Examples of Classical Mechanical Projections

Pavel Bóna

Josh Kirklin

Abstract We describe explicitly how entanglement between quantum mechanical subsystems can lead to emergent gauge symmetry in a classical limit. We first provide a precise characterization of when it is consistent to treat a quantum subsystem classically in such a limit, namely: In any quantum state corresponding to a definite classical state in the classical limit, the reduced density matrix of the subsystem must be approximately proportional to a projection operator, and the projection operators for different classical subsystem states must obey an approximate mutual orthogonality condition. These are strong constraints on the entanglement structure of classical states. They generically give rise to fundamentally nonlocal classical degrees of freedom, which may nevertheless be accounted for using a completely local kinematical description, if one gauges this description in the right way. The mechanism we describe is very general, but for concreteness we exhibit a toy example involving three entangled spins at high angular momentum, and we also describe a significant group-theoretic generalization of this toy example. Finally, we give evidence that this phenomenon plays a role in the emergence of bulk diffeomorphism invariance in gravity.

Matthias Kleinmann

We define a simple rule that allows us to describe sequences of projective measurements for a broad class of generalized probabilistic models. This class embraces quantum mechanics and classical probability theory, but, for example, also the hypothetical Popescu–Rohrlich box. For quantum mechanics, the definition yields the established Lüders rule, which is the standard rule for updating the quantum state after a measurement. For the general case, it can be seen as being the least disturbing or most coherent way of performing sequential measurements. We show, as an example, that the Spekkens toy model (Spekkens 2007 Phys. Rev. A 75 032110 ) provides an instance of our definition. We also demonstrate the possibility of strong postquantum correlations as well as the existence of triple-slit correlations for certain nonquantum toy models.

Mancini, S, Man'ko, O V, Man'ko, V I +1 more

The "marginal" distributions for measurable coordinate and spin projection is introduced. Then, the analog of the Pauli equation for spin-1/2 particle is obtained for such probability distributions instead of the usual wave functions. That allows a classical-like approach to quantum mechanics. Some illuminating examples are presented.