# A note on the Hamiltonian as a polymerisation parameter

## TL;DR

This paper examines the use of the Hamiltonian as a phase space dependent polymerisation scale in loop quantum gravity models, correcting previous errors and clarifying its role in quantum black hole spacetimes.

## Contribution

It clarifies the correct implementation of the Hamiltonian as a polymerisation parameter and addresses errors in recent approaches.

## Key findings

- Identifies errors in recent treatments of the Hamiltonian as a polymerisation scale.
- Provides a method to correct these errors.
- Highlights implications for quantum black hole models.

## Abstract

In effective models of loop quantum gravity, the onset of quantum effects is controlled by a so-called polymerisation scale. It is sometimes necessary to make this scale phase space dependent in order to obtain sensible physics. A particularly interesting choice recently used to study quantum corrected black hole spacetimes takes the generator of time translations itself to set the scale. We review this idea, point out errors in recent treatments, and show how to fix them in principle.