Semiclassical analysis of Loop Quantum Gravity by Claudio Perini

This week I have been reading the PhD thesis Semiclassical analysis of Loop Quantum Gravity by Claudio Perini .

Semiclassical states for quantum gravity

The concept of semiclassical state of geometry is a key ingredient in the semiclassical analysis of LQG. Semiclassical states are kinematical states peaked on a prescribed intrinsic and extrinsic geometry of space. The simplest semiclassical geometry one can consider is the one associated to a single node of a spin-network with given spin labels. The node is labeled by an intertwiner, i.e. an invariant tensor in the tensor product of the representations meeting at the node. However a generic intertwiner does not admit a semiclassical interpretation because expectation values of non-commuting geometric operators acting on the node do not give the correct classical result in the large spin limit. For example, the 4-valent intertwiners defined with the virtual spin do not have the right semiclassical behavior; one has to take a superposition of them with a specific weight in order to construct semiclassical intertwiners.

The Rovelli-Speziale quantum tetrahedron  is an example of semiclassical geometry; there the weight in the linear superposition of virtual links is taken as a Gaussian with phase. The Rovelli-Speziale quantum tetrahedron is actually equivalent to the Livine-Speziale coherent intertwiner with valence 4 more precisely, the former constitutes the asymptotic expansion of the latter for large spins. Coherent intertwiners are as the geometric quantization of the classical phase space associated to the degrees of freedom of a tetrahedron.

In  recent graviton propagator calculations of semiclassical states
associated to a spin-network graph Γ have been  considered. The states used in the definition of semiclassical n-point functions ) are labeled by a spin jo and an angle ξe per link e of the
graph, and for each node a set of unit vectors n, one for each link surrounding that node. Such variables are suggested by the simplicial interpretation of these states: the graph Γ is in fact assumed to be dual to a simplicial decomposition of the spatial manifold, the vectors n are associated to unit-normals to faces of tetrahedra, and the spin jo is the average of the area of a face. Moreover, the simplicial extrinsic curvature is an angle associated to faces shared by tetrahedra and is identified with the label ξe. Therefore, these
states are labeled by an intrinsic and extrinsic simplicial 3-geometry. They are obtained via a superposition over spins of spin-networks having nodes labeled by Livine-Speziale coherent intertwiners.
The coefficients cj of the superposition over spins are given by a Gaussian times a phase as originally proposed by Rovelli.

semi269

Such proposal is motivated by the need of having a state peaked both on the area and on the extrinsic angle. The dispersion is chosen to be given by

semi269a

so that, in the large jo limit, both variables have vanishing relative dispersions. Moreover, a recent result of Freidel and Speziale strengthens the status of these classical labels: they show that the
phase space associated to a graph in LQG can actually be described in terms of the labels (jo , ξe, ne, n′) associated to links of the graph. The states have good semiclassical properties and a clear geometrical
interpretation.

Within the canonical framework, Thiemann and collaborators have strongly advocated the use of complexifier coherent states . Such states are labeled by a graph Γ and by an assignment of a SL(2,C) group element to each of its links. The state is obtained from the gauge-invariant projection of a product over links of modified heat-kernels for the complexification of SU(2). Their peakedness properties have been studied in detail.

Perini’s thesis presents the  proposal of coherent spin-network states: the proposal is to consider the gauge invariant projection of
a product over links of Hall’s heat-kernels for the cotangent bundle of SU(2). The labels of the state are the ones used in Spin Foams: two normals, a spin and an angle for each link of the graph. This set of labels can be written as an element of SL(2,C) per link of the graph. Therefore, these states coincide with Thiemann’s coherent states with the area operator chosen as complexifier, the SL(2,C) labels written in terms of the phase space variables (jo , ξe, ne, n′e) and the heat-kernel time given as a function of jo.

The author shows that, for large jo , coherent spin-networks reduce to the semiclassical states used in the spinfoam framework. In particular that they reproduce a superposition over spins of spin-networks with nodes labeled by Livine-Speziale coherent intertwiners and coefficients cj given by a Gaussian times a phase as originally proposed by Rovelli. This provides a clear interpretation of the geometry these states are peaked on.

Livine-Speziale coherent intertwiners
In ordinary Quantum Mechanics, SU(2) coherent states are defined as the states that minimize the dispersion

semi270

of the angular momentum operator J, acting as a generator of rotations on the representation space Hj ≃C2j+1 of the spin j representation of SU(2). On the usual basis |j,m> formed by simultaneous eigenstates of J2 and J3 we have

semi271

so the maximal and minimal weight vectors |j,±j> are coherent states. Starting from |j, j〉, the whole set of coherent states is constructed through the group action

semi272

One can take a subset of them labelled by unit vectors on the sphere S²:

semi273

where n is a unit vector defining a direction on the sphere S² and g(n) a SU(2) group element rotating the direction z ≡ (0, 0, 1) into the direction n. In other words, a coherent states is a state satisfying

semi274

For each n there is a U(1) family of coherent states and they are related one another by a phase factor. The choice of this arbitrary phase is equivalent to a section of the Hopf fiber bundles :

S2 ≃ SU(2)/U(1) → SU(2).

Explicitly, denoting n = (sin θ cos φ, sin θ sin φ, cos θ), a possible section is

semi275

where m ≡ (sin φ,−cos φ, 0) is a unit vector orthogonal both to z andn. A coherent state can be expanded in the usual basis as

semi276

semi277

Coherent states are normalized but not orthogonal, and their scalar product is

semi278

where A is the area of the geodesic triangle on the sphere S² with vertices z, n and n. Furthermore they provide an overcomplete basis for the Hilbert space Hj of the spin j irreducible representation of SU(2), and the resolution of the identity can be written as

semi278a

with d²n the normalized Lebesgue measure on the sphere S².

The Livine-Speziale coherent intertwiners are naturally defined taking the tensor product of V coherent states (V stands for valence of the node) and projecting onto the gauge-invariant subspace:

semi279

Here the projection is implemented by group averaging. They are labeled by V spins j and V unit vectors n. The states |j, n〉o carry enough information to describe a classical geometry associated to the node. Interpret the vectors j,n as normal vectors to triangles, normalized to the areas j of the triangles.

 Coherent tetrahedron

The case of 4-valent coherent intertwiners is of particular importance for LQG and especially for Spin Foam Models. In fact it is the lowest valence carrying a non-zero volume and most SFM’s are build over a simplicial triangulation, so that the boundary state space has only 4-valent nodes, dual to tetrahedra. A 4-valent coherent intertwiner with normals satisfying the closure condition can be interpreted as a semiclassical tetrahedron. In fact expectation values of geometric operators associated to a node give the correct classical quantities in the semiclassical regime. This regime is identified with the large spin (large areas) asymptotics. In the following we give some details.

Define n = jn as the normals normalized to the area. In terms of them, the volume (squared) of the tetrahedron is given by the simple relation:

semi280

The geometric quantization of these degrees of freedom is based on the identification of generators Ji of SU(2) as quantum operators corresponding to the  ni . This construction gives directly
the same quantum geometry that one finds via a much longer path by quantising the phase space of General Relativity, that is via Loop Quantum Gravity. The squared lengths |ni|² are the SU(2) Casimir
operators C²(j), as in LQG. A quantum tetrahedron with fixed areas lives in the tensor product ⊗Hj .The closure constraint reads:

semi281

and imposes that the state of the quantum tetrahedron is invariant under global rotations. The state space of the quantum tetrahedron with given areas is thus the Hilbert space of intertwiners

semi282

The operators Ji · Jj are well defined on this space, and so is the operator

semi283

Its absolute value |U| can immediately be identified with the quantization of the classical squared volume 36V²,  in agreement with standard LQG results.

To find the angle operators,  introduce the quantities  Jij := Ji+  Jj. Given these quantities, the angle operators θij can be recovered from

semi284

The quantum geometry of a tetrahedron is encoded in the operators Ji ², Jij² and U, acting on Ij₁…j. It is a fact that out of the six independent classical variables parametrizing a tetrahedron, only
five commute in the quantum theory. Indeed while we have

[Jk², Ji · Jj] = 0,

it is easy to see that:

semi285

A complete set of commuting operators, in the sense of Dirac, is given by the operators Ji² , J². In other words, a basis for Ij…j4 is provided by the eigenvectors of any one of the operators Jij² . We write the corresponding eigenbasis as |j〉ij . These are virtual links here we are introducing them via a geometric quantization of the classical tetrahedron. For instance, the basis |j〉 diagonalises the four triangle areas and the dihedral angle θ or, equivalently, the area A of one internal parallelogram. The relation between different basis is  obtained from SU(2) recoupling theory: the matrix describing the change of basis in the space of intertwiners is given by the usual Wigner 6j-symbol,

semi286

so that

semi287

Notice that from the orthogonality relation of the 6j-symbol,

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we have

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The states |j〉 are eigenvectors of the five commuting geometrical operators Ji² , J²    so the average value of the operator corresponding to the sixth classical observable, say J² is on these states maximally spread. This means that a basis state has undetermined classical geometry or, in other words, is not an eigenstate of the geometry.  Then led to consider superpositions of states to be able to study the semiclassical limit of the geometry. Suitable superpositions could be constructed for instance requiring that they minimise the uncertainty relations between non–commuting observables, such as

semi290

States minimising the uncertainty above are usually called coherent states. Coherent intertwiners seem not to verify exactly semi290, but they are such that all relative uncertainties h〈Δ²〉Jij/〈J²ij〉, or equivalently 〈Δθij〉/〈θij〉, vanish in the large scale limit. The limit is defined by taking the limit when all spins involved go uniformly to infinity, namely ji = λki with λ → ∞. Because of these good semiclassical properties we can associate to a coherent intertwiner the geometrical interpretation of a semiclassical tetrahedron; an analogous interpretation should be also valid nodes of higher valence.

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