CLASSICAL AND QUANTUM INITIAL-VALUE PROBLEMS FOR MODELS OF CHRONOLOGYVIOLATION

We study the classical and quantum theory of a class of nonlinear diff erential equations on models of chronology-violating spacetimes in whi ch space consists of only finitely many discrete points. Classically, we study the initial value problem for data specified before the nonch ronal region. In the linear and weakly coupled nonlinear regimes, we s how (for generic choices of parameters) that solutions always exist an d are unique; however, uniqueness (but not existence) fails in the str ongly coupled regime. The evolution is shown to preserve the symplecti c 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 th e canonical (anti)commutation relations. Using normal operator orderin g, we construct solutions to the QIVP for both Bose and Fermi statisti cs (again for generic choice of parameters) and prove that these solut ions are unique. For models with two spatial points, the resulting evo lution is unitary; however, for a more general model the evolution fai ls to preserve the (anti)commutation relations and is, therefore, nonu nitary. We show that this nonunitary evolution cannot be described usi ng a superscattering operator with the usual properties. The classical limit is discussed and numerical evidence is presented to show that t he bosonic quantum theory picks out a unique classical solution for ce rtain ranges of the coupling strength, but that the classical limit fa ils to exist for other values of the coupling. We also show that the q uantum 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.