Quantum mechanics over Galois fields

Abstract:

The report examines alternative models of quantum systems that go beyond the standard formulation of quantum theory. To describe quantum states, instead of a continuous projective complex Hilbert space, projective spaces over finite fields F_q are used. Eliminating non-constructive infinities and continuous elements of the theory allows one to focus on the essential features of quantum behavior and makes detailed computer analysis possible.

In particular, it is shown that:

• The fundamental projective properties of quantum systems and the probabilistic nature of their macroscopic behavior (indeterminism) can be explained as emergent properties arising from averaging over gauge degrees of freedom the microscopic interactions of vectors that form projective rays.
• The Pauli exclusion principle and fermion statistics follow naturally from the arithmetic properties of fields of characteristic 2, which may indicate a fundamental role of fields of the form F₂m in the description of fermionic matter.

For computer simulation and full combinatorial analysis of the proposed models, an original high-performance software package in the C language was developed.