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Here, we point out that interactions with time delay can be described at thequantum level using a multi-time wave function $\psi(x_1,...,x_N)$, i.e., awave function depending on one spacetime variable $x_i = (t_i,\mathbf{x}_i)$per particle. In particular, such a wave function makes it possible toimplement direct interaction along light cones (not mediated by fields), as inthe Wheeler-Feynman formulation of electrodynamics. Our results are as follows.(1) We derive a covariant two-particle integral equation and discuss it indetail. (2) It is shown how this integral equation (or equivalently, a systemof two integro-differential equations) can be understood as defining the timeevolution of $\psi$ in a consistent way. (3) We demonstrate that the equationhas strong analogies with Wheeler-Feynman electrodynamics and thereforesuggests a possible new quantization of that theory. (4) We propose two naturalways how the two-particle equation can be extended to $N$ particles. It isshown that exactly one of them leads to the usual Schr\"odinger equation withCoulomb-type pair potentials if time delay effects are neglected.

Direct interaction along light cones at the quantum level