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It is shown that in a quantum theory over a Galois field, the famous Dirac'sresult about antiparticles is generalized such that a particle and itsantiparticle are already combined at the level of irreducible representationsof the symmetry algebra without assuming the existence of a local covariantequation. We argue that the very existence of antiparticles is a strongindication that nature is described by a finite field rather than by complexnumbers.

Existence of Antiparticles as an Indication of Finiteness of Nature