Holomorphic Factorization for a Quantum Tetrahedron by Freidel, Krasnov and Livine

I’ve  been reviewing this paper during the last week or so, it’s quite an advanced treatment of the quantum tetrahedron from the point of view of symplectic manifolds and Kahler manifolds. I’ve felt it was well worth doing this because it adds an deeper level to my understanding of the structure and properties of the quantum Tetrahedron. I’ll be updating this post as I work through the paper!

In this paper the authors provide a holomorphic description of the Hilbert space H j1…jn  of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for  the decomposition of identity in Hj1…jn.

The integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. The results provide a new interpretation for this quantity as being, in the limit of large  conformal  dimensions, the exponential of the Kahler potential of the symplectic manifold whose quantization gives Hj1 ,…,jn .

For the case n=4, the symplectic manifold in question has the interpretation of the space of shapes of a geometric tetrahedron with fixed face areas, and the results provide a description for the quantum tetrahedron in terms of holomorphic coherent states. The authors describe how the holomorphic intertwiners are related to the usual real ones. The semi-classical analysis of  overlap coefficients in the case of large spins allows  an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron to be obtained. The results of this are of direct relevance for the subjects of loop quantum gravity and spin foams and also for the  bulk/boundary correspondence.


The main object of interest in the paper is the space

holomorphic equ 1

of SU(2)-invariant tensors (intertwiners) in the tensor product of n irreducible SU(2) representations, dim(Vj) =dj = 2j + 1. This space is naturally a Hilbert space. It is finite-dimensional, with the dimension given by the classical formula:

holomorphic equ 2

The intertwiners play the central role in the quantum geometry or spin foam approach to quantum gravity. In quantum geometry (loop quantum gravity) approach the space of states of  geometry is spanned by the so-called spin network states based on a graph. This space is obtained by tensoring together the Hilbert spaces L2(G)of square integrable functions on the group G = SU(2) – one for every edge e of the underlying graph – while contracting them at vertices v with invariant tensors to form a gauge-invariant state.

Using the Plancherel decomposition the spin network Hilbert space can therefore be written as:

holomorphic equ 2a

which is the product of intertwiner spaces one for each vertex v of the graph.

The first non-trivial case that gives a non trivial dimension of the Hilbert space of intertwiners is n= 4. A beautiful geometric interpretation of states from Hj1,…,j4 has been proposed as seen in the post: Quantum Tetrahedra and Simplicial Spin Networks, where it was seen that the Hilbert space in this case can be obtained via the process of quantisation of the space of shapes of a geometric tetrahedron in R3 whose face areas are fixed to be equal to j1…j4.This space of shapes is a phase space of real dimension two of finite symplectic volume, and its geometric quantization gives rise to a finite-dimensional Hilbert space Hj1,…,j4.

It is known that the space of shapes of a tetrahedron is a Kahler manifold of complex dimension one that is parametrized by Z. This complex parameter is the cross ratio of the four stereographic coordinates zi labelling the direction of the normals to faces of the tetrahedron.The two possible viewpoints on Hj1,…,j4 i.e. that of SU(2) invariant tensors and that of quantization of the space of shapes of a geometric tetrahedron – are equivalent, in line with the general principle of Guillemin and Sternberg saying that geometric quantization commutes with symplectic reduction. An explicit formula for the decomposition of the identity in Hj1,…,j4 in terms of coherent states also exists.

The main aim of this paper is to further develop the holomorphic viewpoint on Hj, both for n = 4 and more generally. It is shown that the Hilbert space Hj of intertwiners can be obtained by quantization of a finite volume symplectic manifold Sj. Where the phase space Sj.
turns out to be a Kahler manifold, with holomorphic coordinates given by a string of n=3 cross-ratios {Z1,..Zn-3}. The paper uses the methods of geometric quantization to get to Hj up to the metaplectic correction occurring in geometric quantization of Kahler manifolds.

In the context of Kahler geometric quantization, one can introduce the coherent states |zi> such that

holomorphic equ 2b

The inner product formula can then be rewritten as a formula for the decomposition of the identity operator in terms of the coherent

holomorphic equ 3

The first main result in this paper is a version the above formula for the identity operator in the Hilbert space Hj, this is:

holomorphic equ 4

The integration kernel Kj here turns out to be just the n-point function of the AdS/CFT duality.

A comparison between the last two equations  shows that, in the semi-classical limit of all spins becoming large, the n-point function K
can be interpreted as an exponential of the Kahler potential on Sj .

holomorphic equ 4

Takes a particularly simple form of an integral over a single cross ratio coordinate in the case n = 4 of relevance for the quantum tetrahedron. The coherent states  are holomorphic functions of Z,  referred  to as holomorphic intertwiners. The resulting
holomorphic description of the Hilbert space of the quantum tetrahedron justifies the title of this paper.

The second part of the paper  characterizes the n = 4 holomorphic intertwiners |j ,Z> by projecting them onto a more familiar real basis in Hj1,…,j4. The real basis |j,k>can be obtained by considering the eigenstates of the operators J(i) · J(j) representing the scalar product of the area vectors between the faces i and j. The overlap between the usual normalised intertwiners  : |j,k> and the holomorphic intertwiner |k,Z> is denoted by:

holomorphic equ 4a

It is given by the “shifted” Jacobi polynomials P(α,β)n :

holomorphic equ 5

where jij = ji − jj , and Nk  is a normalisation constant.

This result can be used to express the Hj1,…,j4 norm of the holomorphic intertwiner |j,Z> in a holomorphically
factorised form

holomorphic equ 6

The last step of the paper’s analysis is to discuss the asymptotic properties of the (normalized) overlap coefficients

holomorphic equ 4a for large spins and the related geometrical interpretation. This asymptotic analysis gives a relation between the real and holomorphic description of the phase space of shapes of a geometric tetrahedron. The authors report that the normalized overlap coefficient is sharply picked both in k and in Z around a value k(Z) determined by the classical geometry of a tetrahedron.

The results of this paper are important for the field of quantum gravity the n = 4 intertwiner characterized in this paper plays a very important role in both the loop quantum gravity and the spin foam approaches. These intertwiners have so far been characterized using the real basis |j,k> In particular, the main building blocks of the spin foam models – the (15j)symbols and their analogs – arise as simple pairings of 5 of such intertwiners

The main result of this paper is a holomorphic description of the space of intertwiners, and, in particular, an explicit basis in Hj1,…,j4 given by the holomorphic intertwiners |j.Z>. The basis |j,k> being discrete, is convenient for some purposes, but the underlying geometric interpretation is quite hidden. Recalling the interpretation of the intertwiners from Hj1,…,j4
as giving the states of a quantum tetrahedron, the states |j,k>describe a tetrahedron whose shape is maximally uncertain. In contrast, the intertwiners |j.Z> being holomorphic, are coherent states in that they manage to contain the complete information about the shape of the tetrahedron coded into the real and imaginary parts of the cross-ratio coordinate Z. With the holomorphic intertwiners |j,Z> the quantum geometry can be characterized much more completely than it was possible before. It is possible to build the spin networks – states of quantum geometry – using the holomorphic intertwiners, and then the nodes of these spin networks receive a well-defined geometric interpretation of corresponding to tetrahedra of particular shapes. Similarly, the spin foam model simplex amplitudes can now be built using the coherent intertwiners, and then the basic object becomes not the (15j)-symbol of previous studies, but the (10j)-(5Z)-symbol with a well-defined geometrical interpretation.




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