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The continuum limit of loop quantum gravity - a framework for solving the theory

Bianca Dittrich

TL;DR

This paper addresses constructing the continuum dynamics of loop quantum gravity by promoting the kinematical inductive-limit framework to a dynamical one via refined amplitudes and dynamical embedding maps. It introduces an iterative, truncation-based coarse-graining procedure that yields cylindrically consistent amplitude maps, effectively modeling a background-independent renormalization flow and aiming to restore diffeomorphism symmetry in the continuum. The approach connects canonical LQG, spin foams, and tensor-network methods, offering a path to define the continuum physical Hilbert space and a rigorous projector onto physical states. The work highlights how coarse graining, vacuum selection, and renormalization concepts interplay to reveal a potentially unique continuum limit and informs future directions in spin nets, BF vacua, and matter couplings.

Abstract

The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit.

The continuum limit of loop quantum gravity - a framework for solving the theory

TL;DR

This paper addresses constructing the continuum dynamics of loop quantum gravity by promoting the kinematical inductive-limit framework to a dynamical one via refined amplitudes and dynamical embedding maps. It introduces an iterative, truncation-based coarse-graining procedure that yields cylindrically consistent amplitude maps, effectively modeling a background-independent renormalization flow and aiming to restore diffeomorphism symmetry in the continuum. The approach connects canonical LQG, spin foams, and tensor-network methods, offering a path to define the continuum physical Hilbert space and a rigorous projector onto physical states. The work highlights how coarse graining, vacuum selection, and renormalization concepts interplay to reveal a potentially unique continuum limit and informs future directions in spin nets, BF vacua, and matter couplings.

Abstract

The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit.

Paper Structure

This paper contains 8 sections, 18 equations, 1 figure.

Table of Contents

  1. Solving the dynamics of loop quantum gravity
  2. Continuum limit in canonical loop quantum gravity
  3. Kinematical understanding of the continuum limit
  4. Continuum limit for the dynamics of the theory
  5. Renormalization flow and scale in background independent theories
  6. (Decorated) Tensor network renormalization for spin nets and spin foams
  7. Diffeomorphism symmetry in the discrete, constraints and divergencies
  8. Summary and Outlook

Figures (1)

  • Figure 1: Left: Two vertices in a tensor network, encoded in the matrices $M$, are sharing two edges with labels $\{\alpha,\beta\}$, which have a total range of $\chi^2$. Right: From the singular value decomposition we can define the map $V$ depicted as a three--valent vertex, where we restrict the label $i$ of the singular values to be $\leq \chi$.