Feynman Path Integral Formulation

1.8 Supersymmetry 33treated perturbatively. Ultimately the resolution of such delicate and complex issueswould presumably require the development of the perturbative expansion not aroundflat space, but more appropriately around the de Sitter metric, for which R = 2λ 0 /κ 2 .Even then one would have to confront such genuinely non-perturbative issues, suchas what happens to the spin-zero ghost mass, whether the ghost poles gets shiftedaway from the real axis by quantum effects, and what the true ground state of thetheory looks like in the long distance, strong fluctuation regime not accessible byperturbation theory.What is also a bit surprising is that higher derivative gravity, to one-loop order,does not exhibit a nontrivial ultraviolet point in G, even though such a fixed pointis clearly present in the 2 + ε expansion (to be discussed later) at the one- and twolooporder, as well as in the lattice regularized theory in four dimensions (also to bediscussed later). But this could just reflect a limitation of the one-loop calculation;to properly estimate the uncertainties of the perturbative results in higher derivativegravity and their potential physical implications a two-loop calculation is needed,which hopefully will be performed in the near future.To summarize, higher derivative gravity theories based on R 2 -type terms are perturbativelyrenormalizable, but exhibit some short-distance oddities in the tree-levelspectrum, associated with either ghosts or tachyons. Their perturbative (weak field)treatment suggest that the higher derivative couplings are only relevant at short distances,comparable to the Planck length, but the general evolution of the couplingsaway from a regime where perturbation theory is reliable remains an open question,which perhaps will never be answered satisfactorily in perturbation theory, if non-Abelian gauge theories, which are also asymptotically free, are taken as a guide.1.8 SupersymmetryAn alternative approach to the vexing problem of ultraviolet divergences in perturbativequantum gravity (and for that matter, in any field theory) is to build insome additional degree of symmetry between bosons and fermions, such that loopeffects acquire reduced divergence properties, or even become finite. One such proposal,based on the invariance under local supersymmetry transformation, adds tothe Einstein gravity Lagrangian a spin-3/2 gravitino field, whose purpose is to exactlycancel the loop divergences in the ordinary gravitational contribution. This lastresult comes from the well known fact that fermion loops in quantum field theorycarry an additional factor of minus one, thus potentially reducing, or even cancelingout entirely, a whole class of divergent diagrams. The issue then is to specify thenature of such a supersymmetry transformation, and from it deduce an extension ofpure gravity which includes such a symmetry in an exact way. Since ordinary gravityhas a local gauge invariance under the diffeomorphism group, one would expectits supersymmetric extension to have some sort of local supersymmetry.The first step towards defining a theory of supergravity is therefore to introducethe concept of global supersymmetry. Quantum field theory in flat space is invariant