From the discrete to the continuous - towards a cylindrically consistent dynamics

TL;DR

This paper develops a systematic coarse graining method inspired by tensor network techniques to construct cylindrically consistent dynamics in loop quantum gravity, bridging discrete models with continuum physics.

Contribution

It introduces a novel coarse graining scheme using embedding maps to achieve cylindrically consistent transition amplitudes and Hamilton's functions.

Findings

Provides a tensor network inspired coarse graining procedure

Defines embedding maps aligned with system dynamics

Offers a systematic approximation towards continuum limit

Abstract

Discrete models usually represent approximations to continuum physics. Cylindrical consistency provides a framework in which discretizations mirror exactly the continuum limit. Being a standard tool for the kinematics of loop quantum gravity we propose a coarse graining procedure that aims at constructing a cylindrically consistent dynamics in the form of transition amplitudes and Hamilton's principal functions. The coarse graining procedure, which is motivated by tensor network renormalization methods, provides a systematic approximation scheme towards this end. A crucial role in this coarse graining scheme is played by embedding maps that allow the interpretation of discrete boundary data as continuum configurations. These embedding maps should be selected according to the dynamics of the system, as a choice of embedding maps will determine a truncation of the renormalization flow.