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

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236 P. van Nieuwenhuizen, Supergravitvdiscovered by a direct calculation and later explained by various equivalent theoretical proofs, it wasfound that pure simple supergravity as well as the pure-extended supergravities (irreducible supergravitieswith one graviton, N gravitinos and specified numbers of lower spin fields) are one-loop finite.They are even two-loop finite. This is a better result than obtained so far for pure ordinary Einsteingravity which is known to be one-loop finite (but which might be two-loop finite; this constitutes one ofthe major unsolved problems in quantum gravity). Thus supergravity has taken the lead in the loop raceand the reason is that the extra Fermi—Bose symmetry allows one to prove extra cancellations. Even ifultimately pure Einstein gravity will turn out to be two-loop finite, supergravity has the advantage thatthe pure extended supergravities contain not only gravitons but also gravitinos, vectors, spinors andscalars. Thus one has the possibility of a finite quantum theory describing gravity as well as the otherinteractions which might be phenomenologically correct.About the need to quantize gravity, and now also supergravity, various opinions exist. Some peoplefeel that one should not quantize at all. Others believe that Einstein’s theory of gravity is aphenomenological theory such as thermodynamics and that one should try to quantize the underlyingmicroscopical more fundamental theory. In this connection there have been endeavours to consider thegraviton as a bound state of two photons or of two gravitinos [310,311]. Another point of view is thatgravity is something like van der Waals forces, present where matter is but not existing as free radiationmodes. The recent results from a binary pulsar, which seems to lose energy through gravitationalradiation, seem to refute this proposal. Yet other ideas are to replace the points in spacetime by twistorsand to try to quantize those. A point of ordinary spacetime is described by the intersection of twotwistors, and hopefully the quantum fluctuations give rise to “fluttering” lightcones rather than to fuzzypoints in spacetime. Even further goes the proposal to use path-integral quantization in the space of allgeometries. In this connection by far the most appealing has been Hawking’s spacetime foam: verysmall strongly curved regions of spacetime which average out to flat spacetime over larger distances.Interesting as these suggestions are in their own right, it would be preferable if the same ideas ofquantum field theory which have worked so well in quantum electrodynamics and, later, in unifiedtheories of the weak and electromagnetic interactions, would also turn out to describe gravity with thesame success. Of course there are two major differences between gravity and other interactions. First ofall, the metric determines which points are spacelike and which are timelike and it is not at all clear thatone ends up with the same light cones due to the quantized metric as those due to the classical metricone starts with. A second problem is that only in special classical spacetimes (those with a timelikeKilling vector field) can one define the absorption and creation operators of second quantization. Thesetwo, and other related problems are serious, but our strategy is to just go ahead and see how far onegets. As it turns out, and we will see, elegant results are obtained, and sitting at one’s desk and seeingthe cancellations occur, one cannot resist the feeling that something important is going on. Maybe whatfollows is not the ultimate way of interpreting quantum gravity but it may be the solution in disguise.2.2. General covariant quantization of gauge theoriesWe begin by reviewing the covariant quantization of gauge theories. Suppose one has given aclassical action 1(d) which depends on a set of gauge fields ~‘ and which is invariant under local gaugetransformations with parameters ~= R’,~(4)r. (1)
P. van Nieuwenhuizen. Supergravitv 237We use DeWitt’s summation convention in which i and a denote indices as well as spacetime points andrefer to refs. [514,303, 304]. The symbol R may contain derivatives and depend on gauge fields, forexample it could represent the Yang—Mills law 8W,~= (~ô~+gfa~~W,~)flc. The parameters ~ dependon x~’and represent the parameters of general coordinate, local Lorentz and local supersymmetry.One adds a gauge fixing term for each local gauge invariance. We will consider this I(fix) to bequadratic in the gauge choices Fa(~)I(fix) ~F~(~)y~F 0(4). (2)3is taken to beThis independent is not necessary, of the quantum but usefulgauge in order fields to be able to define propagators. The matrix y’~’~. It may depend on external fields, namely when oneconsiders a quantized gravitino in a background gravitational field, but the correct theory for the casethat y’~depends on çb~is not yet known. The gauge fixing term is needed to remove the degeneracy inthe path integral. For practical purposes, its use is that the kinetic terms of the sum I(cl)+I(fix) arenonsingular, so that one can invert them to obtain the propagators of the theory.By varying the gauge choices one obtains the ghost action I(ghost):I(ghost) = ~ (3)Thus one varies the gauge function, replaces the gauge parameter in eq. (1) by a ghost field andmultiplies from the left with another ghost field which is denoted by C*~.In a path-integral formalism,this result is obtained for bosonic symmetries by exponentiating the normalization determinant, seesubsection 4, in front of the gauge-fixing delta function det (Fa,jR’~3)~(F~— b~)as followsdet ~= J [I dC~dCI exp(~~Integration over anticommuting variables being defined by f dC’~=0, f dC’~C~= ô~,one then finds,upon expanding the exponential, that only the term with four C1 and four C2 fields is nonzero and yieldsindeed the determinant of (FajR’~). The normalization factor i/2 cancels in computations.If one has both bosonic and fermionic symmetries, one defines a superdeterminant (see appendix) andstill arrives at eq. (3). Since, as we shall see, ghost fields C~have3opposite in (3) tostatistics the far right. from the parametersThe oneeffective must specify quantum whereaction to put is the thusC~.We given by will always put C’I(qu) = 1(d) +~Fay~’3F~+ C*~~Fc~jR~C’3. (4)It should be observed that the sum over a and f3 is over the whole gauge group. Thus, for example, insimple supergravity Lorentz ghosts interact with supersymmetry antighosts, etc.The quantum equivalent of the classical gauge invariance given by eq. (1) is the global nonlinearBecchi—Rouet—Stora—Tyutin transformations.t This quantum symmetry uses a constant anticommutingt See Proc. 1975 Erice Conf., eds Velo and wightman, pages 331, 332.
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