No abstract

This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a (Wittgensteinian) philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics (like Bohmian mechanics or 't Hooft's Cellular Automaton interpretation) are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low (or even zero) probability, their very existence implies that the nosignaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements (and of the safety of quantum cryptography) should be reinterpreted statistically, like the second law of thermodynamics. In appendices I discuss the Born rule and its status in both single and repeated experiments, and review the notion of 1-randomness introduced by Kolmogorov, Chaitin, Martin-Löf, Schnorr, and others.

Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of [Formula: see text] à la Rieffel, defined by perfectly natural quantization maps [Formula: see text] (where [Formula: see text] is an equally natural dense Poisson subalgebra of [Formula: see text]). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its [Formula: see text] symmetry is spontaneously broken in the thermodynamic limit [Formula: see text]. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space [Formula: see text] (i.e. the unit three-ball in [Formula: see text]). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors [Formula: see text] of this model as [Formula: see text], in which the sequence converges to a probability measure [Formula: see text] on the associated classical phase space [Formula: see text]. This measure is a symmetric convex sum of two Dirac measures related by the underlying [Formula: see text]-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

Spontaneous symmetry breaking (ssb) is mathematically tied to some limit, but must physically occur, approximately, before the limit. Approximate ssb has been independently understood for Schrödinger operators with double well potential in the classical limit (Jona-Lasinio et al, 1981;Simon, 1985) and for quantum spin systems in the thermodynamic limit (Anderson, 1952;Tasaki, 2019). We relate these to each other in the context of the Curie-Weiss model, establishing a remarkable relationship between this model (for finite N ) and a discretized Schrödinger operator with double well potential.

Bell's Theorem from 1964 and the (Strong) Free Will Theorem of Conway and Kochen from 2009 both exclude deterministic hidden variable theories (or, in modern parlance, 'ontological models') that are compatible with some small fragment of quantum mechanics, admit 'free' settings of the archetypal Alice&Bob experiment, and satisfy a locality condition akin to Parameter Independence. We clarify the relationship between these theorems by giving reformulations of both that exactly pinpoint their resemblance and their differences. Our reformulation imposes determinism in what we see as the only consistent way, in which the 'ontological state' initially determines both the settings and the outcome of the experiment. The usual status of the settings as 'free' parameters is subsequently recovered from independence assumptions on the pertinent (random) variables. Our reformulation also clarifies the role of the settings in Bell's later generalization of his theorem to stochastic hidden variable theories.