Feynman Path Integral Formulation

298 8 Numerical Studies
ξ ξ ξ
Fig. 8.11 The lattice quantum continuum limit is gradually approached by considering sequences
of lattices with increasingly larger correlation lengths ξ in lattice units. Such a limit requires the
existence of an ultraviolet fixed point, where quantum field correlations extend over many lattice
spacing.
appears here as a consequence of the relatively small value of the lattice k c in four
dimensions.
The renormalization group invariance of the physical quantity m requires that
the running gravitational coupling G(μ) varies in the vicinity of the fixed point in
accordance with the above equation, with Λ → μ, where μ is now an arbitrary scale,
( ) 1 ν [ ] G(μ) ν
m = μ
− 1 . (8.80)
a 0 G c
The latter is equivalent to the renormalization group invariance requirement
μ d m[μ,G(μ)] = 0 , (8.81)
d μ
provided G(μ) is varied in a specific way. Indeed this type of situation was already
encountered before, for example in Eqs. (3.62) and (3.122). Eq. (8.81) can therefore
be used to obtain, if one so wishes, a β-function for the coupling G(μ) in units of
the ultraviolet cutoff,
μ ∂ G(μ) =β [G(μ)] , (8.82)
∂μ
with β(G) given in the vicinity of the non-trivial fixed point, using Eq. (8.80), by
β(G) ≡ μ ∂
∂μ G(μ)
∼
G→G c
− 1 ν (G − G c)+... (8.83)
The above procedure is in fact in complete analogy to what was done for the nonlinear
σ-model, for example in Eq. (3.64). But the last two steps are not really
necessary, for one can obtain the scale dependence of the gravitational coupling