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

P. van Nieuwenhuizen, Supergravity 235to the i/c field; for spin 5/2 on has2’(mass) = —M(i/i,.4,.,, — ~. ‘y’y~i/iS — — —4~x). (5)Hence here there is already a so-called van Dam—Veltman mass-discontinuity in the action.For spin 3, a massless and massive theory exist, too. In this theory the limit m —*0 cannot even betaken in the sandwiched propagator. (For spin 2 there is a finite discontinuity.)Our conclusion is that it seems, at this moment, that Nature stops at spin 2.2. Quantum supergravity2.1. IntroductionSupergravity can be covariantly quantized by the same methods as any other gauge theory, providedthat the gauge algebra closes. This closure is obtained by adding auxiliary fields to the classical actionand these fields remain auxiliary fields in the effective quantum action, but become propagating in thecounter terms. The covariant quantization methods can be justified by a Hamiltonian path-integralapproach as we shall see, and, once quantized, unitarity and gauge invariance can be proven. Thesimplest way to do so is to use the Becchi—Rouet—Stora—Tyutin symmetry (BRST-symmetry) of thequantum action which is the quantum extension of the classical gauge invariance. We will also considerthe BRST formalism for gauge theories whose gauge algebra does not close, for example simplesupergravity without S. F, Am.Gravitational theories have a dimensional coupling constant, Newton’s constant, and this fact alone issufficient to rule out ordinary renormalizability for the following reasons. Due to the dimensionalcharacter of the gravitational coupling constant, one-loop (and higher loop) divergences have a differentfunctional form than the quantum action so that one cannot absorb these divergences back into theoriginal quantum action by rescaling of the physical parameters. For example, in Einstein gravity theone-loop divergences are on-shell proportional to2 and the two-loop divergences areproportional to (R,.,,aøR~”Rp,.t~) etc., whereas 5di~’= the original erR ~,,,+ /3R action is proportional to ,(2(g112R).Clearly, one cannot reabsorb the two-loop divergences back into 2’ by rescaling of g,.,, and K alone.(Expanding5di~’ into powers of h,.,, where Kh,.,, = g,.,, — ~, and subsequently replacing each field h,.,, byan infinite set of tree graphs with physical momenta and physical polarization tensors at the end of thetrees, one finds the divergences of the n-point Green’s functions on-shell.) One also finds that anyn-point Green’s function is divergent from the two-loop level on. At the one-loop level, i~2=0 sinceR,.,, = 0, where R,.,, = 0 because the fields g,.,, satisfy R,.,,(g,.,,) = 0 since they represent infinite sets oftree graphs with physical end-momenta and polarization tensors. For more details, we refer to ref. [6061,where the relation between normal field theory and the background field method is given.Thus gravitational theories are never renormalizable and only the two extremal cases are possible:finiteness of the S-matrix or the infinities in the S-matrix cannot be removed. The latter case isusually called “non-renormalizability of the S-matrix” but we stress that it is not the counterpart ofrenormalizability but of finiteness. Finiteness of the S-matrix is of course a much stronger property thanrenormalizability and one can expect it to occur only when “miraculous” cancellations of divergencesoccur due to a symmetry of the theory. It is here that supergravity has had impressive successes. First