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

By L. Freidel, Simone Speziale. 2010

260 citations recorded by the source corpus.

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 additional angle related to the extrinsic curvature. These quantities do not define a Regge geometry, since they include extrinsic data, but a looser notion of discrete geometry which is twisted in the sense that it is locally well-defined, but the local patches lack a consistent gluing among each other. We give the Poisson brackets among the new variables, and exhibit a symplectomorphism which maps them into the Poisson brackets of loop gravity. The new parametrization has the advantage of a simple description of the gauge-invariant reduced phase space, which is given by a product of phase spaces associated to edges and vertices, and it also provides an Abelianization of the SU(2) connection.more » The results are relevant for the construction of coherent states and, as a byproduct, contribute to clarify the connection between loop gravity and its subset corresponding to Regge geometries.« less

Subfields: Neurology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ec2a0b408b0def95eb1b83c1b81f29fbdbc284a4

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Citations in the corpus

Twistors to twisted geometries (2010, methodology): “(23) (3)With respect to the opposite orientation taken in [1], this different choice affects the isomorphism with T SU(2) in a minor way, see (34b) below.”
Towards spinfoam cosmology (2010, background): “of Thiemann’s complexifier’s coherent states [9–18], (iii) induce Speziale-Livine coherent tetrahedra [2, 36, 37] on the nodes, and (iv) are equal to the the Freidel-Speziale coherent states [19, 20] for large spins.”
Zakopane lectures on loop gravity (2011, background): “Freidel and Speziale give a slightly different definition of coherent states [88].”
A new look at loop quantum gravity (2010, background): “Freidel and Speziale give a slightly different definition of coherent states [21].”
Loop Quantum Gravity (2003, background): “Such discretized metrics are called ‘twisted geometries’ [25].”