Deformations of Polyhedra and Polygons by the Unitary Group by Livine

This week whilst preparing some calculations I have been looking at the paper ‘Deformations of Polyhedra and Polygons by the Unitary Group’. In the paper the author inspired by loop quantum gravity, the spinorial formalism and the structures of twisted geometry discusses the phase space of polyhedra in three dimensions and its quantization, which serve as basic building of the kinematical states of discrete geometry.

They show that the Grassmannian space U(N)/(U(N − 2) × SU(2)) is the space of framed convex polyhedra with N faces up to 3d rotations. The framing consists in the additional information of a U(1) phase per face. This provides an extension of the Kapovich-Milson phase space  for polyhedra with fixed number of faces and fixed areas for each face – see the post:

Polyhedra in loop quantum gravity

They describe the Grassmannian as the symplectic quotient C2N//SU(2), which provides canonical complex variables for the Poisson bracket. This construction allows a natural U(N) action on the space of polyhedra, which has two main features. First, U(N) transformations act non-trivially on polyhedra and change the area and shape of each individual face. Second, this action is cyclic: it allows us to go between any two polyhedra with fixed total area  – sum of the areas of the faces.

On quantization, the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners, which is interpreted as the space of quantum polyhedra. By performing a canonical quantization from the complex variables of C2N//SU(2) all the classical features are automatically exported to the quantum level. Each face carries now a irreducible representation of SU(2) – a half-integer spin j, which defines the area of the face. Intertwiners are then SU(2)-invariant states in the tensor product of these irreducible representations. These intertwiners are the basic
building block of the spin network states of quantum geometry in loop quantum gravity.

The U(N) action on the space of intertwiners changes the spins of the faces and each Hilbert space for fixed total area defines an irreducible representation of the unitary group U(N). The U(N) action is cyclic and allows us to generate the whole Hilbert space from the action of U(N) transformation on the highest weight
vector. This construction provides coherent intertwiner states peaked on classical polyhedra.

At the classical level, we can use the U(N) structure of the space of polyhedra to compute the averages of polynomial observables over the ensemble of polyhedra and  to use the Itzykson-Zuber formula from matrix models  as a generating functional for these averages. It computes the integral over U(N) of the exponential of the matrix elements of a unitary matrix tensor its complex conjugate.

At the quantum level the character formula, giving the trace of unitary transformations either over the standard basis or the coherent intertwiner basis, provides an extension of the Itzykson-Zuber formula. It allows us in principle to generate the expectation values of all polynomial observables and so their spectrum.

This paper defines and describe the phase space of framed polyhedra, its parameterization in terms of spinor variables and the action of U(N) transformations. Then it shows  how to compute the averages and correlations of polynomial observables using group integrals over U(N) and the Itzykson-Zuber integral as a generating function. It discusses the quantum case, with the Hilbert space
of SU(2) intertwiners, coherent states and the character formula.

The paper also investigates polygons in two dimensions and shows that the unitary group is replaced by the orthogonal group and that
the Grassmannian Ø(N)/(Ø(N −2)×SO(2)) defines the phase space for framed polygons. It then discusses the issue of gluing such polygons together into a consistent 2d cellular decomposition, as a toy model for the gluing of framed polyhedra into 3d discrete manifolds. These constructions are relevant to quantum gravity in 2+1 and 3+1 dimensions, especially to discrete approaches based on a description of the geometry using glued polygons and polyhedra such as loop quantum gravity  and dynamical
triangulations.

The paper’s goal is to clarify how to parametrize the set of polygons or polyhedra and their deformations, and to introduce mathematical tools to compute the average and correlations of observables over the ensemble of polygons or polyhedra at the classical level and then the spectrum and expectation values of geometrical operators on
the space of quantum polygons or polyhedra at the quantum level.

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