Feynman Path Integral Formulation

140 4 Hamiltonian and Wheeler-DeWitt Equation
{
−ḠΔ q − 1 }
3 R(q)+1 Ψ[q] =0 , (4.181)
Ḡ
with Δ q a discretized form of the covariant Laplacian, acting locally on the function
space of the q = l 2 variables; on near-transverse traceless modes it is expected to
have positive eigenvalues. Furthermore, at this point the similarity with the lattice
Hamiltonian for non-abelian gauge theories in Eq. (4.157) has become evident.
Due to the triangle inequalities, finding a solution of Eq. (4.180) for all lattice
points might not be easy; in principle it could be done numerically. For N lattice q
variables, the solution for Ψ(q) is expected to be in general a linear combination of
N wave functions. But if one is only interested in the lowest p 2 ≈ 0 excitations of the
theory, one could perhaps approximate the Laplacian term by its lowest eigenvalue
∼ (π/L) 2 where L is the linear size of the spatial system (for a spatial volume 3 V
one would use L ≃ ( 3 V ) 1/3 ). Furthermore the local three-curvature operator 3 R(q)
involves an elementary loop on the lattice, with size of the order of the average
lattice spacing l 0 . From dimensional arguments one would expect this term on the
average to contribute 3 R ≃ c 0 /l 2 0 + c 1/L 2 , the first piece representing a subtraction,
and the second one a correction dependent on the boundary conditions in x. Inserting
this expression into Eq. (4.181) one finds
c 0 = l 2 0 Ḡ c 1 = π 2 Ḡ 2 . (4.182)
The first condition amounts to requiring a critical value for Ḡ, reminiscent of the
ultraviolet fixed point condition for G in the 2 + ε expansion. If the theory develops
self-consistently a non-perturbative scale ξ by a mechanism analogous to dimensional
transmutation, then one would replace L → ξ in the above expressions.