Feynman Path Integral Formulation

8.8 Numerical Determination of the Scaling Exponents 291
Fig. 8.7 Average curvature R(k) on a 16 4 lattice, raised to the third power. If ν = 1/3, the data
should fall on a straight line. The continuous line represents a linear fit of the form A (k c − k).The
small deviation from linearity of the transformed data is quite striking.
which is obtained by substituting Eq. (8.50) into Eq. (8.53). The correct dimensions
have been restored in this last equation by supplying appropriate powers of the
Planck length l P = G 1/(d−2)
phys
, which involves the ultraviolet cutoff Λ. Forν = 1/3
the result of Eq. (8.62) becomes particularly simple
< ∫ dx √ gR(x) >
< ∫ dx √ g >
∼
G→G c
const.
1
l P ξ . (8.63)
Note that a naive estimate based on dimensional arguments would have suggested
the incorrect result ∼ 1/lP 2 . Instead the above expression actually vanishes at the critical
point. This shows that ν plays the role of an anomalous dimension, determining
the magnitude of deviations from naive dimensional arguments.
Since the critical exponents play such a central role in determining the existence
and nature of the continuum limit, it appears desirable to have an independent way
of estimating them, which either does not depend on any fitting procedure, or at least
analyzes a different and complementary set of data. By systematically studying the
dependence of averages on the physical size of the system, one can independently
estimate the critical exponents.
Reliable estimates for the exponents in a lattice field theory can take advantage
of a comprehensive finite-size analysis, a procedure by which accurate values for
the critical exponents are obtained by taking into account the linear size dependence
of the result computed in a finite volume V .
In practice the renormalization group approach is brought in by considering the
behavior of the system under scale transformations. Later the scale dependence is
applied to the overall volume itself. The usual starting point for the derivation of