Random World and Quantum Mechanics

Quantum mechanics (QM) predicts probabilities on the fundamentallevel which are, via Born probability law, connected to the formal randomnessof infinite sequences of QM outcomes. Recently it has been shown thatQM is algorithmic 1-random in the sense of Martin-L¨of. We extend this resultand demonstrate that QM is algorithmic ω-random and generic, precisely asdescribed by the ’miniaturisation’ of the Solovay forcing to arithmetic. Thisis extended further to the result that QM becomes Zermelo–Fraenkel Solovayrandom on infinite-dimensional Hilbert spaces. Moreover, it is more likely thatthere exists a standard transitive ZFC model M, where QM is expressed in reality,than in the universe V of sets. Then every generic quantum measurementadds to M the infinite sequence, i.e. random real r ∈ 2ω, and the model undergoesrandom forcing extensions M[r]. The entire process of forcing becomesthe structural ingredient of QM and parallels similar constructions applied tospacetime in the quantum limit, therefore showing the structural resemblanceof both in this limit. We discuss several questions regarding measurability andpossible 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 questionsof QM. When one works in a constant model of ZFC or in axiomatic ZFCitself, the issues considered here remain hidden to a great extent.