The symmetrical foundation of Measure, Probability and Quantum theories

TL;DR

This paper reveals that measure, probability, and quantum theories all fundamentally rely on simple symmetries and sum-product rules, with quantum mechanics extending these principles into complex numbers to describe binary interactions.

Contribution

It demonstrates that quantum theory is a natural extension of measure and probability theories based on elementary symmetries, clarifying its foundations.

Findings

Quantum formalism is based on simple symmetries similar to measure and probability theories.

Quantum mechanics extends these symmetries into two dimensions using complex numbers.

Observable probabilities in quantum theory follow the Born rule, derived from these symmetries.

Abstract

Quantification starts with sum and product rules that express combination and partition. These rules rest on elementary symmetries that have wide applicability, which explains why arithmetical adding up and splitting into proportions are ubiquitous. Specifically, measure theory formalises addition, and probability theory formalises inference in terms of proportions. Quantum theory rests on the same simple symmetries, but is formalised in two dimensions, not just one, in order to track an object through its binary interactions with other objects. The symmetries still require sum and product rules (here known as the Feynman rules), but they apply to complex numbers instead of real scalars, with observable probabilities being modulus-squared (known as the Born rule). The standard quantum formalism follows. There is no mystery or weirdness, just ordinary probabilistic inference.