Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU

Tim Palmer (Oxford)

## Abstract

In this seminar, I show how what might appear the tiniest change imaginable to the postulates of quantum mechanics, could lead to a radically new interpretation of quantum physics, one much more compatible with the nonlinear locally causal determinism of general relativity theory.

Despite Planck’s proposal that the energy of black-body radiation is discretised, the state space of quantum mechanics is itself a continuum. Applying Planck’s insight again, we discretise complex Hilbert Space thereby only allowing certain “rational” complex Hilbert states. We now no longer have algebraic closure on state space. But what we lose is more than compensated by what we gain. In particular, in “rational quantum mechanics", the wavefunction is interpretable as an ensemble of deterministic states. Moreover, quantum properties such as contextuality, complementarity and non-commutativity can be simply derived from number theoretic properties of trigonometric functions (Niven’s Theorem). Above all, it becomes possible to explain the violation of Bell’s inequality without giving up local causality or determinism. All these properties hold *no matter how fine is the discretisation* as long as it is non-zero: quantum mechanics is a singular and not a smooth limit of rational quantum mechanics. For a plausible discretisation scale, the following experimentally testable consequence of rational quantum mechanics is proposed: the exponential speed up of a quantum computer will “max out” at around 100 qubits, no matter how well the qubits are shielded from their environment.

Based on these results, we conclude that a theory of quantum gravity may only be possible by marrying general relativity with rational and not regular quantum mechanics.