An Intensional Probability Theory: Investigating the Link between Classical and Quantum Probabilities

Abstract

The link between classical and quantum theories is discussed in terms of extensional and intensional viewpoints. The paper aims to bring evidence that classical and quantum probabilities are related by intensionalization, which means that by abandoning sets from classical probability one should obtain quantum theory. Unlike the extensional concept of a set, the intensional probability is attributed to the quantum ensemble, which is contextually dependent. The contextuality offers a consistent realization of the measurement problem, which should require the existence of the time operator. The time continuum by Brouwer has satisfied such a requirement, which makes it fundamental to mathematical physics. The statistical model it provides has been proven tremendously useful in a variety of applications.

The Measurement Problem in Statistical Signal Processing

“…obtained by translation and normalized dilatation of a mother wavelet Ψ. They reappear in the signal space L 2 µ due to ψ j,k (x) = ∑ n Ψ j,k (x + n), which gives rise to the periodization axiom ψ j,k = ψ j,k+2 j (10) and also the annihilation…”

“…Projectors constitute the Boolean algebra, which is isomorphic to an algebra of sets due to the Stone representation theorem. It is the measurable space corresponding to devices, which an observable has been defined on [9,10]. A measurement state on the other hand corresponds to a density ρ = FF † which is defined upon the same domain.…”

“…m ], respectively, values ψ l,m ψ o j,k are negligible if supports do not intersect. This implies an approximate decorrelation of the ensemble, which should mean that correlation between detail coefficients is predominantly concerned by inheritance along branches of the binary tree [9,10].…”

“…The measurement problem is therefore a predominantly mathematical issue which is related to the very foundation of geometry, analysis, probability and other topics. It concerns intentionality that is the manner in which mathematics has always applied [10]. This was the reason for it to be termed the reality problem by Philip Pearle [11].…”

Measuremnt Problem in Statistical Signal Processing

“…The measurement problem is therefore a predominantly mathematical issue which is related to the very foundation of geometry, analysis, probability and other topics. It concerns intentionality that is the manner in which mathematics has always applied [10]. This was the reason for it to be termed the reality problem by Philip Pearle [11].…”