Causal set as a discretized phase spacetime

Abstract

The first goal of this paper is to show that discreteness, locality, and relativistic covariance can peacefully coexist if the ordinary spacetime (OST) is replaced with phase spacetime (PST) as a geometric background of a Poisson process, where PST is a spacetime generalization of a notion of phase space (this is a 7-dimensional version of the 8-dimensional structure proposed by Caianiello). Furthermore, Caianiello's idea of finite acceleration is implemented. After this is done, the paper then goes on to generalize the geometric notions obtained from the intuition of PST to a general discrete causal set, without any geometric background. It then takes advantage of the absence of lightcone singularity to attempt to tackle the definition of PST-like causal set; that is, a discrete system that approximates the geometrical properties that we would expect from continuum PST. Finally, the paper proceeds to introduce quantum field theory on a causal set, and shows that the locality gained by switching from OST to PST brings us one step closer to be able to treat quantum field theory on a causal set analytically rather than numerically.