Quantum geometry from phase space reduction by Conrady and Freidel

In this paper the authors give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin–Sternberg and Hall that describe the commutation of quantization and reduction.This resulted to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.

Recently following the proposal of new spin foam models for gravity a number of important developments have occurred. These models allow the inclusion of a nontrivial Immirzi parameter and have been shown to satisfy two very important consistency requirements:
firstly, in the semiclassical limit they are asymptotically equivalent to the usual Regge discretisation of gravity, independently of the complexity of the underlying cell complex. Secondly, they have been shown to possess SU(2) spin network states as boundary states.

These new developments link with the evolution of the understanding of the simplicity constraints. These simplicity constraints state in any dimension that the bivector used to
contract the curvature tensor in the Einstein action comes from a frame field. In a simplicial context this constraint splits into three classes: there are the face simplicity constraints (among bivectors associated to the same face), the cross–simplicity constraints (among
bivectors associated to faces of the same tetrahedron) and the volume constraints (among opposite bivectors). All these constraints are quadratic, but the volume constraint, because it depends on different tetrahedra, also involves the connection and is therefore extremely hard to quantise. The first major advance came from the work of Baez, Barrett and Crane who showed that it is possible to linearise the volume constraint and replace it by a constraint living only at one tetrahedron. This new constraint is the closure
constraint and the discrete analogue of the Gauss law generating gauge transformations. The second key insight in this direction came from the work of Engle, Pereira and Rovelli who showed that one can linearise the simplicity and cross simplicity constraints, opening
the path toward a new way of constructing spin foam models and allowing the incorporation of the Immirzi parameter.
The purpose of the paper is to show that the new spin foam model can be written explicitly in terms of a sum of amplitudes, where all simplicity constraints are imposed strongly in the path integral, and where all the boundary spin networks have a geometrical interpretation even before taking the semiclassical limit.

Along with the construction of new models there is also a merging of two lines of thought on spin foam models that developed in parallel for a long time and are now  intersecting.

One line of thought, can be traced back to the seminal work of Barbieri – see post: ‘Quantum tetrahedra and simplicial spin networks, who realized that spin network states of loop quantum gravity can be understood by applying the quantization procedure to a collection of geometric tetrahedra in 3 spatial dimensions. This important work suggested that spin network states and quantum gravity may be about quantizing geometric structures. This idea was then quickly applied to the problem of quantization of a geometric simplex in R4, with the result being the construction of the Barrett–Crane model. This line of thought was also developed by Engle, Livine, Pereira, Rovelli. The main problem in this approach is that the geometry associated with the quantum tetrahedra is fuzzy and cannot be resolved sharply due to the noncommutativity of the geometric ‘flux’ operators. This problem prevents a priori a sharp coupling between neighboring 4d building blocks, and that was the main issue with the Barrett-Crane model.

The second line of thought can be traced back to the work of Reisenberger who proposed to think about quantum gravity directly in terms of a path integral approach in which we integrate over classical configurations living on a 2–dimensional spine and where
spin foam models arise from a type of bulk discretization. This line of reasoning is related to the Plebanski reformulation of gravity as a constraint SO(4) BF theory. In this approach it is quite easy to get a consistent gluing condition, however, it is much less trivial to find models which have a natural and simple algebraic expression in terms
of recoupling coefficients and it is even more of a challenge to get SU(2) boundary spin networks as boundary states.

The merging between these two approaches starts with a work of Livine and Speziale who proposed to label spin network states not with the usual intertwiner basis, but with ‘coherent intertwiners’ that are labelled by four vectors whose norm is fixed to be the area
of the faces of the tetrahedron. This has led to a very efficient and geometrical way of deriving the new spin foam models
There is however a caveat – the coherent intertwiner resolves the fuzzyness of the quantum space of a tetrahedron by construction, but the classical configuration no longer satisfies the closure constraint unless one takes the semiclassical limit and it is therefore not geometrical.

The resolution of these problems proposed by the authors in ths paper here is to use coherent states that are associated to the geometrical configuration – see post:’ A semiclassical tetrahedron‘ and relate these ‘geometrical’ states to the usual spin network
states or the coherent intertwiner states.

This paper gives a construction of such states and shows that they form an over-complete basis of the space of four–valent intertwiners and gives the explicit isomorphism between this new basis of states, where the closure constraint is imposed strongly, and the coherent intertwiner basis. The proof of this isomorphism amounts essentially to showing  that ‘quantisation commutes with reduction’ and utilises heavily the work of Guillemin and  Sternberg on geometrical quantisation.

The main formula of the paper expresses the decomposition of the identity in the space of four–valent intertwiners Hj= (Vj1 .· · · .Vj4)as an integral over coherent states satisfying the closure constraint, and so as an integral over the space of classical tetrahedra:

quantum geometry


This identity can be applied it to the spin foam models: it allows the formulation the path integrals in terms of variables on which the closure constraints are imposed strongly. These variables are well–suited to analyze the asymptotic large spin behaviour of amplitudes, and are used in this paper to derive the asymptotics of the FK vertex amplitude.

The paper describes in detail the phase space of the classical tetrahedron and shows that it can be obtained as a symplectic quotient. It states the classical part of the Guillemin–Sternberg isomorphism, which is a isomorphism between a constrained phase space divided by the action of the gauge group and the unconstrained phase space divided by the complexification of the group. It also gives a detailed discussion of coherent states and their link with geometrical quantisation and a construction of the isomorphism between the new geometrical basis and the coherent intertwiner basis.This enables the FK. spin foam model to be written in terms of the new tetrahedral states and use this to derive the asymptotics of the vertex amplitude.




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