The WKB method, well-known from quantum mechanics, has been sucessfully applied to the problem of non-perturbative pair production in time-dependent background fields, such as in cosmological particle creation or in the Sauter-Schwinger effect, i.e. particle creation due to a strong electromagnetic field.
It can be used to calculate the pair production probability for a wide range of field profiles and to obtain the momentum spectrum of the produced pairs as well.

We present a perturbative approach to the dynamically-assisted Sauter-Schwinger effect, i.e. exponential enhancement of the Sauter-Schwinger effect by an additional weak and fast electric field. In this scenario, we treat only the strong field non-perturbatively using the WKB method while the weak field enters via a perturbation series.
This approach can be used to explain qualitative differences in the enhancement mechanism when comparing different field profiles for the weak field. We find that these differences can be attributed to different forms of the weak field's Fourier transform.

Furthermore, we propose another setup which gives exponential enhancement that consists of a strong, slowly varying electric field, a weak, fast varying electric field and a high-energy photon. We again employ the WKB method; this time treating strong and weak field non-perturbatively. The exponential enhancement always turns out to be larger than when any of the three ingredients is missing.
In this case, the considered electromagnetic field is spacetime-dependent but this dependency enters the calculation only perturbatively (via the photon).

Truly spacetime-dependent background fields, however, can not be considered using the WKB method in its original form. As the main result of this thesis, we propose a new method based on the relativistic eikonal (or Hamilton-Jacobi) equation that reduces to the WKB method in time-dependent backgrounds but can also be employed for spacetime-dependent fields.
We calculate corrections to the locally homogeneous field approximation for a weakly space-dependent mass and find that the spatial inhomogeneity decreases the pair production density exponent for the chosen field profile.