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Multi-Time Wave Functions for Quantum Field Theory

Multi-time wave functions such as $ϕ(t_1,x_1,\ldots,t_N,x_N)$ have one time variable $t_j$ for each particle. This type of wave function arises as a relativistic generalization of the wave function $ψ(t,x_1,\ldots,x_N)$ 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.