  We investigate fermion--anti-fermion production in 1+1 dimensional QED using
real-time lattice techniques. In this non-perturbative approach the full
quantum dynamics of fermions is included while the gauge field dynamics can be
accurately represented by classical-statistical simulations for relevant field
strengths. We compute the non-equilibrium time evolution of gauge invariant
correlation functions implementing 'low-cost' Wilson fermions. Introducing a
lattice generalization of the Dirac-Heisenberg-Wigner function, we recover the
Schwinger formula in 1+1 dimensions in the limit of a static background field.
We discuss the decay of the field due to the backreaction of the created
fermion--anti-fermion pairs and apply the approach to strongly inhomogeneous
gauge fields. The latter allows us to discuss the striking phenomenon of a
linear rising potential building up between produced fermion bunches after the
initial electric pulse ceased.
