U(N) tools for Loop Quantum Gravity: The Return of the Spinor

TL;DR

The paper recasts the SU(2) intertwiner framework of loop quantum gravity in terms of a classical spinor phase space governed by a U(N) structure, showing that quantization recovers the intertwiner Hilbert spaces of holomorphic functionals and enabling the construction of spin-network states by gluing intertwiners on graphs. It derives a classical action principle from a pair of matrices $M$ and $Q$ constrained to reproduce the quantum algebras, then presents two equivalent quantization pictures: a holomorphic polynomial (anti-holomorphic) picture and a dual coherent-state (holomorphic) picture, both yielding the familiar ${E_{ij}}$, ${F_{ij}}$, and ${F^{\dagger}_{ij}}$ operators. The authors extend the framework to full spin networks by reconstructing edge holonomies from spinors and expressing loop holonomies as products of edge group elements encoded in $M$ and $Q$, culminating in generalized holonomies ${\widehat{\mathcal M}}_{\mathcal L}^{\{r_i\}}$ that shift edge spins by $\pm\tfrac{1}{2}$. Finally, they formulate a classical dynamics for spin networks via an action on graphs and analyze a tractable 2-vertex matrix-model, revealing a rich phase-space structure with elliptic, hyperbolic, and parabolic regimes that resemble loop quantum cosmology big-bounce behavior.

Abstract

We explore the classical setting for the U(N) framework for SU(2) intertwiners for loop quantum gravity (LQG) and describe the corresponding phase space in terms of spinors with appropriate constraints. We show how its quantization leads back to the standard Hilbert space of intertwiner states defined as holomorphic functionals. We then explain how to glue these intertwiners states in order to construct spin network states as wave-functions on the spinor phase space. In particular, we translate the usual loop gravity holonomy observables to our classical framework. Finally, we propose how to derive our phase space structure from an action principle which induces non-trivial dynamics for the spin network states. We conclude by applying explicitly our framework to states living on the simple 2-vertex graph and discuss the properties of the resulting Hamiltonian.

Figures (4)

• Figure 1: Focusing on the four legs $(i,j,k,l)$ of the intertwiner, the Plücker relation $Q_{ij}Q_{kl}=Q_{il}Q_{kj}+Q_{ik}Q_{jl}$ on the $Q$-variables becomes the (standard) recoupling relation for $\mathrm{SU}(2)$ intertwiners (more precisely, for holonomy operators acting on $\mathrm{SU}(2)$ intertwiners). This relation is often used in Loop Quantum Gravity when still using states defined as products of Wilson loops instead of spin network states.
• Figure 2: The loop ${\mathcal{L}}=\{e_1,e_2,..,e_n\}$ on the graph $\Gamma$.
• Figure 3: The 2-vertex graph with vertices $\alpha$ and $\beta$ and the $N$ edges linking them.
• Figure 4: We plot the behavior of $\phi(t)$ and $\lambda(t)$ (given by the equations \ref{['conicparam']} and \ref{['regions']}) in the three different regimes for $\gamma=1$ and respectively $\gamma^0=4$ (elliptic regime), $\gamma^0=1$ (hyperbolic regime) and finally $\gamma^0=2$ (parabolic regime). In the first column, we give the polar plots constructed by taking as polar coordinates $(2 \phi,\lambda(\phi))$. The second column gives for $\phi(t)$ and the third one $\lambda(t)$. We observe in those plots the periodical behavior of $\lambda$ (interpreted as the total area of the model) as a function of time in the elliptic case and a behavior analogous to a cosmological big bounce in the other two cases.

Theorems & Definitions (4)

• proof
• proof
• Conjecture 1
• Conjecture 2