U(N) Intertwiners, Part 2 - Schwinger Boson Representation and Intertwiner Algebra, May 20, 2021
Channel: Deepak Vaid
Date: 2021-05-21

This is the second in a series of lectures on U(N) coherent states in Loop Quantum Gravity which describe semiclassical states of geometry corresponding to a classical polyhedra with N faces.

In the first part I explained the basic ideas behind the geometry of a tetrahedron and general polyhedra in the framework of Loop Quantum Gravity.

In this lecture I talk about the Schwinger boson representation of the angular momentum algebra associated with the Hilbert space of a single edge. I explain how this is extended to the Hilbert space of an intertwiner and introduce operators with correspond interaction terms between faces and are invariant under global SU(2) transformations acting on the intertwiner Hilbert space. These interaction terms can then be shown to form a Lie algebra which is that of U(N) - the group of unitary N x N matrices.

Upcoming lectures will complete the discussion of the intertwiner operators and their algebra, introduce the spinorial representation of the LQG phase space, introduce SU(2) coherent states, intertwiner coherent states which will then lead us to see how the Grassmannian arises in the description of the intertwiner coherent state space.

May 21, 2021