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The new vertices and canonical quantization

Sergei Alexandrov

TL;DR

The paper investigates two key facets of the new spin-foam models. First, it shows that a slightly refined restriction on EPRL representations can produce the Ashtekar-Barbero (AB) connection, potentially bringing spin-foam states into closer alignment with LQG, especially when fully projected holonomies along edges are employed. Second, it argues that the standard quantization procedure, which quantizes BF theory first and then imposes simplicity constraints, is inconsistent due to the presence of second-class constraints and the resulting Dirac-bracket symplectic structure, casting doubt on the viability of the EPRL and FK models except possibly in the γ=∞ FK limit. Together, these results imply a nuanced relationship between spin-foam and LQG formalisms and call for a revised quantization approach that respects Dirac bracket structure, while highlighting the FK γ=∞ case as a potential, though still not fully satisfactory, path forward.

Abstract

We present two results on the recently proposed new spin foam models. First, we show how a (slightly modified) restriction on representations in the EPRL model leads to the appearance of the Ashtekar-Barbero connection, thus bringing this model even closer to LQG. Second, we however argue that the quantization procedure used to derive the new models is inconsistent since it relies on the symplectic structure of the unconstraint BF theory.

The new vertices and canonical quantization

TL;DR

The paper investigates two key facets of the new spin-foam models. First, it shows that a slightly refined restriction on EPRL representations can produce the Ashtekar-Barbero (AB) connection, potentially bringing spin-foam states into closer alignment with LQG, especially when fully projected holonomies along edges are employed. Second, it argues that the standard quantization procedure, which quantizes BF theory first and then imposes simplicity constraints, is inconsistent due to the presence of second-class constraints and the resulting Dirac-bracket symplectic structure, casting doubt on the viability of the EPRL and FK models except possibly in the γ=∞ FK limit. Together, these results imply a nuanced relationship between spin-foam and LQG formalisms and call for a revised quantization approach that respects Dirac bracket structure, while highlighting the FK γ=∞ case as a potential, though still not fully satisfactory, path forward.

Abstract

We present two results on the recently proposed new spin foam models. First, we show how a (slightly modified) restriction on representations in the EPRL model leads to the appearance of the Ashtekar-Barbero connection, thus bringing this model even closer to LQG. Second, we however argue that the quantization procedure used to derive the new models is inconsistent since it relies on the symplectic structure of the unconstraint BF theory.

Paper Structure

This paper contains 12 sections, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. EPRL vs. LQG
  3. EPRL constraints and AB connection
  4. Lorentzian theory
  5. Euclidean theory
  6. Casimir operators
  7. Projected holonomies and projected connections
  8. Simplicity constraints revisited
  9. A simple example
  10. Simplicity constraints in spin foam models
  11. Conclusions
  12. Covariant AB connection

Figures (1)

  • Figure 1: Projected spin network and the structure of its intertwiners.