Abstract. We discuss the properties of Galois Field Quantum Mechanics constructed on a vector space over the finite Galois field GF (q). In particular, we look at 2-level systems analogous to spin, and discuss how SO(3) rotations could be embodied in such a system. We also consider two-particle 'spin' correlations and show that the Clauser-Horne-Shimony-Holt (CHSH) inequality is nonetheless not violated in this model.

We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of non-perturbative string theory.

We offer an interpretation of super-quantum correlations in terms of a "doubly" quantum theory. We argue that string theory, viewed as a quantum theory with two deformation parameters, the string tension α ′ and the string coupling constant gs, is such a super-quantum theory, one that transgresses the usual quantum violations of Bell's inequalities. We also discuss the → ∞ limit of quantum mechanics in this context. As a super-quantum theory, string theory should display distinct experimentally observable super-correlations of entangled stringy states.

We construct a discrete quantum mechanics using a vector space over the Galois field GF (q). We find that the correlations in our model do not violate the Clauser-HorneShimony-Holt (CHSH) version of Bell's inequality, despite the fact that the predictions of this discrete quantum mechanics cannot be reproduced with any hidden variable theory.

We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by [x,p] 2 ). This deformed commutation relation leads to the minimal length uncertainty relation ∆x ≥ ( /2)(1/∆p + β∆p), which implies that ∆x ∼ 1/∆p at small ∆p while ∆x ∼ ∆p at large ∆p. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m > 0), derived in Ref.[1], only populate the ∆x ∼ 1/∆p branch. The other branch, ∆x ∼ ∆p, is found to be populated by the energy eigenstates of the 'inverted' harmonic oscillator (m < 0). The Hilbert space in the 'inverted' case admits an infinite ladder of positive energy eigenstates provided that ∆xmin = √ β > √ 2 [ 2 /k|m| ] 1/4 . Correspondence with the classical limit is also discussed.

We argue that quantum gravity should be a super-quantum theory, that is, a theory whose non-local correlations are stronger than those of canonical quantum theory. As a super-quantum theory, quantum gravity should display distinct experimentally observable super-correlations of entangled stringy states.

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.

We construct a 'mutant' form of quantum mechanics on a vector space over the finite Galois field GF(q). We find that the correlations in our model do not violate the Clauser-Horne-Shimony-Holt (CHSH) version of Bell's inequality, despite the fact that the predictions of this discretized quantum mechanics cannot be reproduced with any hidden variable theory. An alternative 'mutation' is also suggested.