SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

P. van Nieuwenhuizen, Supergravity 233spin 3/2 on-shell. Similar results hold for spin 1, 2, 5/2, etc. This is a general result, valid for any massivespin. (Note that (/+ M)I = (I + M)M if El = M 2.)Massless fields: For massless fields there are local gauge invariances and hence some of the spinblock matrices are singular. One finds the propagator which is sandwiched between sources by invertingthe maximal nonsingular submatrices of the spin blocks. Each gauge invariance implies a sourceconstraint and these source constraints are needed to cancel ghosts in the propagator. To prove that thelinearized Rarita—Schwinger action for real massless gravitinos is the only action (up to fieldredefinitions as in eq. (15)) which is free from ghosts, one must consider all possible cases contained ineq. (11): the rank of the 2 X 2 spin 1/2 matrix 0, 1, 2 and the rank of the 1 x 1 spin 3/2 matrix being 0 or1. If one requires that at the pole k2 = 0 the residue is positive definite, then only the Rarita—Schwingeraction is found. We shall not do this straightforward but tedious computation. Instead we show that (2)is without ghosts.From eq. (17) with M = 0 one finds the field equation with external source J,.(F3”2—2P~2)1i/i,.=J,.. (21)Clearly, there is the gauge invariance Si/i,. = (P~ where x~is arbitrary, since replacing i/i,. by Si/i,.the left hand side vanishes. Thus there is the gauge invariance Si/i,. = 8,. . Acting with P~2on thisequation, the left hand side again vanishes, and one finds the source constraint PJT~J= 0 (for j = 1, 2),which is equivalent to= 0. (22)Inverting the spinblocks, the propagator becomes in k-spaceiJ~(—k)C(P3”2—4F~2),.,, ~J,,(k) = 1. ~ (23)For a Majorana source 1(x) = JTC, one has JT(_k)c= ~J(k))ty4which one may define to be 1(k). Toshow that the residue J,.(k)y,.k’y,.J,,(k) is positive definite, we use the spin 1/2 result that X =u~u~+ ui1, and the decomposition of the Kronecker delta function5,.,, = ~±(~±)*+ ~~(~)* + (k,.le,, +1~,,k,,)(k . 1~’) (24)2 = 0. We rewrite the residue as ,..o,..,.y,,k’y,.&,,J,,. The terms in eq. (24) with k,,or valid k,. on-shell do not contribute where k since Ku~= 0 and k J = 0. For the rest one finds, usingcu4=0, ~u=O (25)that the residue is given byJ12 + ü~% J~2 (26)Hence, there are no ghosts in linearized Rarita—Schwinger theory, and two physical modes.