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On the regularization ambiguities in loop quantum gravity

Alejandro Perez

TL;DR

The paper investigates the m-ambiguity in loop quantum gravity, the freedom to choose the SU(2) representation used to regularize the curvature in Thiemann's Hamiltonian constraint, and the consequences for the continuum limit. It analyzes 2+1 gravity (canonical and covariant/Spin Foam) and 3+1 gravity, showing that in 2+1 gravity only the fundamental representation ($m=1/2$) yields a consistent, topological continuum theory, with linear combinations reproducing the same physics; higher representations introduce spurious local DOF and disrupt the continuum limit. In 3+1 gravity, the evidence points in the same direction: regularizations with $m>1$ produce extra local excitations (e.g., spin-2 modes for $SO(3)$) that are undesirable and likely vanish in the physical inner product, though a complete proof awaits a well-defined H_phys. Overall, the work argues for a highly constrained quantum dynamics in LQG where only a restricted set of regularizations produce physically meaningful continuum physics, with important implications for spin foam models and non-perturbative gravity. The results contribute to narrowing the space of admissible quantum theories and motivate further study of the physical inner product and covariant formulations to achieve robust predictive power.

Abstract

One of the main achievements of LQG is the consistent quantization of the Wheeler-DeWitt equation which is free of UV problems. However, ambiguities associated to the intermediate regularization procedure lead to an apparently infinite set of possible theories. The absence of an UV problem is intimately linked with the ambiguities arising in the quantum theory. Among these ambiguities there is the one associated to the SU(2) unitary rep. used in the diffeomorphism covariant pointsplitting regularization of nonlinear funct. of the connection. This ambiguity is labelled by a halfinteger m and, here, it is referred to as the m-ambiguity. The aim of this paper is to investigate the important implications of this ambiguity./ We first study 2+1 gravity quantized in canonical LQG. Only when the regularization of the quantum constraints is performed in terms of the fundamental rep. of the gauge group one obtains the usual TQFT. In all other cases unphysical local degrees of freedom arise at the level of the regulated theory that conspire against the existence of the continuum limit. This shows that there is a clear cut choice in the quantization of the constraints in 2+1 LQG./ We then analyze the effects of the ambiguity in 3+1 gravity exhibiting the existence of spurious solutions for higher unit. rep. quantizations of the Hamiltonian constraint. Although the analysis is not complete in D=3+1--due to the difficulties associated to the definition of the physical inner product--it provides evidence supporting the definitions quantum dynamics of loop quantum gravity in terms of the fundamental representation of the gauge group as the only consistent possibilities. If the gauge group is SO(3) we find physical solutions associated to spin-two local excitations.

On the regularization ambiguities in loop quantum gravity

TL;DR

The paper investigates the m-ambiguity in loop quantum gravity, the freedom to choose the SU(2) representation used to regularize the curvature in Thiemann's Hamiltonian constraint, and the consequences for the continuum limit. It analyzes 2+1 gravity (canonical and covariant/Spin Foam) and 3+1 gravity, showing that in 2+1 gravity only the fundamental representation () yields a consistent, topological continuum theory, with linear combinations reproducing the same physics; higher representations introduce spurious local DOF and disrupt the continuum limit. In 3+1 gravity, the evidence points in the same direction: regularizations with produce extra local excitations (e.g., spin-2 modes for ) that are undesirable and likely vanish in the physical inner product, though a complete proof awaits a well-defined H_phys. Overall, the work argues for a highly constrained quantum dynamics in LQG where only a restricted set of regularizations produce physically meaningful continuum physics, with important implications for spin foam models and non-perturbative gravity. The results contribute to narrowing the space of admissible quantum theories and motivate further study of the physical inner product and covariant formulations to achieve robust predictive power.

Abstract

One of the main achievements of LQG is the consistent quantization of the Wheeler-DeWitt equation which is free of UV problems. However, ambiguities associated to the intermediate regularization procedure lead to an apparently infinite set of possible theories. The absence of an UV problem is intimately linked with the ambiguities arising in the quantum theory. Among these ambiguities there is the one associated to the SU(2) unitary rep. used in the diffeomorphism covariant pointsplitting regularization of nonlinear funct. of the connection. This ambiguity is labelled by a halfinteger m and, here, it is referred to as the m-ambiguity. The aim of this paper is to investigate the important implications of this ambiguity./ We first study 2+1 gravity quantized in canonical LQG. Only when the regularization of the quantum constraints is performed in terms of the fundamental rep. of the gauge group one obtains the usual TQFT. In all other cases unphysical local degrees of freedom arise at the level of the regulated theory that conspire against the existence of the continuum limit. This shows that there is a clear cut choice in the quantization of the constraints in 2+1 LQG./ We then analyze the effects of the ambiguity in 3+1 gravity exhibiting the existence of spurious solutions for higher unit. rep. quantizations of the Hamiltonian constraint. Although the analysis is not complete in D=3+1--due to the difficulties associated to the definition of the physical inner product--it provides evidence supporting the definitions quantum dynamics of loop quantum gravity in terms of the fundamental representation of the gauge group as the only consistent possibilities. If the gauge group is SO(3) we find physical solutions associated to spin-two local excitations.

Paper Structure

This paper contains 13 sections, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The m-ambiguity in quantum canonical 2+1 Riemannian gravity
  3. Linear combinations
  4. Covariant spin foams
  5. The $m$-ambiguity in 3+1 gravity
  6. Quantization of the Hamiltonian constraint
  7. Constructing solutions
  8. Solutions from an algebraic argument
  9. Example in quantum mechanics
  10. Solutions of Thiemann's Hamiltonian
  11. Linear combinations
  12. Discussion
  13. Acknowledgement

Figures (2)

  • Figure 1: Infinitesimal plaquette-delta-distributions can be integrated and fussioned with the corresponding modification of the face amplitude. The lines in the previous figure represent the holonomy around plaquettes of the regulator in the representation denoted by the latin index $k$ and $j$ in this case. the dark boxes denote integration of the generalized connection associated to the corresponding edge. The previous equation is a trivial consequence of the orthogonality of unitary irreducible representations of $SU(2)$. The plaquettes in the picture are square as a matter of simplicity.
  • Figure 2: Interpretation of the solutions (\ref{['final']}) for $m=1$ as graviton excitations. Starting from a solution to the constraints given by a diff-invariant spin network with a vertex with no exceptional edge we can construct a new solution as explained in Section \ref{['pilin']} and illustrated here. The solution space is parametrized by the quantum numbers $\alpha$ and $\beta$ in this figure. The dotted region corresponds to a single point in the spin network graph.