U(N) Tools for Loop Quantum Gravity

Etera R. Livine

I will present the recently developed U(N) formalism for loop quantum gravity (LQG). It is based on the fact that the space of N-valent intertwiners for the SU(2) Lie group carries a representation of the unitary group U(N). Since the intertwiners are the main building blocks of LQG's spin network states, this new U(N) action turns out to be very interesting from various perspectives. For instance, it leads to the construction of semi-classical states coherent under the U(N) action, which can be interpreted in term of classical polyhedra with N faces. This allows to truly interpret the corresponding spin network states as discrete classical geometries. This is relevant to both the understanding of LQG and to the construction of spin foam amplitudes. We will also see how this U(N) symmetry has a physical meaning and allows for example to select isotropic spin network states, thus allowing in some precise setting to map LQG to loop quantum cosmology.