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Quantum geometry from phase space reduction

Florian Conrady, Laurent Freidel

TL;DR

This paper builds a bridge between geometric tetrahedra and quantum spin networks by proving an explicit isomorphism between the usual SU(2) intertwiner space and the quantized reduced phase space of tetrahedra, via a Guillemin–Sternberg–Hall construction. It derives a central identity expressing the SU(2) invariant space as an integral over coherent states constrained by the closure condition, effectively integrating over classical tetrahedra, and uses this to reformulate FK spin-foam amplitudes with strongly imposed closure. The authors also perform a saddle-point analysis of a single 4-simplex vertex amplitude to establish the emergence of the 4D Regge action in the large-spin limit and discuss implications for perturbation theory and the geometry of quantum states in LQG. Overall, the work advances a geometrical, reduction-first perspective that unifies spin networks with classical tetrahedral geometry and clarifies asymptotics in spin-foam models.

Abstract

In this work we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin--Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.

Quantum geometry from phase space reduction

TL;DR

This paper builds a bridge between geometric tetrahedra and quantum spin networks by proving an explicit isomorphism between the usual SU(2) intertwiner space and the quantized reduced phase space of tetrahedra, via a Guillemin–Sternberg–Hall construction. It derives a central identity expressing the SU(2) invariant space as an integral over coherent states constrained by the closure condition, effectively integrating over classical tetrahedra, and uses this to reformulate FK spin-foam amplitudes with strongly imposed closure. The authors also perform a saddle-point analysis of a single 4-simplex vertex amplitude to establish the emergence of the 4D Regge action in the large-spin limit and discuss implications for perturbation theory and the geometry of quantum states in LQG. Overall, the work advances a geometrical, reduction-first perspective that unifies spin networks with classical tetrahedral geometry and clarifies asymptotics in spin-foam models.

Abstract

In this work we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin--Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.

Paper Structure

This paper contains 17 sections, 1 theorem, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On the space of shape of tetrahedra
  3. The classical phase space of the polygon
  4. The space of shape of tetrahedra
  5. The quantum geometry of coherent states
  6. Geometric quantisation
  7. Quantisation commutes with reduction
  8. 3-- and 4--valent section
  9. The Quantum equivalence
  10. Some properties of invariant coherent states
  11. Back to the Guillemin--Sternberg isomorphism
  12. The main correspondence
  13. FK model and asymptotics for a single 4--simplex
  14. FK$\gamma$ model
  15. Asymptotics of vertex amplitude
  16. ...and 2 more sections

Key Result

Proposition 5.1

Suppose that $(z_i)$ is a point in $P_{\vec{\jmath}}$ with nonzero norm $||N(z_{i})||_{\vec{\jmath}}$. 1- The first derivatives of $||N(z_{i})||_{\vec{\jmath}}$ for an anti-hermitian $X= X^{a}\tau_{a}$ are given by where $H_{X}(N_{i}) = \sum_{i} j_{i} N_{i}$ is the Hamiltonian constraint and $G=\sum_{i} g_{j_{i}}$ is the metric on the orbits $\mathrm{SL(2},\mathbb{C}) \cdot (z_{i})$. Explicitly,

Figures (1)

  • Figure 1: Dual of a 4--simplex with boundary: the boundary of the 4--simplex intersects dual edges at their center. The intersection points provide the base point for boundary data $n_{e_if_{ij}}, u_{e_i}$.

Theorems & Definitions (1)

  • Proposition 5.1