Feynman Path Integral Formulation

274 8 Numerical Studies
are not affected by uncontrollable errors, such as for example the 2 + ε expansion
of Sect. 3.5. Since the lattice cutoff and the method of dimensional regularization
cut the theory off in the ultraviolet in rather different ways, one needs to compare
universal quantities which are cutoff-independent. One example is the critical exponent
ν, as well as any other non-trivial scaling dimension that might arise. Within
the 2 + ε expansion only one such exponent appears, to all orders in the loop expansion,
as ν −1 = −β ′ (G c ). Therefore one central issue in the lattice regularized
theory is the value of the universal exponent ν.
Knowledge of ν would allow one to be more specific about the running of the
gravitational coupling. One purpose of the discussion in Sect. 3.3 was to convince
the reader that the exponent ν determines the renormalization group running of
G(μ 2 ) in the vicinity of the fixed point, as in Eq. (3.22) for the non-linear σ-model,
and more appropriately in Eq. (3.117) for quantized gravity. From a practical point
of view, on the lattice it is difficult to determine the running of G(μ 2 ) directly from
correlation functions , since the effects from the running of G are generally small.
Instead one would like to make use of the analog of Eqs. (3.29), (3.59) and (3.60)
for the non-linear σ-model, and, again, more appropriately of Eqs. (3.121) and possibly
(3.127) for gravity to determine ν, and from there the running of G. Butthe
correlation length ξ = m −1 is also difficult to compute, since it enters the curvature
correlations at fixed geodesic distance, which are hard to compute for (genuinely
geometric) reasons to be discussed later. Furthermore, these generally decay exponentially
in the distance at strong G, and can therefore be difficult to compute due
to the signal to noise problem of numerical simulations.
Fortunately the exponent ν can be determined instead, and with good accuracy,
from singularities of the derivatives of the path integral Z, whose singular part is
expected, on the basis of very general arguments, to behave in the vicinity of the
fixed point as F ≡− 1 V lnZ ∼ ξ −d where ξ is the gravitational correlation length.
From Eq. (3.121) relating ξ (G) to G − G c and ν one can then determine ν, aswell
as the critical coupling G c .
8.2 Observables, Phase Structure and Critical Exponents
The starting point is once again the lattice regularized path integral with action as
in Eq. (6.43) and measure as in Eq. (6.76). Then the lattice action for pure fourdimensional
Euclidean gravity contains a cosmological constant and Regge scalar
curvature term as in Eq. (6.90)
I latt = λ 0 ∑
h
V h (l 2 ) − k∑δ h (l 2 )A h (l 2 ) , (8.1)
h
with k = 1/(8πG), and leads to the regularized lattice functional integral
∫
Z latt = [dl 2 ] e −λ 0 ∑ h V h +k ∑ h δ h A h
, (8.2)