Quantum Theory over a Galois Field and Applications to Gravity and Particle Theory

We argue that the main reason of crisis in quantum physics is that nature, which is fundamentally discrete, is described by continuous mathematics. Moreover, no ultimate physical theory can be based on continuous mathematics because, as follows from Gödel’s incompleteness theorems, that mathematics is not self-consistent. In the first part of the work we discuss inconsistencies in standard approach to quantum theory and reformulate the theory such that it can be naturally generalized to a formulation based on discrete mathematics. It is shown that the cosmological acceleration and gravity can be treated simply as kinematical manifestations of de Sitter symmetry on quantum level (i.e. for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed). In the second part of the work we argue that fundamental quantum theory should be based on a Galois field with a large characteristic p. In this approach the de Sitter gravitational constant depends on p and disappears in the formal limit p → ∞, i.e. gravity is a consequence of finiteness of nature. The application of the approach to particle theory gives the following results: a) no neutral elementary particles can exist; b) the electric charge and the baryon and lepton quantum numbers can be only approximately conserved (i.e. the notion of a particle and its antiparticle is only approximate). We also consider a possibility that only Dirac singletons can be true elementary particles. PACS: 02.10.Hh, 11.30.Fs, 11.30.Ly, 12.90.+b