Hamiltonian flows of Lorentzian polyhedra: Kapovich-Millson phase space and 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 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 threedimensional Minkowski space 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 gl N (R) Lie algebra and that they generate a GLN (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 GLN (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. I. INTRODUCTIONThe question of quantum gravity is a great physical motivation to explore the fundamental structures of geometry. From a conservative perspective, its goal can be understood as defining a intrinsically discrete notion of geometry, due to the introduction of the Planck length, while still carrying an action of the continuous group of diffeomoprhisms. This would achieve the quantization of geometry. Following this line of research, the loop quantum gravity framework proposes quantum states of 3d geometry and aims at describing their evolution thereby generating the 4d space-time (see [1-3] for reviews). For space-like 3d hypersurfaces, those spin network states are graphs dressed with algebraic data from the representation theory of the Lie group SU(2). These can be interpreted as discrete geometries, named "twisted geometries" generalizing Regge triangulations [4,5]. Their fundamental building blocks are SU(2) intertwiners, i.e. SU(2)-invariant states in the tensor product of SU(2) representations, that are understood as the quantum counterpart of 3d polyhedra [6][7][8]. These quantum polyhedra are then glued together to form a discrete quantum 3d geometry. The purpose of the present paper is to investigate the extension of the standard framework to time-like hypersurfaces, with spin network states made from SU(1, 1) intertwiners representing quantized Lorentzian polyhedra. This is directly applicable to loop quantum ...