# Direct interaction along light cones at the quantum level. (arXiv:1801.00060v1 [quant-ph])

In this paper, we point out that interactions with time delay can be
described at the quantum level using a multi-time wave function
$\psi(x_1,...,x_N)$, i.e., a wave function depending on one spacetime variable
$x_i = (t_i,\mathbf{x}_i) \in \mathbb{R}^4$ per particle. In particular, such
wave functions (first suggested by Dirac in 1932) make it possible to implement
direct interaction along light cones (not mediated by fields), as in the
Wheeler-Feynman (WF) formulation of electrodynamics. Our results are as
follows. (1) We derive a covariant two-particle integral equation and discuss
it in detail. (2) It is shown how this integral equation as well as an
equivalent system of integro-differential equations can be understood as time
evolution equations. As an important step, we extract a condition to extend
solutions in time: the "super consistency condition". (3) Two different ways
how to extend the two-particle equation to $N$ particles are presented. The
first is based on the super consistency condition, the second on an analogy of
the two-particle equations with classical WF electrodynamics. This analogy is
interesting in its own right as it suggests a possible new quantization of WF
electrodynamics. (4) Finally, we demonstrate that both $N$-particle equations
reduce to the usual Schr\"odinger equation with Coulomb pair potentials if time
delay effects are neglected. The equations therefore have the correct limiting
behavior.