Polyhedral quantum geometry

This week I’ve been reading Spinfoams: Simplicity Constraints and Correlation Functions PhD thesis by Ding. I’m reviewing the  section on Polyhedral quantum geometry  which is relevant to my work on the quantum tetrahedron.

Consider the truncation of the LQG Hilbert space HLQG and restrict ourself to a single graph Hilbert space H(Γ) and decompose it in terms of SU(2)-invariant spaces Hn associated to each node n. Here I’ll briefly review the state that this node space Hn is the quantization of the space of shapes of the geometry of solids figures tetrahedra, or more general polyhedra . See the posts:

Let’s start with the classical phase space of shapes of a flat polyhedron in Rwith fixed area. A classically flat three-dimensional polyhedron can be described by a set of L vectors Al, l = 1…L, satisfying the following closure constraint:

ding131

Here the L vectors Al can be interpreted as the vectorial areas of the L triangles in the boundary of the polyhedron, in the sense that the norm al = |Al| is the area of the polygon l and normalized vector nl = Al/|Al| is the normal when embedded in to a R3 Euclidean space.
To introduce a symplectic structure, one can associate to each normal Aia generator of the algebra of SO(3).

ding132 A quantum representation of this Poisson algebra is precisely defined by the generators of SU(2) on the space Hn for a 4-valent node n. The operator corresponding to the area al = |Al| is the Casimir of the representation jl, therefore the space quantizes
the space of the shapes of the tetrahedron with areas jl(jl +1). Furthermore, the Hamiltonian flow of G, generates the rotations of the tetrahedron in R3.

By imposing

ding131

and factoring out the orbits of this flow, one obtains the intertwiner space Kn.

In this way, one gives an intertwiner a geometrical interpretation in terms of quantum polyhedron.

There is a  relation among spinfoam formalism, kinematical Hilbert space and polyhedral quantum geometry. For example the boundary space of the simplicial EPRL spinfoam model can be obtained from simplicity constraints, which is the simplicial truncation of LQG kinematical Hilbert space and the boundary state has a geometrical interpretation in terms of quantum tetrahedron geometry. This consistent picture can be generalized into an arbitrary-valence spinfoam formalism. It is also possible to compute the two-point correlation function of Lorentian EPRL spinfoam model and show it matches the one from Regge geometry.

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