Pregeometric quantum lattice: A general discussion

Abstract

We put forward an idea that the fundamental, i.e. pregeometric, structure of spacetime is given by an abstract set, so called abstract simplicial complex ASC. Thus, at the pregeometric level there is no (smooth) spacetime manifold. However, we argue that the structure described by an abstract simplicial complex is dynamical. This dynamics is then assumed to ensure that ASC can be realized as a lattice on a four-dimensional manifold with the simplest topologies dominating. We rewrite the pregeometric model, which is quantized using euclidean path-integral formalism, in an exact way so that as a four-dimensional manifold with the simples topologies dominating. is done by definition. The first step in bringing the continuum into the arena is to build up a lattice on a four-dimensional manifold from a given ASC. In fact, we choose a specific lattice: The Regge calculus lattice, i.e. a piecewise linear (flat) metric spacetime manifold. Secondly, we introduce a smooth (C ∞) manifold (described by a metric tensor g μν ) to approximate the Regge calculus manifold (described by a metric tensor g μν RC). It turns out that after integrating (and summing) out all other degrees of freedom than the metric tensor field g μν , the resulting continuum theory is nonlocal (as would be expected). However, it is our main point to show that the nonlocality is not very severe since it is only of finite range. We argue that the points in the introduced continuum which represent lattice points have so great quantum fluctuations that they are in a high temperature phase with no long-range correlations. In other words, although the effective action for the continuum formulation is not totally local, it is effectively so because it has only finite range nonlocalities. We can prove this kind of weak locality of the effective action by means of a general high-temperature theorem. Then we claim that the resulting local (or rather almost local) model with reparametrization invariance and g μν as a field gives essentially the ordinary Einstein's gravity theory in the long wavelength limit.