Twisted geometries: A geometric parametrisation of SU(2) phase space

  title={Twisted geometries: A geometric parametrisation of SU(2) phase space},
  author={Laurent Freidel and Simone Speziale},
  journal={Physical Review D},
A cornerstone of the loop quantum gravity program is the fact that the phase space of general relativity on a fixed graph can be described by a product of SU(2) cotangent bundles per edge. In this paper we show how to parametrize this phase space in terms of quantities describing the intrinsic and extrinsic geometry of the triangulation dual to the graph. These are defined by the assignment to each face of its area, the two unit normals as seen from the two polyhedra sharing it, and an… 
Null twisted geometries
We define and investigate a quantisation of null hypersurfaces in the context of loop quantum gravity on a fixed graph. The main tool we use is the parametrisation of the theory in terms of twistors,
Spinning geometry = Twisted geometry
It is well known that the SU(2)-gauge invariant phase space of loop gravity can be represented in terms of twisted geometries. These are piecewise-linear-flat geometries obtained by gluing together
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The twisted geometries of spin network states are described by simple twistors, isomorphic to null twistors with a time-like direction singled out. The isomorphism depends on the Immirzi parameter,
On Geometry and Symmetries in Classical and Quantum Theories of Gauge Gravity
Spin Foam and Loop approaches to Quantum Gravity reformulate Einstein's theory of relativity in terms of connection variables. The metric properties are encoded in face bivectors/conjugate fluxes
Polyhedra in loop quantum gravity
Interwiners are the building blocks of spin-network states. The space of intertwiners is the quantization of a classical symplectic manifold introduced by Kapovich and Millson. Here we show that a
Encoding Curved Tetrahedra in Face Holonomies: Phase Space of Shapes from Group-Valued Moment Maps
We present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a
SU(2) graph invariants, Regge actions and polytopes
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Hamiltonian flows of Lorentzian polyhedra: Kapovich-Millson phase space and SU(1, 1) intertwiners
  • E. Livine
  • Mathematics
    Journal of Mathematical Physics
  • 2019
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
At the Corner of Space and Time
We perform a rigorous piecewise-flat discretization of classical general relativity in the first-order formulation, in both 2+1 and 3+1 dimensions, carefully keeping track of curvature and torsion
1 8 Ju l 2 01 8 Deformations of Lorentzian Polyhedra : Kapovich-Millson phase space and SU ( 1 , 1 ) Intertwiners
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


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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
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The symplectic geometry of polygons in Euclidean space
We study the symplectic geometry of moduli spaces M r of polygons with xed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is
Twistors to twisted geometries
In a previous paper we showed that the phase space of loop quantum gravity on a fixed graph can be parametrized in terms of twisted geometries, quantities describing the intrinsic and extrinsic
Quantum theory of geometry: III. Non-commutativity of Riemannian structures
The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures - such as triad and area operators
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We introduce a new spinfoam vertex to be used in models of 4d quantum gravity based on SU(2) and SO(4) BF theory plus constraints. It can be seen as the conventional vertex of SU(2) BF theory, the