Boost modes for a massive fermion field and the Unruh quantization

We show that the Wightman function of a free quantum field generates any complete set of solutions of the relativistic wave equations. Using this approach, we construct the complete set of solutions of the two-dimensional Dirac equation consisting of eigenfunctions of the generator of Lorentz rotations (boost operator). We show that at the surface of the light cone, the boost modes for a fermion field contain the Gelfand delta function of a complex argument. Because of the presence of such a singularity, excluding even a single mode with an arbitrary value of the boost quantum number makes the set of boost modes incomplete. This results in the nonapplicability of the Unruh quantization scheme to a massive fermion field in the two-dimensional Minkowski space–time. Hence, in full accordance with the boson case, the Unruh procedure for a fermion field cannot be used to prove the existence of the Unruh effect.