Space of the vertices of relativistic spin networks by A. Barbieri

This week I been reading quite a short paper about three and four dimensional vertices in LQG.

In this paper the general solution to the constraints that define relativistic spin neworks vertices is given and their relations with 3-dimensional quantum tetrahedra are discussed.

Quantum  Tetrahedra

The author’s basic idea is to use as fundamental variables the bivectors defined by 2-dimensional faces of the tetrahedron rather than the edges. Since the set of bivectors in n dimensions is isomorphic to the dual of the Lie algebra of SO(n), the Hilbert space of a quantum bivector is defned as the orthogonal sum of the  irreducible representations of the universal covering of SO(n);

If n = 3 the covering group is SU(2),

while

if n = 4 it is Spin(4) ≃ SU(2) x SU(2).

The author focuses on these two subcases i.e n=3 and n=4. The next step  is to take the tensor product of four such spaces (one for each face) and impose some constraints which so that the faces have suitable properties, which in terms of bivectors. These constraints are that

  • The bivectors defined by the faces must add up to zero.
  • The bivectors are all simple i.e. they can be written as wedge products of two vectors.
  • The sum of two bivectors is also simple.

In three dimensions all bivectors are simple and the only constraint is that the faces close: the resulting space is that of 4-valent SU(2)-spin network vertices. Whilst in n four dimensions there are non-simple bivectors

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