Quantum indeterminism over Galois fields

In the standard formalism of quantum mechanics, pure states are identified with points in projective space—rays in the underlying Hilbert space representing equivalence classes of vectors under multiplication by non-zero complex scalars. Traditionally, the theory postulates “local” projective invariance, assuming that all physical observables, including probabilities derived from Born’s rule, are independent of the choice of specific representative vectors within these classes.
The report puts forward the hypothesis that the indeterminism of individual quantum events at the “microscopic” level stems from the lack of local invariance in the interaction laws of specific vectors. In this framework, the probabilistic nature of quantum behavior emerges naturally during the transition to “global” invariance at the level of equivalence classes through averaging over gauge parameters.
This approach is explored using vector spaces over finite fields to circumvent the mathematical complexities of defining an averaging procedure over continuous, non-compact complex fields. The Galois field setting enables the efficient use of computer algebra algorithms for model verification. We compare the results of two models: one serving as a direct analogue of continuous quantum mechanics with a locally invariant Born’s rule, and another employing a non-invariant local interaction law that accounts for the interference of representative vectors.