# Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics

## TL;DR

This paper develops a relational framework for understanding quantum observables as symmetry-invariant quantities relative to reference systems, clarifying debates on reference frames and superselection rules.

## Contribution

It introduces a rigorous mathematical formalism for quantum reference systems and demonstrates how orthodox absolute quantities relate to observable relative quantities.

## Key findings

- Orthodox absolute quantities are good representatives of relative observables with localized reference states.
- The relational formalism clarifies the role of reference frames in quantum superselection rules.
- The paper settles a long-standing debate on the relationship between reference frames and superselection rules.

## Abstract

We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.