Feynman Path Integral Formulation

264 7 Analytical Lattice Expansion Methods
Fig. 7.12 Cross polytope
β n with n = 8and2n = 16
vertices, whose surface can
be used to define a simplicial
manifold of dimension d =
n − 1 = 7. For general d, the
cross polytope β d+1 will have
2(d + 1) vertices, connected
to each other by 2d(d + 1)
edges.
δ d = 0 + 4 d − ( ε d,d+1 + 3terms + ε 1,d ε 1,d+1 + ... ) + ... (7.144)
and therefore close to flat in the large d limit (due to our choice of an equilateral
starting configuration). Indeed if the choice of triangulation is such that the deficit
angle is not close to zero, then the discrete model leads to an average curvature
whose magnitude is comparable to the lattice spacing or ultraviolet cutoff, which
from a physical point of view does not seem very attractive: one obtains a spacetime
with curvature radius comparable to the Planck length.
When evaluated on such a manifold the lattice action becomes
√
d 2
d/2
2 ( λ 0 − kd 3) [ 1 − 1 d!
8 ∑ εij 2 + 1 ( 1
d 4 ∑ ε ij + 1 ) ]
8 ∑ ε ij ε ik + O(1/d 2 ) .
(7.145)
Dropping the 1/d correction the action is proportional to
(
λ0 − kd 3) ∑εij 2 . (7.146)
− 1 2
Since there are 2d(d + 1) edges in the cross polytope, one finds therefore that, at
the critical point kd 3 = λ 0 , the quadratic form in ε, defined by the above action,
develops 2d(d + 1) ∼ 2d 2 zero eigenvalues.
This result is quite close to the d 2 /2 zero eigenvalues expected in the continuum
for large d, with the factor of four discrepancy presumably attributed to an underlying
intrinsic ambiguity that arises when trying to identify lattice points with points
in the continuum.
It is worth noting here that the competing curvature (k) and cosmological constant
(λ 0 ) terms will have comparable magnitude when
k c = λ 0 l 2 0
d 3 . (7.147)