A note on coarse graining and group representations

TL;DR

This paper explains the group theoretical basis of a coarse graining method for quantum states, generalizes previous results, and shows how these techniques can be extended to other Lie groups.

Contribution

It clarifies the group theoretical origin of a coarse graining procedure and demonstrates its generalization to arbitrary Lie groups.

Findings

The method is rooted in SU(1,1) Lie group structure.

Generalization to other Lie groups is straightforward.

Provides a deeper understanding of the mathematical foundation of coarse graining.

Abstract

A coarse graining operation of spatially homogeneous quantum states based on an SU(1,1) Lie group structure has recently been proposed in [1] and used in [2] to compute an explicit renormalisation group flow in the context of loop quantum cosmology. In this note, we explain the group theoretical origin of this procedure and generalise previous results based on these insights. We also highlight how the group theoretical origin of these techniques implies their immediate generalisation to other Lie groups.