Classical and Quantum Initial Value Problems for Models of Chronology Violation

Abstract

We study the classical and quantum theory of a class of nonlinear differential equations on chronology violating spacetime models in which space consists of finitely many discrete points. Classically, in the linear and weakly nonlinear regimes (for generic choices of parameters) we prove existence and uniqueness of solutions corresponding to initial data specified before the dischronal region; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution preserves 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 with initial data representing 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. We present numerical evidence to show that the bosonic quantum theory can pick out a unique classical limit for certain ranges of the coupling strength, even when there are many classical solutions. We show that the quantum theory depends strongly on the choice of operator ordering. In addition, we show that our results differ from those obtained using the ``self-consistent path integral''. It follows that the path integral evolution does not correspond to a solution of the equation of motion.

39 pages, REVTeX 3.0, 2 figures included

References in corpus (7)

• Unitarity and Causality in Generalized Quantum Mechanics for Non-Chronal Spacetimes
• Quantum Coherence and Closed Timelike Curves
• Path Integrals, Density Matrices, and Information Flow with Closed Timelike Curves
• Unitarity Restoration in the Presence of Closed Timelike Curves
• Nonlinearity in Quantum Theory and Closed Timelike Curves
• Unitarity of Quantum Theory and Closed Time-Like Curves
• The Quantum Propagator for a Nonrelativistic Particle in the Vicinity of a Time Machine

Cited by in corpus (6)

• Time machines and the Principle of Self-Consistency as a consequence of the Principle of Stationary Action (II): the Cauchy problem for a self-interacting relativistic particle
• Topological censorship and chronology protection
• Bisolutions to the Klein-Gordon Equation and Quantum Field Theory on 2-dimensional Cylinder Spacetimes
• Quantum physics, fields and closed timelike curves: The D-CTC condition in quantum field theory
• Emergent Time and Time Travel in Quantum Physics
• Quantum state tomography on closed timelike curves using weak measurements