The Born rule as a statistics of quantum micro-events

Abstract

We deduce the Born rule from a purely statistical take on quantum theory within minimalistic math-setup. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics—a linear, not Hilbert’, vector space—and empirical notion of the Statistical Length of a state. Its statistical nature comes from the lab micro-events (detector-clicks) being formalized into the C -coefficients of quantum superpositions. We also comment that not only has the use not been made of quantum axioms (scalar-product, operators, interpretations , etc.), but that the involving thereof would be, in a sense, inconsistent when deriving the rule. In point of fact, the quadratic character of the statistical length, and even not (the ‘physics’ of) Born’s formula, represents a first step in constructing the mathematical structure we name the Hilbert space of quantum states.

Why and whence the Hilbert space in quantum theory?

“…Therefore it is permissible to rely only on coefficients a j in |α -expansions (2), i. e., on bases, to which the f-statistics has been attached. This is implemented by a core object-the statistical length [6] StatLength…”

“…The function ] is utterly minimalistic in its derivation-it requires neither Hilbert' space nor physics, is unique, and has a sum-of-squares form [6] N…”

“…The numerical nature of Born's rule-function StatLength on abstract LVS-has already been described in [6], and we may therefore characterize subsequent actions in 'the vein of the numerical ideology' over the C N -model (a 1 , a 2 , . .…”

“…It is admitted only to such bases or, to be precise, it is through this nature that determines them [6]. One notices that we do not assume a priori that the belonging to a basis of an observable should be numerical in character and binary, i. e., that it should be given by some numerical relation between its two members.…”

“…now, without reference to the concept of a basis. But since the N was being derived as an A -invariant construct-this is one of its axioms [6]-the change of bases * should be carried over to the space H A . This amounts to a transition to the same but one more space H B .…”