We discuss properties of a noncompact formulation of gauge theories with fermions on a momentum ($k$) lattice. (a) This formulation is suitable to build in Fourier acceleration in a direct way. (b) The numerical effort to compute the action (by fast Fourier transform) goes essentially like $VlogV$ with the lattice volume $V$. (c) For the Yang-Mills theory we find that the action conserves gauge symmetry and chiral symmetry in a weak sense: On a finite lattice the action is invariant under infinitesimal transformations with compact support. Under finite transformations these symmetries are approximately conserved and they are restored on an infinite lattice and in the continuum limit. Moreover, these symmetries also hold on a finite lattice under finite transformations, if the classical fields, instead of being $c$-number valued, take values from a finite Galois field. (d) There is no fermion doubling. (e) For the ${\ensuremath{\varphi}}^{4}$ model we investigate the transition towards the continuum limit in lattice perturbation theory up to second order. We compute the two- and four-point functions and find local and Lorentz-invariant results. (f) In QED we compute a one-loop vacuum polarization and find in the continuum limit the standard result. (g) As a numerical application, we compute the propagator $〈\ensuremath{\varphi}(k)\ensuremath{\varphi}({k}^{\ensuremath{'}})〉$ in the ${\ensuremath{\varphi}}^{4}$ model, investigate Euclidean invariance, and extract ${m}_{R}$ as well as ${Z}_{R}$. Moreover we compute $〈{F}_{\ensuremath{\mu}\ensuremath{\nu}}(k){F}_{\ensuremath{\mu}\ensuremath{\nu}}({k}^{\ensuremath{'}})〉$ in the SU(2) model.