  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.
