Multi-time Schrödinger equations cannot contain interaction potentials

Abstract

Multi-time wave functions are wave functions that have a time variable for every particle, such as ϕ(t{sub 1},x{sub 1},...,t{sub N},x{sub N}). They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in order to be consistent. While this condition is automatically satisfied for non-interacting particles, it is a challenge to set up consistent multi-time equations with interaction. We prove for a wide class of multi-time Schrödinger equations that the presence of interaction potentials (given by multiplication operators) leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles, which, in fact, can be done consistently [S. Petrat and R. Tumulka, “Multi-time wave functions for quantum field theory,” Ann. Physics (to be published)]. We also prove the following result: When a cut-off length δ > 0 is introduced (in the sense that the multi-time wave function is defined only on a certain set of spacelike configurations, thereby breaking Lorentz invariance), then the multi-time Schrödinger equations with interaction potentials of range δ are consistent; however, in the desired limit δ → 0 of removing the cut-off, the resulting multi-time equations are interaction-free, which supports the conclusion expressed in the title.