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Classical and Quantum Polyhedra

Quantum polyhedra constructed from angular momentum operators are the building blocks of space in its quantum description as advocated by Loop Quantum Gravity. Here we extend previous results on the semiclassical properties of quantum polyhedra. Regarding tetrahedra, we compare the 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 we formulate a set of quantum operators fulfilling in the semiclassical regime the standard commutation relations between momentum and position. Differently from previous formulations, the position variable here is chosen to have dimension of (Planck) length squared which facilitates the identification of quantum corrections. Finally, we provide expressions for the pentahedral volume in terms of Kapovich-Millson variables.


I. INTRODUCTION
The quantum volume operator is among the most intensively investigated items in the field of Loop Quantum Gravity and pivotal for the construction of space-time dynamics within this theoretical framework [1][2][3] . Traditionally two versions of such an operator are discussed, due to Rovelli and Smolin 4 , and to Ashtekar and Lewandowski 5 , respectively, and considerable attention has been devoted to their properties and interrelations [6][7][8][9][10][11][12][13][14][15][16][17][18][19] . More recently, Bianchi, Dona, and Speziale 20 offered a third proposal for a volume operator which is closer to the concept of spin foams 3 . It relies on an older geometric theorem due to Minkowski 21 stating that N face areas A i , i ∈ {1, . . . , N }, with normal vectors n i such that for A i = n i A i uniquely define a convex polyhedron of N faces with areas A i . The approach of Ref. 20 amounts in 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 the above route to a volume operator is to actually find the shape of a general polyhedron given its face areas and face normals 20,[22][23][24][25] . 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 11,26 . 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 27 results in just one pair of canonical variables, and the (square of the) volume operator can explicitly formulated in terms of these quantities 23,28 . Moreover , 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 22,23 . The above observations make clear that classical tetrahedra, arguably 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" 25 .
Most recently, the present author has put forward yet another approach to the semiclassical regime of quantum tetrahedra 29 . 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 the present work is to demonstrate the relation between the different treatments of quantum tetrahedra sketched above.
The outline of this paper is as follows. In section II we first summarize the Kapovich-Millson phase space parametrization of general classical polyhedra (section II A) before reviewing and extending in section II B results for the classical tetrahedron. In particular, we derive an expansion of the volume squared around its maximum and minimum in up to quadrilinear order. Section III A is devoted to the quantum tetrahedron. We first outline in section III A 1 elementary facts about the volume operator and its Hilbert space, and we point out several relations between appropriate quantum operators which have analogs in the classical tetrahedron. Next the analysis of Ref. 29 is extended to higher corrections to the resulting harmonic oscillator of up to fourth order. The results are compared with the outcome of a canonical quantization of the classical volume expression. Finally we construct in section III B a set of quantum operators for general polyhedra whose commutation relations approach in the semiclassical limit the standard commutators between momentum and position.

II. CLASSICAL POLYHEDRA
Let us first recall the essentials of the polyhedral phase space parametrization due to Kapovich and Millson 27 .

A. Kapovich-Millson Phase Space Variables
Viewing the vectors A i as angular momenta, the Poisson bracket of arbitrary functions of these variables read In order to implement the closure relation (1) one defines for i ∈ {1, . . . , N − 3} resulting in N − 3 momenta p i = | p i |. The canonical conjugate variables q i are then given be the angle between the vectors and a straightforward calculation shows that these quantities fulfil indeed the canonical Poisson relations 23,27

B. The Tetrahedron
The classical volume of a tetrahedron can be expressed as suggesting to investigate the quantity The latter can indeed easily be expressed in terms of the phase space variables p 1 , q 1 using the observation 23 Moreover, it is easily seen that where ∆(a, b, c) is the area of a triangle with edges a, b, c expressed via Heron's formula, Analogously, using the closure relation (1), such that In order to make closer contact to the quantum tetrahedron to be discussed below, let us introduce the notation fulfilling {p, A} = 1 and withβ where A varies according to A min ≤ A ≤ A max with An expression close to (14) was also found in Ref. 28 in the semiclassical limit of a quantum tetrahedron. Obviously,β(A) is a nonnegative function with β(A min ) =β(A max ) = 0, and it is not difficult to verify that it has a unique maximum at some A =Ā between A min and A max29 . Thus, Q has a unique maximum at A =k and p = 0 while the unique minimum lies at p = π. Expanding around the maximum gives (x := A −Ā, |x| 1, |p| 1) The analogous expansion around the minimum reads Concentrating in both cases on the quadratic contributions, one obtains two harmonic oscillators, III. QUANTUM POLYHEDRA

A. The Quantum Tetrahedron
We begin by reviewing and extending general results of quantum tetrahedra.

General Properties
A quantum tetrahedron is defined by four angular momentum operatorsˆ j i , i ∈ {1, 2, 3, 4}, representing its faces and coupling to a total singlet 11,12,22,23,26 , i.e. the Hilbert space consists of all states |k fulfilling A usual way to construct this space is to couple first the pairsˆ j 1 ,ˆ j 2 andˆ j 3 ,ˆ j 4 to two irreducible SU(2) representations of dimension 2k + 1 each. Forˆ j 1 ,ˆ j 2 this standard construction reads explicitlŷ such thatk where j 1 m 1 j 2 m 2 |km are Clebsch-Gordan coefficients following their usual phase convention 32 . Defining analogous states |km 34 forˆ j 3 ,ˆ j 4 , the quantum number k becomes restricted by k min ≤ k ≤ k max with The two multiplets |km 12 , |km 34 are then coupled to a total singlet, where the phase factor in front will become useful shortly below. The states |k span a Hilbert space of dimension d = k max − k min + 1.
The volume operator of a quantum tetrahedron can be formulated asV where the operatorsˆ i ∈ {1, 2, 3, 4} represent the faces of the tetrahedron with 2 P = G/c 3 being the Planck length squared. Usually the operatorsˆ E i are defined with additional prefactors proportional to the Immirzi parameter on the r.h.s. of Eq. (33). This establishes contact to the general formalism of loop quantum gravity 1-3 but is unnecessary for our purposes here. What will become important, however, is that 2 P is proportional to . As a result, one is led to consider the operator which reads in the basis of the states |k as 12,23,29-32 with α(k) = 2 Note the close similarity of the expressions (36) and (15). Moreover, in the above basisQ couples only states with neighboring labels and is represented by a real matrix. The latter fact depends on the phase factor in the first line of Eq. (31). Indeed, upon striping this factor (which is a unitary operation)R becomes antisymmetric and purely imaginary. Thus, for even d, the eigenvalues of Q come in pairs q, (−q), and since , the corresponding eigenstates |φ q , |φ −q fulfill i.e. eigenvectors of eigenvalues differing just in sign are related to each other by changing the sign of any other component. For odd d an additional zero eigenvalue occurs 13 .
To make further contact between the classical and the quantum tetrahedron we define in analogy to Eqs.

Rescaling to Dimensionful Variables
So far we 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 let us rescale the involved quantum numbers by the Planck length squared according to to quantities having also dimension of area. As we shall see below, this step will also provide a close analogy to standard quantum mechanics in the Schrödinger representation. The analog of the classical expression (7) readŝ = 2 The latter quantity shares the essential properties of β(A) in Eq. (15). In particular β(a) has a unique maximum at some a =ā.

Large Volumes
In Ref. 29 the present author has shown how to accurately describe the large-volume (semiclassical) regime of Q (orR) by a quantum harmonic oscillator in real-space representation with respect to a (or k, respectively). Here we shall extend this analysis taking into account higherorder corrections within the rescaled variables introduced in the previous section.
Let us label the eigenstates ofQ 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 Thus, taking the view of the standard Schrödinger formalism of elementary quantum mechanics, the coefficients a|n are the "wave function" of the state |n with respect to the "coordinate" a. The approach of Ref. 29 starts from evaluating matrix elements Φ|Q|Ψ = a β(a) Φ|a a − 2 P |Ψ between states lying predominantly in the sector of large eigenvalues by approximating the sum by an integral introducing the integration variable x := a −ā, withΦ(x) = ā + x|Φ ,Ψ(x) = ā + x|Ψ . Expanding now β(ā + x) around its maximum atā and the wave functionsΦ * (x − 2 P ),Ψ(x − 2 P ) around x, one obtains in up to fourth order in the expansions In calculating the r.h.s. of Eq. (53) we have repeatedly performed integrations by parts and assumed the boundary terms to vanish. Introducing now the operatorŝ one easily reads off the effective operator expression This result extends the findings of Ref. 29 to higher corrections in the operatorsp,x. The contribution in Eq. (53) involving only derivatives with respect to x can be viewed as the result of a continuum approximation according to Note also that the symmetric operator ordering in the last term of the second line in Eq. (53) (i.e. the middle contribution in the second line in Eq. (57)) emerges from the calculation and not an additional assumption. As a result, the operator (57) perfectly matches the classical expression (18) taking into account the correct operator ordering and the vanishing of the commutators which are indeed small compared to the other contributions in (57) as they are proportional to . Alternatively, the matrix elements of such commutators can be viewed to be of higher order in derivatives since where the r.h.s contains in total three derivatives with respect to a or x. Finally the coefficients in the expansions (57) and (18) obviously coincide in the limit of large quantum volumes,ā 2 P . In summary, up to the commutators discussed above, the operator (57) is the result of the canonical quantization of the classical expression (18) via the standard operator replacement (56).
When concentrating on the quadratic contributions in Eq. (57) one recovers the harmonic-oscillator expression of Ref. 29 and corresponding eigenfunctions where H n (x) are the usual Hermite polynomials. We note that ω has dimension of inverse area while 2 P ω is dimensionless and can be computed via Eqs. (54) using α(k) given in Eq. (36) instead of β(a), As stated in Ref. 29 , the expressions (62) and (63) are excellent approximations to the eigenstates and eigenvalues of (square of the) the volume operator for already intermediate lengths of the involved spins. This fact is illustrated again in Fig. 1 for a typical typical choice of angular momentum quantum numbers all being of order a few ten. In addition to Ref. 29 we also plot there the wave function within the lowest-order correction in Eq. (57 arising from cx 3 /3 accounted for by first-order perturbation theory. Fig. 2 shows similar data but for smaller spin lengths j i ≡ 4. Here the oscillator-like features of the wave functions noticeably disappear with increasing n, and the corrections from cubic term are clearly more substantial. In both Figs. 1 and 2 we have used the expression a −ā + 2 P /2 as the argument for the wave functions where the additional increment 2 P takes into account that β(ā) couples states of the form |ā − 2 P and |ā and facilitates comparison with finite size data. With increasing angular momentum quantum numbers, this shift becomes more and more obsolete.

Negative Eigenvalues ofQ
So far we have concentrated on the large and positive eigenvalues of the operatorQ. The regime of negative eigenvalues of large modulus can be explored by canonically quantizing the classical expression (23) according to the standard recipe (56), where we have put for simplicityq =q,ω =ω. This operator is related to Q osc given in Eq. (61) by a gauge transformation along with a change in sign, such that the eigenfunctions are related by where the phase factor mimics the change in sign stated in the strict relation (38) between eigenvectors ofQ to eigenvalues differing in sign only. In fact, based on this analogy, Eq. (67) and, as a consequence, Eqs. (65),(66) have already been given in Ref. 29 . Here we have provided a more profound derivation based on the canonical quantization of the classical expression (23).

Canonical Operators in the Discrete Case
In the operators (56) the variable x (and, in turn, a) is considered to be a continuous quantity. Therefore the question remains how to possibly construct a pair of canonical operators retaining the discrete character of a = k/ 2 P with k being (half-)integer. As a step towards this goal we propose the operatorŝ A = a max a=a min a|a a| . (68) For large volumes, the r.h.s. approaches the unit operator acting on states whose components vary only little on the scale set by 2 P , In fact, the expression (69) is obviously a discretization of a differential operator. However, as such discretizations are by no means unique, the question remains open whether there are operatorsP ,Â which (i) act on the original discretely-labeled quantum states, (ii) turn intô p,x at large volumes, and (iii) fulfill as an exact equation on the entire Hilbert space.

B. General Polyhedra
In full analogy to the Kapovich-Millson variables we define for a quantum polyhedron of N faces (angular momenta) the operatorsˆ for i ∈ {1, . . . , N − 3}. As the squares of these quantities commute with each other, orthonormal basis states of the Hilbert space can be labeled by quantum numbers k i according tô The closure relation (1) i.e. the angular momentum operatorˆ k N −3 couples with the remaining spinsˆ j N −1 ,ˆ j N to a total singlet implying k min Consider now two total singlet states with k i = k and all other quantum numbers k j , j = i identical. Then states with − 1 (and all other k j the same as before) are also singlets, sinceˆ k i−1 andˆ j i+1 can couple to these values of k i , andˆ k i with the above quantum numbers andˆ j i+2 can couple to the given value of k i+1 . Thus, also the other quantum quantum numbers k i , i < N − 3, vary within intervals, k min i ≤ k i ≤ k max i , and the representation theory of the angular momentum algebra implies with k 0 = j 1 for i = 1. Without the additional conditions (78),(79) the inequalities (80),(81) would hold as equalities. We note, however, that the structure of the quantum numbers k i is in general quite complex. For instance, the limiting values k min i , k max i can depend on other quantum numbers k j , j = i, and the entire structure depends obviously also on the coupling scheme, i.e. the labeling of the operators j 1 , . . . , j N . A very simple example is provided by the case N = 4 in Eqs. (29), (30); for N > 4 explicit expressions for k min i , k max i become increasingly tedious. Now rescaling the quantum numbers k i to dimensionful quantities as in section III A 2, k i → a i = k i 2 P , we define analogous to Eqs. (68),(69) the operatorŝ fulfilling the commutation relations In the limit of large volumes, the above r.h.s. approaches the unit operator in just the same way as in Eq. (70).

IV. CONCLUSIONS AND OUTLOOK
The investigation of the semiclassical limit of Loop Quantum Gravity is one of the key issues in that approach towards a quantum theory of gravitation. In the present work we have focussed semiclassical properties of quantum polyhedra. Regarding tetrahedra as their simplest examples, we have explicitly established the connection between a canonical quantization of the classical system with a recent wave function based approach by the present author 29 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. As regards higher orders, the approach of Ref. 29 leads to additional corrections in terms of commutators which are naturally absent in the classical expressions. Including the third-order correction perturbatively leads to improvements of the approximate wave functions. Furthermore, the comparison of both methods leads also to a full wave function description of the eigenstates of negative eigenvalues of large modulus, a result which could only be conjectured in Ref. 29 .
Differently from previous formulations, the position variable used here is chosen to have dimension of (Planck) length squared, 2 P = G/c 3 . This definitional detail is by no means necessary but facilitates the identification of quantum corrections. The ultimate reason for the latter observation is the fact that Planck's constant itself is dimensionful.
A further interesting point is the zero eigenvalue occurring for tetrahedra with odd Hilbert space dimension d where the eigenstate can be given, up to normalization, in a closed form 13 . Moreover, the Bohr-Sommerfeld quantization carried out by Bianchi and Haggard 22,23 yields also surprisingly accurate results for eigenvalues of such small modulus. Thus the question arises whether the eigenstates corresponding to zero eigenvalues can also be cast, for large angular momenta j i 1, in a wave function of a continuous variable.
For general quantum polyhedra described by discrete angular momentum quantum numbers we have formulated a set of quantum operators fulfilling in the semiclassical regime the standard commutation relations between momentum and position. Indeed a major challenge is of course the analysis of the volume operator(s) for higher polyhedra. Results towards this goal were obtained in Refs. 24,25 for pentahedra, and for the general case the operators constructed in the present paper in analogy to the Kapovich-Millson variables of the classical phase space might, although fairly straightforward, provide a useful step.