Classical and quantum initial value problems for models of chronology violation

We study the classical and quantum theory of a class of nonlinear differential equations on models of chronology-violating spacetimes in which space consists of only finitely many discrete points. Classically, we study the initial value problem for data specified before the nonchronal region. In the linear and weakly coupled nonlinear regimes, we show (for generic choices of parameters) that solutions always exist and are unique; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution is shown to preserve the symplectic structure. The quantum theory is approached via the quantum initial value problem (QIVP), that is, by seeking operator-valued solutions to the equation of motion whose initial data form a representation of the canonical (anti)commutation relations. Using normal operator ordering, we construct solutions to the QIVP for both Bose and Fermi statistics (again for generic choice of parameters) and prove that these solutions are unique. For models with two spatial points, the resulting evolution is unitary; however, for a more general model the evolution fails to preserve the (anti)commutation relations and is, therefore, nonunitary. We show that this nonunitary evolution cannot be described using a superscattering operator with the usual properties. The classical limit is discussed and numerical evidence is presented to show that the bosonic quantum theory picks out a unique classical solution for certain ranges of the coupling strength, but that the classical limit fails to exist for other values of the coupling. We also show that the quantum theory depends strongly on the choice of operator ordering. In addition, we compare our results with those obtained using the "self-consistent path integral" and find that they differ. It follows that the evolution obtained from the path integral does not correspond to a solution to the equation of motion.