Quantum Theory on a Galois Field

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. PACS: 02.10.De, 03.65.Ta, 11.30.Fs, 11.30.Ly