Is Quantum Mechanics an Exotic Probability Theory?

Saul Youssef

Richard Healey

Daniel I. Fivel

Conventional quantum mechanics with a complex Hilbert space and the Born Rule is derived from five axioms describing properties of probability distributions for the outcome of measurements. Axioms I,II,III are common to quantum mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first noted by Turing and von Neumann, in which the increase in entropy resulting from a measurement is reduced by a suitable intermediate measurement. This is shown to be impossible for local hidden variable theories. Axiom IV, together with the first three, almost suffice to deduce the conventional rules but allow some exotic, alternatives such as real or quaternionic quantum mechanics. Axiom V recognizes a property of the distribution of outcomes of random measurements on qubits which holds only in the complex Hilbert space model. It is then shown that the five axioms also imply the conventional rules for all dimensions. 1

Antony Valentini

Abstract ‘Saved by Gravity’ shows how exotic gravitational effects could enable our escape from quantum death, by creating particles that violate quantum mechanics and Born’s formula. We begin with a review of quantum gravity and of the notorious ‘problem of time’. We explain why the theory is unable to give a mathematically consistent account of quantum probability. We propose that the problem can be solved if we accept that there is no quantum death, and no Born’s formula, at the deepest level of quantum gravity. The chapter ends by showing how quantum-gravitational effects can create violations of Born’s formula, in particular for radiation from exploding primordial black holes.

Peter Mittelstaedt

Itamar Pitowsky