This week I have been reading a paper by Freidel and Livine which investigates the geometry of the space of N-valent SU(2)intertwiners. In this paper, the authors propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian GrN,2, and have a geometrical interpretation in terms of framed polyhedra and are related to coherent intertwiners – see the post
Loop quantum gravity is a canonical quantization of general relativity where the quantum states of geometry are the so-called spin network states. A spin network is based on a graph dressed up with half-integer spins je on its edges and intertwiners iv on its vertices. The spins define quanta of area while the intertwiners describe chunks of space volume. The dynamics then acts on the spins je and intertwiners iv, and can also deform the underlying graph .
In this paper, the authors focus on the structure of the space of intertwiners describing the chunk of space. They focus their study on a region associated with a single vertex of a graph and arbitrary high valency. Associated with this setting there is a classical geometrical description. To each edge going out of this vertex there is associated a dual surface element or face whose area is given by the spin label. The collection of these faces encloses a 3-dimensional volume whose boundary forms a 2-dimensional polygon with the topology of a sphere. This 2d-polygon is such that each vertex is trivalent. At the quantum level the choice of intertwiner iv attached to the vertex describes the shape of the full dual surface and gives the volume contained in that surface.
The space of N-valent intertwiners carries an irreducible representation of the unitary group U(N). These irreducible representations of U(N) are labeled by one integer: the total area of the dual surface – defined as the sum of the spins coming
through this surface. The U(N) transformations deform of the shape of the intertwiner at fixed area. This provides a clean geometric interpretation to the space of intertwiners as wavefunctions over the space of classical N-faced polyhedron. It also leads to a clearer picture of what the discrete surface dual to the intertwiner should look like in the semi-classical regime.
In this work the authors present the explicit construction for new coherent states which are covariant under U(N), then compute their norm, scalar product and show that they provide an overcomplete
basis. They also compute their semi-classical expectation values and uncertainties and show that they are simply related to the Livine-Speziale coherent intertwiners used in the construction of the Engle-Pereira-Rovelli-Livine (EPRL) and Freidel-Krasnov (FK) spinfoam models and their corresponding semi-classical boundary states. These new coherent states confirm the polyhedron interpretation of the intertwiner space and show the relevance of the U(1) phase/frame attached to each face, which appears very similar to the extra phase entering the definition of the discrete twisted geometries for loop gravity.
Considering the space of all N-valent intertwiners, they decompose it separating the intertwiners with different total area :
Each space H(J)N at fixed total area carries an irreducible representation of U(N). The u(N) generators Eij are quadratic operators in the harmonic oscillators of the Schwinger representation of the su(2)-algebra. the full space HN as a Fock space by introducing annihilation and creation operators
The full space HN is established as a Fock space by introducing annihilation and creation operators Fij , F†ij , which allow transitions between intertwiners with different total areas. These creation operators can be used to define U(N) coherent states |J, zi 〉 ∝ F†z| 0〉labeled by the total area J and a set of N spinors zi . These states turn out to have very interesting properties.
- They transform simply under U(N)-transformations u |J, zi 〉 = |J(u z)i 〉
- They get simply rescaled under global GL(2,C) transformation acting on all the spinors: In particular, they are invariant under global SL(2,C) transformations.
- They are coherent states and are obtained by the action of U(N) on highest weight states. These highest weight vectors correspond to bivalent intertwiners such as the state defined by
- For large areas J, they are semi-classical states peaked around the expectation values for the u(N) generators:
- The scalar product between two coherent states is easily computed:
- They are related to the coherent and holomorphic intertwiners, writing |j,zi〉 for the usual group-averaged tensor product of SU(2) coherent states defining the coherent intertwiners, we have:
The authors believe that this U(N) framework opens the door to many applications in loop quantum gravity and spinfoam models.
- Deformations of polyhedra and polygons by the unitary group
- Quantum geometry from phase space reduction
- Holomorphic Factorization for a Quantum Tetrahedron
- A new spinfoam vertex for quantum gravity