respectively. In the limit R -m we obtain [P,, K,] = 0, instead of LP,, K , 1 = 2i k,,D -M,,). (We need not consider any central extension of 11.) All the other commutation relations between the 15 gen- era tors a r e unaltered.

Now we make a basis transformation in the Lie algebra 11:

-goo =g,, =g,, =g,, = 1, g,, = 0 for pfv.

The Casimir operators of this algebra are P2, E , , ~ ~ ~ ~ M , ~ PpKor and p2KZ - (PaK)'. (Obviously, the final algebra IV could have been obtained by a contraction I-111-IV, which corresponds to the contraction de Sitter group - PoincarB group - Galilei group, o r of course directly I- IV.)

It has been pointed out4 that one can construct a ser ies of unitary representations of U(2, 2 j by means of para-Bose operators. The 16 generators of U(2, 2) a r e defined by $(T,, +T:, ), (l/22)(TpU - 1,

The commutation relations of the para-Bose operators a r e given by

In addition one postulates

We replace PC, D, and KA by cP,, cD, and K,, La,aV+a,a,,apl = O , [aLaf, +aLa:,a;T]=o. and finally obtain in the limit c - .o the commutation relations The Hilbert space is defined by

[ P , , K , I = o , C K , , K , I = O , LD, P , I - 0 , L D , K , I =IP,, Q, 10) =a2 10) =aJ10) =a;[o) = o ,

besides

[ M ~ ~ , M ~ ~ I = ~ kppMUo-gpoMup - g v p ~ ~ o + g v a ~ ~ p ) , aaatg/O) =pba8/0), ~ L @ B $ I O ) ' P ~ a 1 6 ' I O ~ ,

[D,M, , I = O , IM,~, ppI = i k w P V -gupP , ) , a,a,, 10) =a: la: 10) = 0,

1Jf,,,,KP~ = i (g,,K, -gUpK,), LP,, P,] = 0, for

12 - 3 994

S U P E R S Y M M E T R Y AND L O C A L GAUGE S Y M M E T R Y 3995

y 4 = 111 : 1 J 0 0 - 1 0

0 0 0 - 1

where a, /3 = 1 ,2 and a', 0' = 3,4. The order of the para-Bose statistics5 is given by p = 1, 2, 3, ... . For p = 1 one can derive Bose statistics [a,, a:] = ( Y ~ ) ~ " , lap, aol = l a \ , a:] = 0. The representations of SU(2,2) with Bose operators lead to mass-zero particles with a certain helicityS6 The proper para-Bose representations4 a r e representations of SU(2,2) with a continuous mass spectrum O<m2 <a.

It is not difficult to s ee that the following r e - lations hold,4 since they follow from (1) and (2):

where p , v=O,1,2,3, iip=af,(y4),p, and ~ = ~ ( a , a , + i i , a , ) is the generator of the center of

U(2, 2). The f i r s t contraction (replacing a p and iip by G a p and 6 2,) leads to

The second contraction (replacing up and Zp by 6 a, and 6 a,) leads to the supersymmetry r e - lation''')

In addition we have

So we have obtained the following result: The representation of the Lie algebra of the relativistic local gauge group by contracted para-Bose op- era tors restricted to the 10-dimensional Poincar6 subalgebra i s isomorphic to the supersymmetry algebra. Moreover, we have derived an extension of the supersymmetry algebra to the full local relativistic gauge symmetry.

'J. Wess and B. Zumino, Nucl. Phys. z, 39 (1974). 2 ~ . Roman and J . P. Leveille, J. Math. Phys. lJ, 2053

(1974); P . Roman, in Quantum Theory and the S t m c - tures of Time and Space, edited by L. Castell, M. Drieschner, and C. F . von Weizszcker (Hanser, Iffinchen, 1975); J. J. Aghassi, P. Roman, and R. M. Santilli, Phys. Rev. D 1, 2753 (1970).

3 ~ . Castell, Nuovo Cimento %, 285 (1967); P . L. Huddleston, M. Lorente, and P . Roman, Found.

Phys. 5, 75 (1975). 4 ~ . Castell, Nuovo Cimento g, 445 (1975). 50. W. Greenberg and A . M. L. Messiah, Phys. Rev. 138,

B1155 (1965). 6 ~ . Castell, Nucl. Phys. E, 231 (1969). 'B. Zumino, in Proceedings of the XVII International Con-

ference on High Energy Physics, London, 1974, edited by J. R. Smith Butherford Laboratory, Chilton, Didcot, Berkshire, England, 1974), p. 1-254.