Multi-time wave functions for quantum field theory

•Multi-time wave functions are manifestly Lorentz-covariant objects.•We develop consistent multi-time equations with interaction for quantum field theory.•We discuss in detail a particular model with particle creation and annihilation.•We show how multi-time wave functions are related to the Tomonaga–Schwinger approach.•We show that they have a simple representation in terms of operator valued fields.

Multi-time wave functions such as ϕ(t1,x1,…,tN,xN) have one time variable tjtj for each particle. This type of wave function arises as a relativistic generalization of the wave function ψ(t,x1,…,xN) of non-relativistic quantum mechanics. We show here how a quantum field theory can be formulated in terms of multi-time wave functions. We mainly consider a particular quantum field theory that features particle creation and annihilation. Starting from the particle–position representation of state vectors in Fock space, we introduce multi-time wave functions with a variable number of time variables, set up multi-time evolution equations, and show that they are consistent. Moreover, we discuss the relation of the multi-time wave function to two other representations, the Tomonaga–Schwinger representation and the Heisenberg picture in terms of operator-valued fields on space–time. In a certain sense and under natural assumptions, we find that all three representations are equivalent; yet, we point out that the multi-time formulation has several technical and conceptual advantages.