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