Feynman Path Integral Formulation

6.5 Invariant Lattice Action 181
The structure of the gravitational action of Eq. (6.43) leads naturally to some
rather general observations, which we will pursue here. The first, cosmological constant,
term represents the total four-volume of space-time. As such, it does not contain
any derivatives (or finite differences) of the metric and is completely local; it
does not contribute to the propagation of gravitational degrees of freedom and is
more akin to a mass term (as is already clear from the weak field expansion of
∫ √ g in the continuum). In an ensemble in which the total four-volume is fixed in
the thermodynamic limit (number of simplices tending to infinity) one might in fact
take the lattice coupling λ 0 = 1, since different values of λ 0 just correspond to a trivial
rescaling of the overall four-volume (of course in a traditional renormalization
group approach to field theory, the overall four-volume is always kept fixed while
the scale or q 2 dependence of the action and couplings are investigated). Alternatively,
one might even want to choose directly an ensemble for which the probability
distribution in the total four-volume V is
P(V) ∝ δ(V −V 0 ) , (6.45)
in analogy with the microcanonical ensemble of statistical mechanics.
The second, curvature contribution to the action contains, as in the continuum,
the proper kinetic term. This should already be clear from the derivation of the lattice
action given above, and will be made even more explicit in the section dedicated
to the lattice weak field expansion. Such a term now provides the necessary coupling
between neighboring lattice metrics, but the coupling still remains local. Geometrically,
it can be described as a sum of elementary loop contributions, as it contains
as its primary ingredient the deficit angle associated with an elementary parallel
transport loop around the hinge h. When k = 0 one resides in the extreme strong
coupling regime There the fluctuations in the metric are completely unconstrained
by the action, insofar as only the total four-volume of the manifold is kept constant.
At this point it might be useful to examine some specific cases with regards to
the overall dimensionality of the simplicial complex. In two dimensions the Regge
action reduces to a sum over lattice sites p of the 2 − d deficit angle, giving the
discrete analog of the Gauss-Bonnet theorem
∑ δ p = 2πχ , (6.46)
sites p
where χ = 2 − 2g is the Euler characteristic of the surface, and g the genus (the
number of handles). In this case the action is therefore a topological invariant, and
the above lattice expression is therefore completely analogous to the well known
continuum result
∫
1
2
d 2 x √ gR = 2πχ . (6.47)
This remarkable identity ensures that two-dimensional lattice R-gravity is as trivial
as in the continuum, since the variation of the local action density under a small
variation of an edge length l ij is still zero. Of course there is a much simpler formula
for the Euler characteristic of a simplicial complex, namely