  Systems of free particles in a quantum theory based on a Galois field (GFQT)
are discussed in detail. In this approach infinities cannot exist, the
cosmological constant problem does not arise and one irreducible representation
of the symmetry algebra necessarily describes a particle and its antiparticle
simultaneously. As a consequence, well known results of the standard theory
(spin-statistics theorem; a particle and its antiparticle have the same masses
and spins but opposite charges etc.) can be proved without involving local
covariant equations. The spin-statistics theorem is simply a requirement that
quantum theory should be based on complex numbers. Some new features of GFQT
are as follows: i) elementary particles cannot be neutral; ii) the Dirac vacuum
energy problem has a natural solution and the vacuum energy (which in the
standard theory is infinite and negative) equals zero as it should be; iii) the
charge operator has correct properties only for massless particles with the
spins 0 and 1/2. In the AdS version of the theory there exists a dilemma that
either the notion of particles and antiparticles is absolute and then only
particles with a half-integer spin can be elementary or the notion is valid
only when energies are not asymptotically large and then supersymmetry is
possible.
