Feynman Path Integral Formulation

4.14 Lattice Hamiltonian for Quantum Gravity 139It is important to note that the squared edge lengths take on only positive valuesli 2 > 0, a fact that would seem to imply that the wavefunction has to vanish whenthe edge lengths do, Ψ(l 2 = 0) ≃ 0. In addition one has some rather complicatedconstraints on the squared edge lengths, due to the triangle inequalities. These ensurethat the areas of triangles and the volumes of tetrahedra are always positive. Asa result one would expect an average soft local upper bound on the squared edgelengths of the type li2 < ∼ 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 smallphase space available in this region.These considerations have some consequences already in the strong couplinglimit of the theory. For sufficiently strong coupling (large Newton constant G) thefirst term in Eq. (4.175) is dominant, which shows again some similarity with whatone finds for non-abelian gauge theories for large g, Eq. (4.157). It is then easy tosee, both from the constraint l i > 0 and the triangle inequalities, that the spectrum ofthis operator is discrete. In particular the mass gap, the spacing between the lowesteigenvalue 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 takenin the vicinity of a non-trivial ultraviolet fixed point, if such a fixed point can befound. One would then anticipate that the excitation spectrum would become denseras one approaches the lattice continuum limit, in accordance with the existence of amassless spin two particle in this limit.Note that in the lattice theory the operator ordering ambiguity has not gone awayeither: in principle one would have to check that different orderings give the samephysical results, whichever way those are defined (for example in terms of vacuumexpectation values of invariant operators, or quantum correlations of invariant operatorsat fixed geodesic distance along the spatial directions).Irrespective of its specific form, it is in general possible to simplify the latticeHamiltonian constraint in Eqs. (4.175) and (4.176) by using scaling arguments, asone does often in ordinary non-relativistic quantum mechanics. After setting for thescaled cosmological constant λ = 8πGλ 0 and dividing the equation out by commonfactors, 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 jwhere one finds it useful to define a dimensionless Newton’s constant, as measuredin 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 . Furthermorethe edge lengths have been rescaled so as to be able to set λ 0 = 1 in lattice units (it isclear from the original gravitational action that the cosmological constant λ 0 simplymultiplies the total spacetime volume, thereby just shifting around the overall scalefor the problem). Schematically Eq. (4.176) is therefore of the form