Feynman Path Integral Formulation

8.2 Observables, Phase Structure and Critical Exponents 275
where, as customary, the lattice ultraviolet cutoff is set equal to one (i.e. all length
scales are measured in units of the lattice cutoff). The lattice measure is given in
Eq. (6.76) and will be therefore of the form
∫ ∫ ∞
[dl 2 ]= ∏
0 s
[V d (s)] σ ∏ dlij 2 Θ[lij] 2 , (8.3)
ij
with σ a real parameter given below.
Ultimately the above lattice partition function Z latt is intended as a regularized
form of the continuum Euclidean Feynman path integral of Eq. (2.34),
∫
Z cont =
with functional measure over the g μν (x)’s of the form
∫
[dg μν ] e −λ ∫ √
0 dx g+ 1 ∫ √
16πG dx gR
, (8.4)
[dg μν ] ≡ ∏ [g(x)] σ/2 ∏ dg μν (x) , (8.5)
x
μ≥ν
where σ is a real parameter constrained by the requirement σ ≥−(d + 1). For
σ = 1 2
(d − 4)(d + 1) one obtains the De Witt measure of Eq. (2.18), while for σ =
−(d + 1) one recovers the original Misner measure of Eq. (2.22). In the following
we will mostly be interested in the four-dimensional case, for which d = 4 and
therefore σ = 0 for the DeWitt measure.
It is possible to add higher derivative terms to the lattice action and investigate
how the results are affected. The original motivation was that they would improve
the convergence properties of functional integral for the lattice theory, but extensive
numerical studies suggest that they don’t seem to be necessary after all. In any
case, with such terms included the lattice action for pure gravity acquires the two
additional terms whose lattice expressions can be found in Eqs. (6.111) and (6.122),
[
]
I latt = ∑ λ 0 V h − k δ h A h − bA 2 h δ h 2 /V h
h
[ δ
[ δ
] ) 2
+ 1 3 (a + 4b) ∑ V s ∑ ε h,h ′
(ω h
s
A C
]h − ω h ′ . (8.6)
A C h ′
h,h ′ ⊂s
The above action is intended as a lattice form for the continuum action
∫
I = dx √ [
g λ 0 − 1 2 kR− 1 4 bR μνρσR μνρσ + 1 2 (a + 4b) C μνρσC μνρσ] , (8.7)
and is therefore of the form in Eqs. (1.137) and (6.123). Because of its relative
complexity, in the following the Weyl term will not be considered any further, and b
will chosen so that b = − 1 4a. Thus the only curvature term to be discussed here will
be a Riemann squared contribution, with a (small) positive coefficient + 1 4 a → 0.