Expectation values, experimental predictions, events and entropy in quantum gravitationally decohered quantum mechanics

We restate Kay’s 1998 hypothesis which simultaneously offers an objective definition for the entropy of a closed system, a microscopic foundation for the Second Law, a resolution of the Information Loss (and other) Black-Hole Puzzle(s) as well as an objective mechanism for decoherence. The hypothesis presupposes a conventional unitary theory of low-energy quantum gravity and offers all this by taking the physical density operator of a closed system to be the partial trace of its total density operator (assumed pure) over gravity and by defining its physical entropy to be its ‘matter-gravity entanglement entropy’. We further recall Kay’s 1998 modification of non-relativistic (many-body) quantum mechanics based on Kay’s hypothesis with a Newtonian approximation to quantum gravity. In this modification, we find formal expectation values for certain ‘observables’ such as momentum-squared and Parity are altered but those for functions of positions are unaltered. However, by arguing that every real measurement can ultimately be taken to be a position measurement, we prove that, in practice, it is impossible to detect any alteration at all and, in particular, we predict no alteration for an experiment recently proposed by Roger Penrose. Nevertheless, Kay’s modification contains no Schrödinger Cat-like states, and also allows a possible ‘events’ interpretation which we tentatively propose and begin to explore. We also obtain a first result of Second-Law type for a non-relativistic toymodel closed system and argue that similar results will apply for a wide class of model Newtonian and post-Newtonian closed systems although we argue that ordinary actual laboratory-sized systems can never be treated as closed for the purpose of calculating their entropy. Compared with ‘collapse models’ such as GRW, Kay’s Newtonian theory does a similar job while being free from ad hoc assumptions. PACS numbers: 03.65.Yz, 03.65.Ta. 04.60.-m, 04.70.Dy