Consistency Proof for Multi-Time Schrodinger Equations with Particle Creation and Ultraviolet Cut-Off

Abstract

For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

53 pages LaTeX, 10 figures; v2: Sec. 1.3 added, minor improvements throughout the paper

References in corpus (4)

• Photon wave function
• A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1+1 dimensions
• Dirac Equation with External Potential and Initial Data on Cauchy Surfaces
• Multi-Time Wave Functions versus Multiple Timelike Dimensions

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• Boundary Conditions that Remove Certain Ultraviolet Divergences
• Singular light cone interactions of scalar particles in 1+3 dimensions
• Multi-Time Version of the Landau-Peierls Formulation of Quantum Electrodynamics
• The multi-time propagators and the consistency condition