Deformations of polyhedra and polygons by the unitary group
We introduce the set of framed (convex) polyhedra with N faces as the symplectic quotient C{sup 2N}//SU(2). A framed polyhedron is then parametrized by N spinors living in C{sup 2} satisfying suitable closure constraints and defines a usual convex polyhedron plus extra U(1) phases attached to each face. We show that there is a natural action of the unitary group U(N) on this phase space, which changes the shape of faces and allows to map any (framed) polyhedron onto any other with the same total (boundary) area. This identifies the space of framed polyhedra to the Grassmannian space U(N)/ (SU(2)×U(N−2)). We show how to write averages of geometrical observables (polynomials in the faces' area and the angles between them) over the ensemble of polyhedra (distributed uniformly with respect to the Haar measure on U(N)) as polynomial integrals over the unitary group and we provide a few methods to compute these integrals systematically. We also use the ItzyksonZuber formula from matrix models as the generating function for these averages and correlations. In the quantum case, a canonical quantization of the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners (or, in other words, SU(2)invariant states in tensor productsmore »
 Authors:

^{[1]}
 Laboratoire de Physique, ENS Lyon, CNRSUMR 5672, 46 Allée d'Italie, Lyon 69007, France and Perimeter Institute, 31 Caroline St N, Waterloo, Ontario N2L 2Y5 (Canada)
 Publication Date:
 OSTI Identifier:
 22217742
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 54; Journal Issue: 12; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ALGEBRA; GEOMETRY; HILBERT SPACE; IRREDUCIBLE REPRESENTATIONS; MATRICES; PHASE SPACE; POLYNOMIALS; QUANTIZATION; QUANTUM GRAVITY; SPIN; SU2 GROUPS; TRANSFORMATIONS; UNITARY SYMMETRY; VECTORS