Deformations of Lorentzian Polyhedra: Kapovich-Millson phase space and SU(1,1) Intertwiners

TL;DR

This paper develops a Lorentzian phase space for polyhedra with space-like faces using SU(1,1) algebra, revealing symmetries and deformations relevant for quantum gravity and holography.

Contribution

It introduces a Lorentzian analog of the Kapovich-Millson phase space, identifying invariant observables and their algebraic structure, and extends these concepts to quantum intertwiners in loop quantum gravity.

Findings

The phase space is constructed using SU(1,1) spinors.

Area-preserving transformations form a gl_N(R) Lie algebra.

Quantum Lorentzian polyhedra are described as SU(1,1) intertwiners.

Abstract

We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with $N$ faces. Starting with the Schwinger representation of the $\mathfrak{su}(1,1)$ Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for a space-like vectors in the three-dimensional Minkowski space $\mathbb{R}^{1,2}$. Considering $N$ copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with $N$ faces whose normal vectors are space-like, up to Lorentz transformations. We identify a generating set of $SU(1,1)$-invariant observables, whose flow by the Poisson bracket generate both area-preserving and area-changing deformations. We further show that the area-preserving observables form a $\mathfrak{gl}_{N}(\mathbb{R})$ Lie algebra and that they generate a $GL_{N}(\mathbb{R})$ action on Lorentzian polyhedra at fixed total area. That action is cyclic and all Lorentzian polyhedra can be obtained from a totally squashed polyhedron (with only two non-trivial faces) by a $GL_{N}(\mathbb{R})$ transformation. All those features carry on to the quantum level, where quantum Lorentzian polyhedra are defined as $SU(1,1)$ intertwiners between unitary $SU(1,1)$-representations from the principal continuous series. Those $SU(1,1)$-intertwiners are the building blocks of spin network states in loop quantum gravity in 3+1 dimensions for time-like slicing and the present analysis applies to deformations of the quantum geometry of time-like boundaries in quantum gravity, which is especially relevant to the study of quasi-local observables and holographic duality.