Feynman Path Integral Formulation

8.10 Curvature Scales 301
for gravity, QED (massive via the Higgs mechanism) and a self-interacting scalar
field, respectively.
A third argument suggesting the identification of the scale ξ with large scale curvature
and therefore with the observed scaled cosmological constant goes as follows.
Observationally the curvature on large scale can be determined by parallel transporting
vectors around very large loops, with typical size much larger than the lattice
cutoff l P . In gravity, curvature is detected by parallel transporting vectors around
closed loops. This requires the calculation of a path dependent product of Lorentz
rotations R, in the Euclidean case elements of SO(4), as discussed in Sect. 6.4. On
the lattice, the above rotation is directly related to the path-ordered (P) exponential
of the integral of the lattice affine connection Γμν λ via
R α β = [
P e
∫
path
between simplices
Γ λ dx λ
] α
β . (8.88)
Now, in the strongly coupled gravity regime (G > G c ) large fluctuations in the gravitational
field at short distances will be reflected in large fluctuations of the R matrices.
Deep in the strong coupling regime it should be possible to describe these
fluctuations by a uniform (Haar) measure. Borrowing from the analogy with Yang-
Mills theories, and in particular non-Abelian lattice gauge theories with compact
groups [see Eq. (3.145)], one would therefore expect an exponential decay of nearplanar
Wilson loops with area A of the type
[ ∫
W(Γ ) ∼ trexp
S(C)
R·· μν A μν
C
]
∼ exp(−A/ξ 2 ) , (8.89)
where A is the minimal physical area spanned by the near-planar loop. A derivation
of this standard result for non-Abelian gauge theories can be found, for example, in
the textbook (Peskin and Schroeder, 1995).
In summary, the Wilson loop in gravity provides potentially a measure for the
magnitude of the large-scale, averaged curvature, operationally determined by the
process of parallel-transporting test vectors around very large loops, and which
therefore, from the above expression, is computed to be of the order R ∼ 1/ξ 2 .
One would expect the power to be universal, but not the amplitude, leaving open
the possibility of having both de Sitter or anti-de Sitter space at large distances
(as discussed previously in Sect. 8.8, the average curvature describing the parallel
transport of vectors around infinitesimal loops is described by a lattice version of
Euclidean anti-de Sitter space). A recent explicit lattice calculation indeed suggests
that the de Sitter case is singled out, at least for sufficiently strong copuling (Hamber
and Williams, 2007). Furthermore one would expect, based on general scaling
arguments, that such a behavior would persists throughout the whole strong coupling
phase G > G c , all the way up to the on-trivial fixed point. From it then follows
the identification of the correlation length ξ with a measure of large scale curvature,
the most natural candidate being the scaled cosmological constant λ phys ,asin
Eq. (8.86). This relationship, taken at face value, implies a very large, cosmological