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The EPRL intertwiners and corrected partition function

Wojciech Kamiński, Marcin Kisielowski, Jerzy Lewandowski

Abstract

Do the SU(2) intertwiners parametrize the space of the EPRL solutions to the simplicity constraint? What is a complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective for n-valent vertex in case when it is a map from SO(3) into SO(3)xSO(3) representations. We find, however, that the EPRL map is not isometric. In the consequence, in order to be written in a SU(2) amplitude form, the formula for the partition function has to be rederived. We do it and obtain a new, complete formula for the partition function. The result goes beyond the SU(2) spin-foam models framework.

The EPRL intertwiners and corrected partition function

Abstract

Do the SU(2) intertwiners parametrize the space of the EPRL solutions to the simplicity constraint? What is a complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective for n-valent vertex in case when it is a map from SO(3) into SO(3)xSO(3) representations. We find, however, that the EPRL map is not isometric. In the consequence, in order to be written in a SU(2) amplitude form, the formula for the partition function has to be rederived. We do it and obtain a new, complete formula for the partition function. The result goes beyond the SU(2) spin-foam models framework.

Paper Structure

This paper contains 17 sections, 5 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our proposal for a partition function of the EPRL model
  3. Partition functions for the spin-foam models of 4-gravity: definition
  4. Partition functions for the spin-foam models of 4-gravity: the amplitude form
  5. The EPRL map
  6. The problem with the EPRL intertwines
  7. A solution
  8. Example of the edge amplitude $h^{\bar{b}a}$ showing that the EPRL map is not isometry
  9. Injectivity of the map ${\cal I}\mapsto\iota^{\rm EPRL}({\cal I})$
  10. Statement of the result
  11. Proof of the theorem
  12. Auxiliary definitions
  13. Useful inequalities
  14. Important maps
  15. Relation among intertwiners
  16. ...and 2 more sections

Key Result

Theorem 1

For any sequence $(k_1,\ldots, k_n)\in {\mathbb N}$ such that $(j_1^+,\ldots, j_n^+)\in {\mathbb N}$, the map is injective.

Figures (5)

  • Figure 1: According to this rule, given an edge $e$ ( $e'$ ) contained in incoming (outgoing) face $f$, the indices of $P_e$ ( $P_{e'}$ ) corresponding to ${\cal H}(f)$ are assigned to the beginning and, respectively, to the end of the edge. The oriented arc only marks the orientation of the polygonal face $f$.
  • Figure 2: An intertwiner proportional to $C^{k_1}_{k_2k_3}$, $k_{12}=k_1+k_2-k_3$ and etc.
  • Figure 3: An equality between ${\mathcal{H}}_{1/2}^{2k_1}\otimes{\mathcal{H}}_{1/2}^{2k_2}$ and ${\mathcal{H}}_{1/2}^{2k_1+2k_2}$.
  • Figure 4: An equality $P^{k_1+k_2+k_3}\circ P^{k_1+k_2}\otimes {\mathbb I}=P^{k_1+k_2+k_3}$.
  • Figure 5: Intertwiner proportional to $C^{k_1 A_1}_{j^+_1 B_1 j^-_1 C_1} C^{j^+_2 B_2 j^-_2 C_2}_{k_2 A_2}C^{j^+_1 B_1 }_{j^+_2 B_2j^+_\alpha B} C^{j^-_1 C_1 }_{j^-_2 C_2j^-_\alpha C}C_{k_\alpha A}^{j^+_\alpha B j^-_\alpha C}$.

Theorems & Definitions (11)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 1 more