U(N) Coherent States for Loop Quantum Gravity
Laurent Freidel, Etera R. Livine
TL;DR
The paper addresses the structure of the space of N-valent SU(2) intertwiners in loop quantum gravity by building a holomorphic, U(N)-covariant framework. It introduces a Fock-space interpretation with creation/annihilation operators $F_{ij},F_{ij}^ -$ and defines a family of U(N) coherent states $|J,z_i)$ labeled by spinors $z_i$, which transform covariantly under ${\mathrm{SL}}(N,{\mathbb C})$ and are labeled by the Grassmannian Gr_{2,N}. These states have norms, scalar products, and expectation values that reproduce a semi-classical polyhedral geometry: the normals ${\vec J}(z_i)$ define a framed $N$-faced polyhedron with total area $2J$, and the closure constraint governs their classical limit. A new resolution of identity on intertwiners is given, and a precise link to the standard coherent intertwiners used in EPRL-FK spinfoam models is established, suggesting broad applicability to spin-network dynamics, twisted geometries, and semi-classical boundary states. Overall, the work provides a robust, geometrically transparent, and technically powerful handle on intertwiner spaces with potential impact on spinfoam amplitudes and the semiclassical regime of loop quantum gravity.
Abstract
We investigate the geometry of the space of N-valent SU(2)-intertwiners. We propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian Gr(N,2), they possess a direct geometrical interpretation in terms of framed polyhedra and are shown to be related to the well-known coherent intertwiners.