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 

quantum tetrahedron

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

quantaequ1where 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

quantaequ2

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

quantaequ3

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

quantaequ4

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

quantaequ5

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.

quantafig1 We define the area of the big link by

quantaequ17

and the volume of the big node by

quantaequ18

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

quantafig2A  set of small links l that are contained in a single large link L.

quantafig3Any general coarse-graining is a combination of collecting nodes and summing links

quantafig4

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,

quantaequ43

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 byquantaequ48a

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

quantaequ49

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

quantaequ50

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:

quantaequ51

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

quantaequ7

Consider the net of a polyhedron:

quantafig6

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

quantaequ52

Then the area operator on the base is

quantaequ53

and we can define the coarse-grained area as:

quantaequ54

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:

quantaequ58

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

quantaequ59and

quantaequ63where 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

quantafig7

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,

quantaequ63a

 

we obtain the fine-grained volume, which is

quantaequ64

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

quantaequ65

so the Regge curvature is

quantaequ67

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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Classical and Quantum Polyhedra by John Schliemann

This week I have been  looking again at the Quantum Tetrahedron and quantum polyhedra in general. I’ll be doing further  numerical studies to add to the numerical work done in earlier posts:

Quantum polyhedra constructed from angular momentum operators are the building blocks of space in its quantum description by Loop Quantum Gravity. The author extends results on the semiclassical properties of quantum polyhedra. They compare results from a canonical quantization of the classical system with a recent wave function based approach to the large-volume sector of the quantum system. Both methods agree in the leading order of the resulting effective operator given by an harmonic oscillator, while minor differences occur in higher corrections. Perturbative inclusion of such corrections improves the approximation to the eigenstates. Moreover, the comparison of both methods leads also to a full wave function description of the eigenstates of the (square of the) volume operator at negative eigenvalues of large modulus.

For the case of general quantum polyhedra described by discrete angular momentum quantum numbers the authors formulate a set of quantum operators fullling in the semiclassical regime the standard commutation relations between momentum and position. The position variable here is chosen to have dimension of Planck length squared which facilitates the identication of quantum corrections.

Introduction
The quantum volume operator is pivotal for the construction of space-time dynamics within this Loop Quantum Gravity. Traditionally two versions of such an operator are discussed, due to Rovelli and Smolin, and to Ashtekar and Lewandowski, and more recently, Bianchi, Dona, and Speziale offered a third proposal for a volume operator which is closer to the concept of spin foams. It relies on an older geometric theorem due to Minkowski stating that N face areas Ai with normal vectors ni such that

CCqTequ1

uniquely define a convex polyhedron of N faces with areas Ai.

The approach amounts to expressing the volume of a classical polyhedron in terms of its face areas, which are in turn promoted to be operators. Minkowski’s proof, however, is not constructive,
and a remaining obstacle of this approach  to a volume operator is to actually find the shape of a general polyhedron given its face areas and face normals. Such difficulties do not occur in the simplest case
of a polyhedron, i.e. a tetrahedron consisting of four faces represented by angular momentum operators coupling to a total spin singlet. Indeed, for such a quantum tetrahedron all three definitions of the volume operator coincide. On the other hand, for a classical tetrahedron the general phase space parametrization
devised by Kapovich and Millson results in just one pair of canonical variables, and the square of the volume operator can explicitly formulated in terms of these . Bianchi and Haggard have performed a Bohr-Sommerfeld quantization of the classical tetrahedron where the role of an Hamiltonian generating classical orbits is played by the volume operator squared. The resulting semiclassical eigenvalues agree extremely well with exact numerical data, see the post

 

The above observations make clear that classical tetrahedra, the simplest structures a volume can be ascribed to, should be considered as perfectly integrable systems. In turn, a quantum tetrahedron can be viewed as the hydrogen atom of quantum spacetime, whereas the next complicated case of a pentahedron might be referred to as the helium atom.

Recently, Schliemann put forward another approach to the semiclassical regime of quantum tetrahedra, see the post

Here, by combining observations on the volume operator squared and its eigenfunctions as opposed to the eigenvalues, an effective operator in terms of a quantum harmonic oscillator was derived providing an accurate as well as transparent description of the the large-volume sector.

One of the purposes of this paper is to demonstrate the relation between the different treatments of quantum tetrahedra sketched above.

The outline of this paper is as follows.

  • Summarize the Kapovich-Millson phase space parametrization of general classical polyhedra.
  • Reviewing  the classical tetrahedron and expand of the volume squared around its maximum and minimum in up to quadrilinear order.
  • The quantum tetrahedron.

 Classical Polyhedra

 Kapovich-Millson Phase Space Variables

Viewing the vectors Ai as angular momenta, the Poisson
bracket of arbitrary functions of these variables read

CCqTequ2v1

To implement the closure relation   define

CCqTequ3

resulting in N -3 momenta pi =|pi|. The canonical conjugate variables qi are then given by the angle between the vector

CCqTequ4

These quantities fulfill the canonical Poisson relations

CCqTequ5

The Tetrahedron

The classical volume of a tetrahedron can be expressed
as

CCqTequ6

Look at the quantity,

CCqTequ7

This can be  expressed in terms of the phase space variables p1, q1 using;

CCqTequ8and with

CCqTequ9where Δ(a, b, c) is the area of a triangle with edges a,b, c expressed via Heron’s formula,

CCqTequ10

and

CCqTequ11such that

CCqTequ12

In order to make closer contact to the quantum tetrahedron introduce the notation

CCqTequ13

fullling {p,A} = 1 and

CCqTequ14

with

CCqTequ15

where A varies according to Amin ≤A ≤Amax with

CCqTequ16

β(A) is a nonnegative function with β(Amin) = β (Amax) = 0, and it has a unique maximum at  A between Amin and Amax. Thus, Q has a maximum at A = k and p = 0 while the unique minimum lies at p = . Expanding around the maximum gives

CCqTequ18

with

CCqTequ19

and

CCqTequ20

 

The analogous expansion around the minimum reads

CCqTequ21

 

Concentrating in both cases on the quadratic contributions,
one obtains two harmonic oscillators,

CCqTequ22

 

The Quantum Tetrahedron

General Properties

A quantum tetrahedron is defined by four angular momentum operators ji representing its faces and coupling to a total singlet the Hilbert space consists of all states |k〉 fulling

CCqTequ24

 

 A usual way to construct this space is to couple first the pairs j1,j2 and j3, j4 to two irreducible SU(2) representations of dimension 2k+1 each. For j1, j2 this standard construction reads explicitly

CCqTequ25

CCqTequ27such that

CCqTequ27

where 〈j1m1j2m2|km〉 are Clebsch-Gordan coefficients

Defining analogous states |km〉34 for j3, j4, the quantum number k  becomes restricted by kmin ≤ k ≤kmax with

CCqTequ29

The two multiplets |km〉12, |km〉34 are then coupled to a
total singlet,

CCqTequ31The states jki span a Hilbert space of dimension d = kmax – kmin + 1.

The volume operator of a quantum tetrahedron can be
formulated as

CCqTequ32

where the operators

CCqTequ33

represent the faces of the tetrahedron with CCqTequ33abeing the Planck length squared. Consider the operator

CCqTequ34

which reads in the basis of the states |k〉 as

CCqTequ35

For even d, the eigenvalues of Q come in pairs (q, -q), and since

CCqTequ37

the corresponding eigenstates fulfill

CCqTequ38

For odd d an additional zero eigenvalue occurs.

To make further contact between the classical and the
quantum tetrahedron define

CCqTequ39

fulfilling

CCqTequ41

also

CCqTequ42

and

CCqTequ43

is the projector onto the singlet space.

So far have followed the formalism common to
the literature and parametrized the Hilbert space of the
quantum tetrahedron by a dimensionless quantum number
k, whereas the phase space variable A of the classical
tetrahedron has dimension of area. In order to establish
closer contact between both descriptions, rescale the
involved quantum numbers by the Planck length squared
according to

CCqTequ44

to quantities having also dimension of area.

This gives,

CCqTequ46

with

CCqTequ48

β(a) has a unique maximum at some a.

The Quantum Tetrahedron at Large Volumes

In the post Large-Volume Limit of a Quantum Tetrahedron is a Quantum Harmonic Oscillator

It was shown how to accurately describe the large-volume semiclassical regime of Q or R by a quantum harmonic oscillator in real-space representation with respect to a or k, respectively.

Here the  analysis is extended by taking into account higher order corrections.

label the eigenstates of Q by |n〉, n ∈ {0,1, 2….}, in descending order of eigenvalues with |0〉 being the state of largest eigenvalue. With respect to the basis states |k〉 they can be expressed as

CCqTequ50

Taking the view of the standard Schrodinger formalism of elementary quantum mechanics, the coefficients 〈a|n〉 are the wave function of the state |n〉 with respect to the coordinate a.

Evaluating the matrix elements

CCqTequ51

one obtains up to fourth order in the expansions

CCqTequ53

CCqTequ54

 

Introducing the operators

CCqTequ56

The effective operator expression is

CCqTequ58

Concentrating on the quadratic contributions in gives the harmonic-oscillator expression

CCqTequ61

with eigenvalues

CCqTequ62

and corresponding eigenfunctions

CCqTequ63

where Hn(x) are the usual Hermite polynomials.

fig1

fig2

CONCLUSIONS

The investigation of the semiclassical limit of Loop
Quantum Gravity is one of the key issues in that approach to  quantum gravity. This paper has focussed  on the semiclassical properties of quantum polyhedra. Regarding tetrahedra as their simplest examples, it has been established that there is a connection
between a canonical quantization of the classical system  and the  wave function based approach  to the large-volume sector of the quantum system. In the leading order both routes concur yielding a quantum harmonic oscillator as an effective
description for the square of the volume operator.

A further interesting point is the zero eigenvalue occurring for tetrahedra with odd Hilbert space dimension d. The Bohr-Sommerfeld quantization carried out by Bianchi and Haggard gives accurate results for eigenvalues.

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