Spekkens’ Toy Model, Finite Field Quantum Mechanics, and the Role of Linearity

Abstract

We map Spekkens' toy model to a quantum mechanics defined over the finite field F5. This allows us to define arbitrary linear combinations of the epistemic states in the model. For Spekkens' elementary system with only 2 2 = 4 ontic states, the mapping is exact and the two models agree completely. However, for a pair of elementary systems there exist interesting differences between the entangled states of the two models.

Interference and Oscillation in Nambu Quantum Mechanics

“…Various proposals for alternative QM theories can be found in the literature. The field/division algebra over which the state space is constructed has been modified from C to R [10], H [11], O [12][13][14], and the finite fields F q [15][16][17][18][19][20][21]. Non-linear corrections to the Schrödinger equation have been considered by Weinberg [22,23].…”

Finite Deformations of Quantum Mechanics

“…On the other hand, the Hamiltonians with which we model just about every physical system known to man, as well as many that belong to the realm of science fiction, have the form 2 . For such Hamiltonians, the Trotter product formula tells us that if we approximate continuous time evolution by evolution over short enough discrete intervals then the short time evolution operators are the product of a function of U and a function of V .…”

“…This is a cheat, because rationals are dense in the real numbers and all we really mean by the real numbers is limits of sequences of rationals. Choosing finite fields [2] does not appear to work because vector spaces over finite fields do not have scalar products. The authors of [2] provide definitions of probability for these vector spaces, but it is not clear whether a consistent formalism including probability conserving time evolution exists or whether their are limits where that formalism can approximate the empirical successes of QM.…”