Rotationally invariant field theory on lattices. III. Quantizing gravity by means of lattices

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Abstract
Whether one views the lattice as merely regulatory or physically real, lattice gravity is different from continuum gravity because the symmetry group of coordinate transformations is destroyed by the lattice. Since negative-norm kinetic terms can no longer be removed by specifying a gauge, the functional integrals will not converge. We suggest including a quartic potential in the action which obliges the action to converge regardless of the sign of those terms. This potential must depend on a background metric in order that a number of desirable features may hold. For example, consider the conformal factor: One needs a quartic term to cancel the wrong-sign kinetic energy. But one also wants the theory to have asymptotically flat manifolds and to admit the flat manifold as a solution. This requires a canceling quadratic. This in turn requires a background structure. This structure is supplied in a natural way by the lattice itself. The approach to the continuum is examined.