Polyhedral quantum geometry

This week I’ve been reading Spinfoams: Simplicity Constraints and Correlation Functions PhD thesis by Ding. I’m reviewing the  section on Polyhedral quantum geometry  which is relevant to my work on the quantum tetrahedron.

Consider the truncation of the LQG Hilbert space HLQG and restrict ourself to a single graph Hilbert space H(Γ) and decompose it in terms of SU(2)-invariant spaces Hn associated to each node n. Here I’ll briefly review the state that this node space Hn is the quantization of the space of shapes of the geometry of solids figures tetrahedra, or more general polyhedra . See the posts:

Let’s start with the classical phase space of shapes of a flat polyhedron in Rwith fixed area. A classically flat three-dimensional polyhedron can be described by a set of L vectors Al, l = 1…L, satisfying the following closure constraint:

Here the L vectors Al can be interpreted as the vectorial areas of the L triangles in the boundary of the polyhedron, in the sense that the norm al = |Al| is the area of the polygon l and normalized vector nl = Al/|Al| is the normal when embedded in to a R3 Euclidean space.
To introduce a symplectic structure, one can associate to each normal Aia generator of the algebra of SO(3).

A quantum representation of this Poisson algebra is precisely defined by the generators of SU(2) on the space Hn for a 4-valent node n. The operator corresponding to the area al = |Al| is the Casimir of the representation jl, therefore the space quantizes
the space of the shapes of the tetrahedron with areas jl(jl +1). Furthermore, the Hamiltonian flow of G, generates the rotations of the tetrahedron in R3.

By imposing

and factoring out the orbits of this flow, one obtains the intertwiner space Kn.

In this way, one gives an intertwiner a geometrical interpretation in terms of quantum polyhedron.

There is a  relation among spinfoam formalism, kinematical Hilbert space and polyhedral quantum geometry. For example the boundary space of the simplicial EPRL spinfoam model can be obtained from simplicity constraints, which is the simplicial truncation of LQG kinematical Hilbert space and the boundary state has a geometrical interpretation in terms of quantum tetrahedron geometry. This consistent picture can be generalized into an arbitrary-valence spinfoam formalism. It is also possible to compute the two-point correlation function of Lorentian EPRL spinfoam model and show it matches the one from Regge geometry.

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Quanta in a Quantum Spacetime

The Frontiers of Fundamental Physics 14 conference again captures my interest this week and I’ve been looking at a paper, ‘ How Many Quanta are there in a Quantum Spacetime?

In this paper the authors develop a technique for describing quantum states of the gravitational field in terms of coarse grained spin networks. They show that the number of nodes and links and the values of the spin depend on the observables chosen for the description of the state. So in order to say how many quanta are in a quantum spacetime further information about what has been measured has to be given.

Introduction

The electromagnetic field can be viewed as formed by individual photons. This is a consequence of quantum theory. Similarly, quantum theory is likely to imply a granularity of the gravitational field, and therefore a granularity of space.

How many quanta form a macroscopic region of space? This question has implications for the quantum physics of black holes, scattering calculations in non perturbative quantum gravity and quantum cosmology. It is related to the question of the number of nodes representing a macroscopic geometry in a spin network state in loop gravity. In this context, it takes the following form: what is the relation between a state with many nodes and small spins, and a state with few nodes but large spins?

Quanta of space

A quanta of space may be  a quanta of energy from the excitation of the gravitational field. In loop quantum gravity, each quanta is a  quantum polyhedron. The geometry of quantum polyhedron defined by graph. We associate a state or element of Hilbert space for each quanta of space. The basis which spanned this Hilbert space is the spin network basis.

A quantum tetrahedron and its dual space geometry: the graph

A graph γ is a finite set N of element n called nodes and a set of L of oriented couples called links l = (n, n’). Each node corresponds to one quantum tetrahedron. Four links pointing out from the node correspond to each triangle of the tetrahedron.

How many quanta in a field?

Consider a free scalar field in a finite box, in a classical configuration φ(x,t). The standard quantum-field-theoretical number operator, which sums the number the quanta on each mode, has a well defined classical limit. The number operator is

where an and an are the annihilation and creation operators for the mode n of the field and the sum is over the modes, namely the Fourier components, of the field. Since the energy can be expressed as a sum over modes as

where ωn is the angular frequency and En its energy of the mode n, it follows that the number of particles is given
by

which is a well defined classical expression that can be directly obtained from φ(x,t) by computing the energy in each mode. Therefore each classical configuration defines a total particle-number N and a distribution of these particles over the modes

Subset graphs

The state space of loop quantum gravity contains subspaces Hγ associated to abstract graphs γ. A graph γ is defined by a finite set N of |N| elements n called nodes and a set L of |L| oriented couples l = (n, n’) called links.

A pure state |ψ〉 determines the density matrix ργ= |ψ〉〈ψ|. A generic state can be written in the form

In the loop gravity the operators defined on Hγ can be interpreted as the description of the geometry of |N| quantum polyhedra connected to one another when there is a link between the corresponding nodes.

Given a graph , define a subset graph Γ which partitions N into subsets N such that each N is a set of nodes connected among themselves by sequences of links entirely formed by nodes in N.

We define the area of the big link by

and the volume of the big node by

where we recall that v is the expression for the classical volume of a polyhedron. The operators AL and VN commute, so they can be diagonalized together. The quantum numbers of the big areas are half integers JL and the quantum numbers of the volume are VN.

Course graining spin networks

Coarse-graining the entire graph into a graph Γ formed by a single node N with legs b

A  set of small links l that are contained in a single large link L.

Any general coarse-graining is a combination of collecting nodes and summing links

The geometry of the subset graph

The geometrical interpretation of the coarse grained states in HΓ is that these describe the geometry of connected polyhedra. The partition that defines the subset graph Γ is a coarse-graining of the polyhedra into larger chunks of space. The surfaces that separate these larger chunks of space are labelled by the big links L and are formed by joining the individual faces labelled by the links l in L.

In general, it is clearly not the case that the area AL is equal to the sum of the areas Al of all l in L. However, this is the case if all these faces are parallel and have the same orientation. Similarly, in general, it is clearly not the case that the volume VN is equal to the sum of the volumes Vn for the n in N. However, this is true if in gluing n polyhedra one obtains a at polyhedron with flat faces.

The two operators,

provide a good measure of the failure of the geometry that the state associates to Γto be flat.

To have a good visualization of the coarse-grained geometries, it is helpful to consider the classical picture. In the 4-dimensional theory, the graph is defined at the boundary of a 3-dimensional hypersurface, the spin operator on the links is related to the area operator by

Given a 3-valent graph with spins operators Jla, Jlb , and Jlc on each link, the dihedral angle between Jlb and Jlc can be obtained from the angle operator, defined by

Applying this operator to the spin network state gives the dihedral angle between Jland Jlc on the internal links lb and lc.

The Regge intrinsic curvature of a discretized manifold is given by the deficit angle on the hinges, the (n -2) dimensional simplices of the n-dimensional simplex. Thus, given a loopgraph with n-external links, the deficit angle for a general n-polytope (n-valent loop graph) is:

Coarse grained area

The boundary of spacetime is a 3-dimensional space. Triangulation on the boundary is defined using flat polyhedra. Every closed, flat, n polyhedron satisfies the closure relation on the node given by

Consider the net of a polyhedron:

Since the interior of polyhedron is at, the closure relation can be written as

Then the area operator on the base is

and we can define the coarse-grained area as:

Thus, for a 2-dimensional surface, we can always think the coarse grained area AL as the area of the base of a polyhedron, while the total sum of area Al is the area around the hat – the area of n triangle which form the net of the polyhedron.

The differences between the coarse-grained and the fine-grained area gives a good measurement on how the space deviates from being flat. It is possible to obtain the explicit relation between the Regge curvature with these area differences in some special cases.

The Regge curvature for a 2-dimensional surface is defined as 2 minus the sum of all dihedral angle surrounding a point of the triangulation, which is n. The Regge curvature as a function of the coarse-grained and the fine-grained area is:

In the classical limit, it is clear that there can be states where ε = 0 or AL = 0. These correspond to geometries where the normals to the facets forming the large surface L are parallel. However, this is only true in the classical limit, namely disregarding Planck scale effects. If we take Planck-scale effects into account, we have the  result that

and

where n + 1 is the number of facets. Therefore the fine grained area is always strictly larger than the coarse grained area. There is a Planck length square contribution for each additional facet. It is as if there was an irreducible Planck-scale fluctuation in the orientation of the facets.

Coarse-grained volume

In the same manner as the surface’s coarse-graining, we triangulate a 3-dimensional chunk of space using n symmetric tetrahedra. The Regge curvature is defined by the dihedral angle on the bones of the tetrahedra. Using the volume of one tetrahedron,

we obtain the fine-grained volume, which is

The coarse-grained volume is the volume of the 3-dimensional base, which is the volume of the n-diamond:

so the Regge curvature is

Notice that this is just a classical example. In the quantum picture, adding two quantum tetrahedra does not gives only a triangular bipyramid, it could give other possible geometries which have 6 facets, i.e., a parallelepiped, or a pentagonal-pyramid.

Conclusion

The number of quanta is not an absolute property of a quantum state: it depends on the basis on which the state is expanded. In turn, this depends on the way we are interacting with the system. The quanta of the gravitational field we interact with, are those described by the quantum numbers of coarse-grained operators like AL and V, not the maximally fine-grained ones.

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