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A state sum for four-dimensional Lorentzian quantum geometry in terms of edge vectors

Roukaya Dekhil, Matteo Laudonio, Daniele Oriti

Abstract

We present the construction of a new state sum model for $4d$ Lorentzian quantum gravity based on the description of quantum simplicial geometry in terms of edge vectors. Quantum states and amplitudes for simplicial geometry are built from irreducible representations of the translation group, then related to the representations of the Lorentz group via expansors, leading to interesting (and intricate) non-commutative structures. We also show how the new model connects to the Lorentzian Barrett-Crane spin foam model, formulated in terms of quantized triangle bivectors.

A state sum for four-dimensional Lorentzian quantum geometry in terms of edge vectors

Abstract

We present the construction of a new state sum model for Lorentzian quantum gravity based on the description of quantum simplicial geometry in terms of edge vectors. Quantum states and amplitudes for simplicial geometry are built from irreducible representations of the translation group, then related to the representations of the Lorentz group via expansors, leading to interesting (and intricate) non-commutative structures. We also show how the new model connects to the Lorentzian Barrett-Crane spin foam model, formulated in terms of quantized triangle bivectors.
Paper Structure (31 sections, 1 theorem, 138 equations, 4 figures)

This paper contains 31 sections, 1 theorem, 138 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Relevant representation theory of translation and Lorentz groups
  3. Infinite dimensional unitary representations of the Lorentz group
  4. The Harmonic oscillator representation of the translation group
  5. The four dimensional Lorentzian harmonic oscillator
  6. Quantum triangle
  7. Classical triangle
  8. Classical triangle.
  9. Classical bivector.
  10. Quantum triangle
  11. Quantum triangle from quantum edge vectors.
  12. Quantum triangle from the quantum bivector.
  13. Wave functions of quantum triangles and bivectors.
  14. Quantum tetrahedron
  15. Classical tetrahedron
  16. ...and 16 more sections

Key Result

Proposition 1

Consider a triplet of vectors $e_1,e_2,e_3$ on Minkowski space parametrized by the coordinates $\zeta,\lambda,\omega$, such that they form the boundary of a triangle: $\zeta + \lambda + \omega=0$. The Hilbert space of the bivector QuantumBiVector does not depend on which pair of edge vectors are use up to a sign that reflects the orientation of the vector normal to the triangle.

Figures (4)

  • Figure 1: Triangle with edge vectors $e_1,e_2,e_3 \in \mathrm{M}^4$ and closure relation \ref{['Traingle_ClosureCondition']}. In blue is the bivector part.
  • Figure 2: Combinatorics of $\tau$ and its closure constraints from edge vectors.
  • Figure 3: 4-simplex boundary construction: five tetrahedra share five vertices. Each of the four faces of each tetrahedron is identified with one of the faces of the other four tetrahedra. We use the same color and a double dotted line for the identified faces.
  • Figure 4: The tetrahedron amplitude of the Barrett-Crane model BarrettCrane:1999BarrettCrane1

Theorems & Definitions (2)

  • Proposition 1: Invariance under the switching operator
  • proof