Feynman Path Integral Formulation

4.14 Lattice Hamiltonian for Quantum Gravity 139
It is important to note that the squared edge lengths take on only positive values
li 2 > 0, a fact that would seem to imply that the wavefunction has to vanish when
the edge lengths do, Ψ(l 2 = 0) ≃ 0. In addition one has some rather complicated
constraints on the squared edge lengths, due to the triangle inequalities. These ensure
that the areas of triangles and the volumes of tetrahedra are always positive. As
a result one would expect an average soft local upper bound on the squared edge
lengths of the type li
2 < ∼ l2 0 where l 0 is an average edge length, 〈li 2〉 = l2 0 .Theterm
“soft” refers to the fact that while large values for the edge lengths are possible,
these should nevertheless enter with a relatively small probability, due to the small
phase space available in this region.
These considerations have some consequences already in the strong coupling
limit of the theory. For sufficiently strong coupling (large Newton constant G) the
first term in Eq. (4.175) is dominant, which shows again some similarity with what
one finds for non-abelian gauge theories for large g, Eq. (4.157). It is then easy to
see, both from the constraint l i > 0 and the triangle inequalities, that the spectrum of
this operator is discrete. In particular the mass gap, the spacing between the lowest
eigenvalue and the first excited state, is of the same order as the ultraviolet cutoff.
One can argue that this is in fact a general feature of the strong coupling theory,
where one is far removed from a lattice continuum limit. The latter has to be taken
in the vicinity of a non-trivial ultraviolet fixed point, if such a fixed point can be
found. One would then anticipate that the excitation spectrum would become denser
as one approaches the lattice continuum limit, in accordance with the existence of a
massless spin two particle in this limit.
Note that in the lattice theory the operator ordering ambiguity has not gone away
either: in principle one would have to check that different orderings give the same
physical results, whichever way those are defined (for example in terms of vacuum
expectation values of invariant operators, or quantum correlations of invariant operators
at fixed geodesic distance along the spatial directions).
Irrespective of its specific form, it is in general possible to simplify the lattice
Hamiltonian constraint in Eqs. (4.175) and (4.176) by using scaling arguments, as
one does often in ordinary non-relativistic quantum mechanics. After setting for the
scaled cosmological constant λ = 8πGλ 0 and dividing the equation out by common
factors, it can be recast in the slightly simpler form
{
}
−α a 6 1 · √
g(l 2 ) G ij(l 2 ∂ 2
) − β a 2 · 3R(l 2 )+1 Ψ[l 2 ]=0 , (4.180)
∂l 2 i ∂l2 j
where one finds it useful to define a dimensionless Newton’s constant, as measured
in units of the cutoff Ḡ ≡ 16πG/a 2 , and a dimensionless cosmological constant λ ¯ 0 ≡
λ 0 a 4 , so that in the above equation one has α = Ḡ/¯λ 0 and β = 1/Ḡλ ¯ 0 . Furthermore
the edge lengths have been rescaled so as to be able to set λ 0 = 1 in lattice units (it is
clear from the original gravitational action that the cosmological constant λ 0 simply
multiplies the total spacetime volume, thereby just shifting around the overall scale
for the problem). Schematically Eq. (4.176) is therefore of the form