Existence of Antiparticles as an Indication of Finiteness of Nature

It is shown that in a quantum theory over a Galois field, the famous Dirac’s result about antiparticles is generalized such that a particle and its antiparticle are already combined at the level of irreducible representations of the symmetry algebra without assuming the existence of a local covariant equation. We argue that the very existence of antiparticles is a strong indication that nature is described by a finite field rather than by complex numbers.