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.

 Related articles

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s