Quantum mechanics as a measurement theory on biconformal space

TL;DR

This paper demonstrates that quantum mechanics naturally emerges from biconformal geometry, deriving key principles like the Schrödinger equation and uncertainty relations without external quantization assumptions.

Contribution

It introduces a geometric framework where quantum mechanics arises from postulates on motion and measurement in biconformal space, eliminating the need for independent quantization procedures.

Findings

Derivation of the Schrödinger equation from biconformal geometry

Explanation of probability amplitudes as measurement standards

Connection between geometric postulates and quantum uncertainty relations

Abstract

Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement. Additionally, we show that a postulate for unique, classical motion yields Hamiltonian dynamics with no measurable size changes, while a postulate for probabilistic evolution leads to physical dilatations manifested as measurable phase changes. Our results lead to the Feynman path integral formulation, from which follows the Schroedinger equation. We discuss the Heisenberg uncertainty relation and fundamental canonical commutation relations.