  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 turns out to be unambiguous,
with no need for normal ordering. The positive-energy requirement imposed on
the second-quantized system leads to equally acceptable Bose and Fermi choices
for particle statistics. This suggests supersymmetric extension of original
Dirac's theory in which Bose and Fermi single-component positive-energy Dirac
fields are combined into a doublet whose members can transform into each other.
Our model includes a Landau-Anderson-Higgs type potential which allows
spontaneous selection of the universe with a given bose-to-fermi ratio of the
positive-energy Dirac particles. The bosonic sector of proposed extension hints
at the possibility of a dark, background Bose-Einstein condensate that endows
the universe with its cosmological temporal asymmetry. In the massless case of
Dirac's original theory, we explore the possibility of allowing the field
expansion to involve both positive- and negative-frequency modes. This leads to
the anticommutation relations for creation and annihilation operators
associated with the negative energy solutions, resulting in supersymmetric
behavior of the single-component field in the ultrarelativistic limit. Finally,
we speculate on the possibility for the positive-energy Dirac particles to obey
some exotic (such as non-abelian, Clifford) statistics in which the particles
are neither created nor destroyed.
