Origin of Complex Quantum Amplitudes and Feynman's Rules

TL;DR

This paper derives the complex nature of quantum amplitudes and Feynman's rules from basic assumptions about measurement outcomes, showing that complex arithmetic naturally emerges from symmetry and probability considerations.

Contribution

It provides a derivation of quantum complex amplitudes and Feynman's rules solely from symmetry and measurement assumptions, without assuming complex numbers a priori.

Findings

Complex numbers arise naturally from measurement outcome assumptions.

Feynman's sum and product rules are derived from symmetry conditions.

Probability is given by the modulus squared of complex amplitudes.

Abstract

Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. In this paper, we show that the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.