Zachary Lewis

On the Minimal Length Uncertainty Relation and the Foundations of String Theory

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 nonperturbative string theory.

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.

GALOIS FIELD QUANTUM MECHANICS

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

Position and momentum uncertainties of a particle in a V-shaped potential under the minimal length uncertainty relation

We calculate the uncertainties in the position and momentum of a particle in the 1D potential V(x)=F|x|, F>0, when the position and momentum operators obey the deformed commutation relation [x,p]=i\hbar(1+\beta p^2), \beta>0. As in the harmonic oscillator case, which was investigated in a previous publication, the Hamiltonian H_1 = p^2/2m + F|x| admits discrete positive energy eigenstates for both positive and negative mass. The uncertainties for the positive mass states behave as \Delta x ~ 1/\Delta p as in the \beta=0 limit. For the negative mass states, however, in contrast to the harmonic oscillator case where we had \Delta x ~ \Delta p, both \Delta x and \Delta p diverge. We argue that the existence of the negative mass states and the divergence of their uncertainties can be understood by taking the classical limit of the theory. Comparison of our results is made with previous work by Benczik.

Quantum Systems based upon Galois Fields - from Sub-quantum to Super-quantum Correlations -

In this talk we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.

Quantum

We argue that the q = 1 limit of Galois field quantum mechanics, which was constructed on a vector space over the Galois field , corresponds to its ?classical limit?, where superposition of states is disallowed. The limit preserves the projective geometry nature of the state space, and can be understood as being constructed on an appropriately defined analogue of a ?vector? space over the ?field with one element? .

{ {\mathbb{F}}_{ {\rm un}}}

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.