A procedure is considered which upgrades the Lagrangian description of quantum relativistic particles to the Lagrangian of a proper field theory in the case that the Klein-Gordon wave equation is classically interpreted in terms of a relativistic sub-quantum potential. We apply the resulting field theory to cosmology and show that the relativistic version of the Bohm's subquantum potential which can be associated with a homogeneous and isotropic distribution of particles behaves like though it was a cosmological constant responsible for the current accelerating expansion of the universe, at least in the limit where the field potential vanishes.

7 pages, RevTex, some minor changes and references added, to appear in Phys. Rev. D