Algebraic quantum kinematics and SR-selection

TL;DR

This paper introduces an operator-algebraic framework connecting non-relativistic quantum mechanics with special relativity, highlighting structural distinctions and obstructions in extending quantum field theories across different spacetime symmetries.

Contribution

It develops the first part of a series establishing an algebraic approach to quantum kinematics and SR-selection, emphasizing the roles of constants c and ħ and structural obstructions in Galilean cases.

Findings

Photon sector modeled by classical Fourier--Maxwell theory with quantum features emerging from a canonical commutator.

Constants c and ħ have distinct roles in the framework, with c intrinsic to Fourier spaces and ħ linking phase rates to observables.

Obstruction to lifting the framework to Galilean microcausality is supported by multiple theoretical strands, including a no-go theorem.

Abstract

We develop, as the first of a six-paper series, an operator-algebraic framework relating non-relativistic quantum mechanics and special relativity. Three structural facts organize the framework. (i)~The photon sector of free QED is a transparent realization: classical Fourier--Maxwell theory supplies a complex Hilbert-space scaffold (inner product, symplectic form, Schr\"odinger-form mode equation, polarization $\mathbb{C}^2$) with no quantum input, and a single canonical commutator with scale $\hbar$ on the mode amplitudes promotes it to single-photon QED, with photon indivisibility, the Planck relation $E=\hbar\omega$, and the spin spectrum $\pm\hbar$ as theorems. (ii)~The constants $c$ and $\hbar$ play non-interchangeable roles: $c$ is intrinsic to each Fourier-conjugate space, while $\hbar$ acts \emph{between} them, converting kinematic phase rates into dynamical observables. (iii)~Lifting the framework to a Haag--Kastler net with sharper-than-equal-time microcausality and positive-energy spectrum is structurally obstructed in the Galilean case; we state this as the SR-selection conjecture and identify three strands of evidence (Hegerfeldt spreading; absence of a known Galilean multi-particle resolution; Reeh--Schlieder failure on Galilean Haag--Kastler nets), the third established as a precise no-go theorem in the second paper of the series. The framework's modular content (Tomita--Takesaki, Bisognano--Wichmann, Unruh as KMS, type-$\mathrm{III}_1$ universality) is collected as the algebraic substrate on which the curved-background, dynamical-metric, and crossed-product extensions of the third through sixth papers operate.