  We consider elementary particles in a quantum theory based on a Galois field.
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. {\it In
other words, the very existence of antiparticles is a strong indication that
nature is described rather by a finite field (or at least a field with a
nonzero characteristic) than by complex numbers.} As a consequence, the
spin-statistics theorem is simply a requirement that standard quantum theory
should be based on complex numbers and elementary particles cannot be neutral.
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.
