Born rule: quantum probability as classical probability

TL;DR

This work argues that quantum probabilities can be understood as classical-like probabilities by positing a universal, continuous ontic basis for the entire universe, realized as classical field configurations within the wavefunctional framework. A central derivation shows that the Born rule emerges from counting ontic states compatible with macrostates, via a continuous limit of partitions and measure densities, yielding probability as a measure ratio over the ontic space. The interpretation integrates a gauge structure that renders phases as U(1) gauge degrees of freedom, and adopts a local-beables ontology that naturally aligns with a many-worlds view in which macro sectors decohere and micro-branches are counted. The framework claims to reproduce standard quantum predictions while providing a classical probabilistic grounding, clarifying the roles of complex numbers, macro observables, and the wavefunctional, and offering a coherent path toward integrating quantum theory with a field-theoretic and gravity-compatible ontology.

Abstract

I provide a simple derivation of the Born rule as giving a classical probability, that is, the ratio of the measure of favorable states of the system to the measure of its total possible states. In classical systems, the probability is due to the fact that the same macrostate can be realized in different ways as a microstate. Despite the radical differences between quantum and classical systems, I show that the same can be applied to quantum systems, and the result is the Born rule. This works only if the basis is continuous (an eigenbasis of observables with continuous spectra), but all known physically realistic measurements involve a continuous basis (the position basis). The continuous basis is not unique, and for subsystems it depends on the observable. But for the entire universe, there are continuous bases that give the Born rule for all measurements, because all measurements reduce to distinguishing macroscopic pointer states, and macroscopic observations commute. This allows for the possibility of a unique ontic basis for the entire universe. In the wavefunctional formulation, the basis can be chosen to consist of classical field configurations, and the coefficients $Ψ[φ]$ can be made real by absorbing them into a global U(1) gauge. For the many-worlds interpretation, this result gives the Born rule from micro-branch counting.

Figures (2)

• Figure 1: The Born rule from "counting" basis states. A. The usual interpretation of a wavefunction as a linear combination of basis state vectors of different amplitudes. B. The interpretation of the wavefunction in terms of basis vectors representing ontic states.
• Figure 2: Interpretation of the wavefunctional. The $\textnormal{U}(1)$ gauge or phase is represented by the pure color hues in the color wheel. The density $r[\phi]$, represented as shades of gray, is the density of state vectors as they combine in $|\Psi\rangle$ (while the probability density over the sample space $\mathcal{C}$ is $r^2[\phi]$). Their combination gives the wavefunctional $|\Psi\rangle=\int_{\mathcal{C}}|\phi\rangle d\widetilde{\mu}[\phi]$ as a set of classical fields with varying densities and gauges.

Theorems & Definitions (11)

• Definition 1: Classical probabilities
• Remark 1
• Theorem 1
• proof
• Example 1
• Proposition 1
• proof
• Proposition 2
• proof
• Remark 2: "Magic" accident