## Multi-Time Schrödinger Equations Cannot Contain Interaction Potentials    [PDF]

Sören Petrat, Roderich Tumulka
Multi-time wave functions are wave functions that have a time variable for every particle, such as $\phi(t_1,x_1,...,t_N,x_N)$. They arise as a relativistic analog of the wave functions of quantum mechanics, and sometimes arise 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 leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles. We also prove a result that seemingly points in the opposite direction: 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.
View original: http://arxiv.org/abs/1308.1065