Feynman Path Integral Formulation

1.8 Supersymmetry 33
treated perturbatively. Ultimately the resolution of such delicate and complex issues
would presumably require the development of the perturbative expansion not around
flat 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, such
as what happens to the spin-zero ghost mass, whether the ghost poles gets shifted
away from the real axis by quantum effects, and what the true ground state of the
theory looks like in the long distance, strong fluctuation regime not accessible by
perturbation 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 point
is clearly present in the 2 + ε expansion (to be discussed later) at the one- and twoloop
order, as well as in the lattice regularized theory in four dimensions (also to be
discussed later). But this could just reflect a limitation of the one-loop calculation;
to properly estimate the uncertainties of the perturbative results in higher derivative
gravity 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 perturbatively
renormalizable, but exhibit some short-distance oddities in the tree-level
spectrum, 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 couplings
away 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 Supersymmetry
An alternative approach to the vexing problem of ultraviolet divergences in perturbative
quantum gravity (and for that matter, in any field theory) is to build in
some additional degree of symmetry between bosons and fermions, such that loop
effects acquire reduced divergence properties, or even become finite. One such proposal,
based on the invariance under local supersymmetry transformation, adds to
the Einstein gravity Lagrangian a spin-3/2 gravitino field, whose purpose is to exactly
cancel the loop divergences in the ordinary gravitational contribution. This last
result comes from the well known fact that fermion loops in quantum field theory
carry an additional factor of minus one, thus potentially reducing, or even canceling
out entirely, a whole class of divergent diagrams. The issue then is to specify the
nature of such a supersymmetry transformation, and from it deduce an extension of
pure gravity which includes such a symmetry in an exact way. Since ordinary gravity
has a local gauge invariance under the diffeomorphism group, one would expect
its supersymmetric extension to have some sort of local supersymmetry.
The first step towards defining a theory of supergravity is therefore to introduce
the concept of global supersymmetry. Quantum field theory in flat space is invariant