On localization and position operators in Moebius-covariant theories

TL;DR

This paper explores how Moebius symmetry in certain theories can intrinsically define notions of locality and position operators, linking symmetry transformations with the concept of space in a mathematically consistent way.

Contribution

It introduces a method to define a position operator associated with modular transformations in Moebius-covariant theories, demonstrating compatibility with locality and providing a concrete example.

Findings

The proposed coordinate operator aligns with the abstract notion of locality.

A concrete example with a quantum particle on a line illustrates the approach.

The approach links symmetry transformations to the concept of position in quantum theories.

Abstract

Some years ago it was shown that, in some cases, a notion of locality can arise from the group of symmetry enjoyed by the theory, thus in an intrinsic way. In particular, when Moebius covariance is present, it is possible to associate some particular transformations to the Tomita Takesaki modular operator and conjugation of a specific interval of an abstract circle. In this context we propose a way to define an operator representing the coordinate conjugated with the modular transformations. Remarkably this coordinate turns out to be compatible with the abstract notion of locality. Finally a concrete example concerning a quantum particle on a line is also given.