The Born Rule -- Axiom or Result?

TL;DR

This paper demonstrates that the Born rule's quadratic probability dependence in quantum mechanics can be derived from a single physical assumption called observable independence, reducing its status from an axiom to a consequence.

Contribution

It introduces the observable independence assumption and shows how it derives the Born rule, providing a more physical basis for this fundamental quantum principle.

Findings

Born rule's quadratic form follows from observable independence

The derivation does not rely on a specific interpretation of quantum mechanics

The approach simplifies understanding the origin of the Born rule

Abstract

The Born rule is part of the collapse axiom in the standard version of quantum theory, as presented by standard textbooks on the subject. We show here that its signature quadratic dependence follows from a single additional physical assumption beyond the other axioms - namely, that the probability of a particular measurement outcome (the state $\phi_k$, say) is independent of the choice of observable to be measured, so long as one of its eigenstates corresponds to that outcome. We call this assumption ``observable independence.'' As a consequence, the Born rule cannot be completely eliminated from the list of axioms, but it can, in principle, be reduced to a more physical statement. Our presentation is suitable for advanced undergraduates or graduate students who have taken a standard course in quantum theory. It does not depend on any particular interpretation of the theory.