Random World and Quantum Mechanics

Abstract

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω, to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

Black Holes and Complexity via Constructible Universe

“…The experimental grasping, along with theoretical understanding, of these kinds of problems within QM is important for designing and manipulating the true random sequences in practice (e.g., [26]). Most of the issues raised above will be addressed in our forthcoming publication [17].…”

“…There are, however, specific sentences independent on the axioms which still can be proved in one forcing extension and its negation in another, and they can make difference between the extended models. This phenomenon is related to randomness in QM-we do not investigate this topic here any further (for more details see [17]), but we do apply it when analyzing computational complexity below. On the set-theoretical side, this is connected with relative consistency and provability by forcings in the extended models.…”