  Lattice field theory is a very powerful tool to study Feynman's path integral
non-perturbatively. However, it usually requires Euclidean background metrics
to be well-defined. On the other hand, a recently developed regularization
scheme based on Fourier integral operator $\zeta$-functions can treat Feynman's
path integral non-pertubatively in Lorentzian background metrics. In this
article, we formally $\zeta$-regularize lattice theories with Lorentzian
backgrounds and identify conditions for the Fourier integral operator
$\zeta$-function regularization to be applicable. Furthermore, we show that the
classical limit of the $\zeta$-regularized theory is independent of the
regularization. Finally, we consider the harmonic oscillator as an explicit
example. We discuss multiple options for the regularization and analytically
show that they all reproduce the correct ground state energy on the lattice and
in the continuum limit. Additionally, we solve the harmonic oscillator on the
lattice in Minkowski background numerically.
