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

• URL: http://arxiv.org/abs/2001.05920v2
• Date: Thu, 29 Oct 2020 10:35:37 GMT
• Title: Consistency Proof for Multi-Time Schrodinger Equations with Particle Creation and Ultraviolet Cut-Off
• Authors: Sascha Lill, Lukas Nickel, Roderich Tumulka
• Abstract summary: We show the existence and uniqueness of a smooth solution for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space. 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schr\"odinger 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.

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