Multi-Time Schr\"odinger Equations Cannot Contain Interaction Potentials

TL;DR

This paper demonstrates that multi-time Schr"odinger equations cannot include interaction potentials without inconsistency, implying that interactions must be modeled via particle creation and annihilation rather than potentials.

Contribution

It proves that multi-time Schr"odinger equations with interaction potentials are inherently inconsistent, supporting the necessity of particle creation and annihilation for interactions.

Findings

Interaction potentials cause inconsistency in multi-time Schr"odinger equations.

Introducing a cut-off length makes the equations consistent with interactions.

Removing the cut-off leads to interaction-free equations, confirming the main conclusion.

Abstract

Multi-time wave functions are wave functions that have a time variable for every particle, such as $\phi(t_1,x_1,\ldots,t_N,x_N)$. 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 Schr\"odinger equations, one for each time variable. These Schr\"odinger 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 Schr\"odinger 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, as we show elsewhere [17]. We also prove the following result: When a cut-off length $\delta>0$ 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 Schr\"odinger equations with interaction potentials of range $\delta$ are consistent; however, in the desired limit $\delta\to 0$ of removing the cut-off, the resulting multi-time equations are interaction-free, which supports the conclusion expressed in the title.