Biorthogonal Quantum Mechanics: Super-Quantum Correlations and   Expectation Values without Definite Probabilities
  We propose mutant versions of quantum mechanics constructed on vector spaces over the finite Galois fields GF(3) and GF(9). The mutation we consider here is distinct from what we proposed in previous papers on Galois field quantum mechanics. In this new mutation, the canonical expression for expectation values is retained instead of that for probabilities. In fact, probabilities are indeterminate. Furthermore, it is shown that the mutant quantum mechanics over the finite field GF(9) exhibits super-quantum correlations (i.e. the Bell-Clauser-Horne-Shimony-Holt bound is 4). We comment on the fundamental physical importance of these results in the context of quantum gravity. 
Lay Nam Chang, Zachary Lewis, Djordje Minic, Tatsu Takeuchi
math-ph gr-qc hep-th math.MP quant-ph