Feynman Path Integral Formulation

220 6 Lattice Regularized Quantum Gravity
(Hasslacher and Perry, 1981) based on quantum spin variables attached to lattice
links. In these models representations of SU(2) label edges. One natural underlying
framework for such theories is the canonical 3 + 1 approach to quantum gravity,
wherein quantum spin variables are naturally related to SU(2) spin connections.
Extensions to four dimensions have been attempted, and we refer the reader to the
recent review of spin foam models in (Perez, 2003).
6.18 Lattice Invariance versus Continuum Invariance
In simplicial lattice gravity, and for that matter, in any theory of lattice gravity,
one could wonder what the effects might be due to the restrictions on the metric
arising from the generalized triangle inequalities of Eq. (6.74). Effectively these
imply a soft cutoff at large edge lengths, and possibly, for scale invariant measures,
a second cutoff at small edge lengths, in addition to the cosmological constant term
of Eq. (6.43), which exponentially suppresses large volumes. 2
On the lattice one would ideally like to preserve all of the symmetries of the
continuum theory, including some version of diffeomorphism invariance. Since a
truly diffeomorphism invariant cutoff does not seem to exists, there might be difficulties
in implementing such a program. The hope therefore is that the lattice theory
has enough symmetry built into it from the start to fully recover the symmetries of
the original theory in some suitable lattice continuum limit. One example of such a
mechanism is the demonstrated restoration of rotational invariance in many examples
of ordinary lattice field theories in the vicinity of ultraviolet fixed points. What
is meant by having enough symmetry built into the lattice theory is the following:
that the unwanted terms arising from the lattice regularization can in some sense be
considered small because they lead to vanishing contributions in the limit of small
momenta and large distances.
For gauge theories the proof that small violations of gauge invariance do not
affect the long distance properties of the theory, which is therefore still described
by a locally gauge invariant effective action, goes as follows (Foerster, Nielsen and
Ninomiya, 1980; Parisi, 1992). One first assumes that under local gauge transformations
of the gauge fields A μ (x)
(A μ ) Ω = Ω −1 A μ Ω + Ω −1 ∂ μ Ω , (6.205)
the action can be decomposed as a gauge invariant part S 0 (A) , plus a small noninvariant
contribution δS(A)
∫
S(A) =S 0 (A)+δS(A) = dx [L 0 [A(x)] + δL [A(x)]] . (6.206)
2 In the continuum such delicate field cutoff do not arise, since in a perturbative calculation the
integration domain for the metric perturbation h μν is generally extended from minus infinity to
plus infinity, leading in the usual way to straightforward Gaussian integrations, without regards to
field constraints such as detg > 0.