Canonical quantization of the dark Dirac field and time asymmetry

We perform canonical quantization of the single-component, spin-zero field that was introduced by Dirac in 1971 and recently suggested as a candidate for dark matter by Bogomolny. The massive and massless cases are treated separately. Since in the massive case only positive-frequency modes are normalizable and regarded as physical, the mode expansion for the field involves annihilation operators only, making the quantization procedure particularly simple. The corresponding Hamiltonian of the system turns out to be unambiguous, with no need for normal ordering. The positive-energy requirement imposed on the second-quantized system leads to bosonic commutation relations, in agreement with the spin-statistics theorem. The boson nature of the positive-energy Dirac particles, which, evidently, have no antiparticles, hints at the possibility of a dark, background Bose-Einstein condensate that drives different parts of the universe in the same temporal direction. In the massless case, on the other hand, we explore the possibility of allowing the mode expansion to involve both positive- and negative-frequency components, which leads to the anticommutation relations, resulting in the fermionic behavior of the field in ultrarelativistic limit.