Multi-Time Wave Functions

Abstract

In non-relativistic quantum mechanics of $N$ particles in three spatial dimensions, the wave function $\psi(q_1,\ldots,q_N,t)$ is a function of $3N$ position coordinates and one time coordinate. It is an obvious idea that in a relativistic setting, such functions should be replaced by $\phi((t_1,q_1),\ldots,(t_N,q_N))$, a function of $N$ space-time points called a multi-time wave function because it involves $N$ time variables. Its evolution is determined by $N$ Schr\"odinger equations, one for each time variable; to ensure that simultaneous solutions to these $N$ equations exist, the $N$ Hamiltonians need to satisfy a consistency condition. This condition is automatically satisfied for non-interacting particles, but it is not obvious how to set up consistent multi-time equations with interaction. For example, interaction potentials (such as the Coulomb potential) make the equations inconsistent, except in very special cases. However, there have been recent successes in setting up consistent multi-time equations involving interaction, in two ways: either involving zero-range ($\delta$ potential) interaction or involving particle creation and annihilation. The latter equations provide a multi-time formulation of a quantum field theory. The wave function in these equations is a multi-time Fock function, i.e., a family of functions consisting of, for every $n=0,1,2,\ldots$, an $n$-particle wave function with $n$ time variables. These wave functions are related to the Tomonaga-Schwinger approach and to quantum field operators, but, as we point out, they have several advantages.