Feynman Path Integral Formulation

198 6 Lattice Regularized Quantum Gravity
6.11 Lattice Higher Derivative Terms
So far only the gravitational Einstein-Hilbert contribution to the lattice action and
the cosmological constant term have been considered. There are several motivations
for extending the discussion to lattice higher derivative terms, which would include
the fact that these terms a) might appear in the original microscopic action, or might
have to be included to cure the classical unboundedness problem of the Euclidean
Einstein-Hilbert action, b) that they are in any case generated by radiative corrections,
and c) that on a more formal level they may shed new light on the relationship
between the lattice and continuum expressions for curvature terms as well as quantities
such as the Riemann tensor on a hinge, Eq. (6.36).
For these reasons we will discuss here a generalization of the Regge gravity
equivalent of the Einstein action to curvature squared terms. When considering
contributions quadratic in the curvature there are overall six possibilities, listed in
Eq. (1.127). Among the two topological invariants, the Euler characteristic χ for
a simplicial decomposition may be obtained from a particular case of the general
formula for the analogue of the Lipschitz-Killing curvatures of smooth Riemannian
manifolds for piecewise flat spaces. The formula of (Cheeger, Müller and Schrader,
1984) reduces in four dimensions to
[
χ = ∑ 1 − ∑
σ 0
]
(0,2) − ∑ (0,4)+ ∑ (0,2)(2,4) , (6.106)
σ 2 ⊃σ 0 σ 4 ⊃σ 0 σ 4 ⊃σ 2 ⊃σ 0
where σ i denotes an i-dimensional simplex and (i, j) denotes the (internal) dihedral
angle at an i-dimensional face of a j-dimensional simplex. Thus, for example, (0,2)
is the angle at the vertex of a triangle and (2,4) is the dihedral angle at a triangle
in a 4-simplex (The normalization of the angles is such that the volume of a sphere
in any dimension is one; thus planar angles are divided by 2π, 3-dimensional solid
angles by 4π and so on).
Of course, as noted before, there is a much simpler formula for the Euler characteristic
of a simplicial complex
χ =
d
∑
i=0
(−1) i N i , (6.107)
where N i is the number of simplices of dimension i. However, it may turn out to be
useful in quantum gravity calculations to have a formula for χ in terms of the angles,
and hence of the edge lengths, of the simplicial decomposition. These expression
are interesting and useful, but do not shed much light on how the other curvature
squared terms should be constructed.
In a piecewise linear space curvature is detected by going around elementary
loops which are dual to a (d − 2)-dimensional subspace. The area of the loop itself
is not well defined, since any loop inside the d-dimensional simplices bordering the
hinge will give the same result for the deficit angle. On the other hand the hinge has
a content (the length of the edge in d = 3 and the area of the triangle in d = 4), and