Reconditioning in Discrete Quantum Field Theory

Abstract

We consider a discrete scalar, quantum field theory based on a cubic 4-dimensional lattice. We mainly investigate a discrete scattering operator $S(x_0,r)$ where $x_0$ and $r$ are positive integers representing time and maximal total energy, respectively. The operator $S(x_0,r)$ is used to define transition amplitudes which are then employed to compute transition probabilities. These probabilities are conditioned on the time-energy $(x_0,r)$. In order to maintain total unit probability, the transition probabilities need to be reconditioned at each $(x_0,r)$. This is roughly analogous to renormalization in standard quantum field theory, except no infinities or singularities are involved. We illustrate this theory with a simple scattering experiment involving a common interaction Hamiltonian. We briefly mention how discreteness of spacetime might be tested astronomically. Moreover, these tests may explain the existence of dark energy and dark matter.