Operational reconstruction of Feynman rules for quantum amplitudes via composition algebras

TL;DR

This paper develops an operational, coordinate-independent framework for quantum transition amplitudes, identifying composition algebras and connecting them to the Born rule, with improvements over previous models.

Contribution

It introduces a coordinate-free approach to quantum amplitudes, deriving algebraic structures from axioms and observer choices, expanding the quantum reconstruction program.

Findings

Allowed amplitude algebras are real associative composition algebras, including complex numbers and quaternions.

Probabilities are quadratic in amplitudes, consistent with the Born rule.

The framework applies broadly to quantum discovery and reformulates observer questions.

Abstract

This article explores an operational model for transition amplitudes between measurements proposed by Goyal et al. within the quantum reconstruction program. To classify suitable amplitude algebras, we distinguish mathematical axioms, physical choices, and their consequences. This leads to several improvements on the published work: Our coordinate-independent approach requires no two-dimensional amplitudes a priori. All scalar field and vector space axioms are traced from model axioms and observer choices, including additive and multiplicative units and inverses. Existing mathematical characterizations identify allowable amplitude algebras as the real associative composition algebras, namely the complex numbers and the quaternions, as well as their split forms. Observed probabilities are quadratic in amplitudes, akin to the Born rule. We examine selected implications of the proposed axioms, reformulate observer questions, and highlight the broad applicability of our framework to subsequent discovery.