Analysis of the expectation value of the volume operator for 8-valent graphs

the case of 4- and 6-valent graphs, we first consider the non-rotated graph. We then analyse the rotational and translational dependence of the expectation value by performing a rotation and then a translation of the graph; we then repeat the calculation. It transpires that, even for the 8-valent graph, the off-diagonal elements of the matrix have non-trivial contributions that cause the expectation value of the volume operator to be translationally and rotationally dependent, for higher orders than the zeroth-one.

3.4.1 Expectation value of the volume operator for a general 8-valent graph

As in the previous cases, we take the (0, 0, 0) point of the lattice to coincide with the (0, 0, 0) point of the plaquette, and each vertex to be symmetric with respect to the axis. The coordinates of the vertices

comprising the periodicity cell are the following:

For the 8-valent lattice we choose V1 as our reference vertex. The allowed value for its x-coordinate is nl <|xV²| < nl +₄^l, where n = [√^δ

3l]. Similarly to the cases of 4- and 6-valent graphs, the allowed positions of the remaining vertices in the periodicity cell can be computed from the allowed positions of V₁. Because of the geometry of the 8-valent graph we obtain

T_e^{x,y}_i = F_e^{x,y}_i = Z_e^{x,y}_i = 1 ∀ei∈ γ (3.57) This results in the following values for the terms √^t_t^eiej

eit_ej as computed for the above five vertices.

V0 all√^teiej_t

3 − nl)^√1_3δ and it is proportional to the off-diagonal entries of the matrix√

A⁻¹ whose values are given in Section A.3.1 in the Appendix.

The expectation value of the volume operator is:

V_R=r 1 By approximating to first-order (see Section 2.2) we obtain

VR=4δ³ In this case, the deviation from the classical value of the volume of a region, R, is of the order four, even to zeroth-order in l/δ.

3.4.2 Expectation value of the volume operator for a rotated 8-valent graph

We will now analyse the expectation value of the volume operator for a rotated 8-valent graph. In order to make the comparison with the 4- and 6-valent graphs as accurate as possible, we will rotate the 8-valent graph by the same amount the other valence graphs were rotated. It follows that the angles of the edges incident at V₀ will satisfy the following conditions:

1) 0 < θ0,2, θe0,8 < ^π₂, ^3π₂ < θ0,3, θ0,7< 2π ^π₂ < θ0,1, θ0,6 < π and π < θ0,4, θ0,5 < ^5π₄ .

2) ^3π₂ < φ_e0,2, φ_e0,3 < 2π− sin^{−1 1}₃, π + sin⁻¹(¹₃) < φ_e0,7, φ_e0,8 < ^3π₂ ,^3π₂ < φ_e0,6, φ_e0,5 < π− sin⁻¹(¹₃) and sin⁻¹(¹₃) < φ_e0,4, φ_e0,1 < ^π₂

The angles for the co-linear edges are defined through the formula θ_e = 180^◦ − θecollinear and φ_e = 180^◦− φecollinear respectively. It follows that the edges e_0,1, e_0,2, e_0,3, e_0,4, e_0,5, e_0,6, e_0,7 and e_0,8 lie in the octants B, A, D, C, G, F , H and E respectively. The coordinates of the rotated vertices are

V₁^′ = 

(−R11− R12+ R₁₃) δ

√3, (−R21− R22+ R₂₃) δ

√3, (−R31− R32+ R₃₃) δ

√3



(3.60) V₉^′ = 

− 2R12

√δ

3,−2R22

√δ

3,−2R32

√δ 3

 (3.61)

V₁₂^′ = 

(−R12+ 2R₁₃) δ

√3, (−R22+ 2R₂₃) δ

√3, (−2R32+ 2R₃₃) δ

√3



(3.62) V_e^′ = 

(−R11− R12+ 3R13) δ

√3, (−R21− R22+ 3R23) δ

√3, (−R31− R32+ 3R33) δ

√3



(3.63) As was done for the 4- and 6-valent graphs, in order to carry out the calculations for the expectation value of the volume operator we would have to specify a particular combinations of angles satisfying conditions 1) and 2) above. However, all combinations satisfying 1) and 2) above lead to the same value, in zeroth-order in _δ^l, of the expectation value of the volume operator. Rotational dependence will only appear for higher orders in _δ^l. Moreover any sub-case of 1) and 2) will lead to the same couples of edges commonly intersecting surfaces s^I_α,t in a given stack. Therefore, to leave the result as general as possible, we will not specify a particular sub-case of 1) and 2) but simply derive a general expression for the expectation value of the volume operator given conditions 1)and 2).

The expectation value of the volume operator for a periodicity region, R, is then computed as

δ³ q

| det E_j^a(u)
| 4

 5 + 1

2

− 1 32

n 2(n−1)

X

i,j=1;j6=i

α²_ji    det

 δX_S^a δ(s, u¹, u²)

  



(3.64)

where the terms α_ij are the off-diagonal entries of the matrix √

A⁻¹ (See Section A.3.2 in the Appendix) Evidently, the higher-order corrections are angle dependent, while the zeroth-ones are not. Therefore, as for the 4- and 6-valent case, the expectation value of the volume operator for the 8-valent graph is rotational invariant, in zeroth-order up to measure zero in SO(3). However, it does not reproduce the correct semiclassical limit.

3.4.3 Expectation value of the volume operator for a translated 8-valent graph

We now consider the translated 8-valent graph. As for the 4- and 6-valent graphs we choose the following conditions on the components of the translation vector:

1) b > ǫ_x> ǫ_y > ǫ_z > 0

2) |Vkⁱ| − |Vk^j| > nⁱkl− n^jkl for all|Vkⁱ| > |Vk^j| Similarly as for the aligned 8-valent graph we have

T_e^{x,y}_i = F_e^{x,y}_i = Z_e^{x,y}_i = 1 ∀ei∈ γ (3.65)

The coordinates of the translated vertices are

The expression for the matrix√

A⁻¹ is given in Section A.3.3 in the Appendix.

The value obtained for the volume of a region R is:

V_R=r 1

Considering only contributions of first-order in _δ^l (see Section 2.2) we obtain

A⁻¹ as defined in Section A.3.3 of the Appendix. Again, translational invariance holds only at zeroth-order up to measure zero in SO(3). However, it does not reproduce the correct semiclassical limit.

4 Summary and Conclusions

We have shown that if we use semiclassical states derived from the area complexifier, then we do not obtain the correct semiclassical value of the volume, operator unless we perform an artificial re-scaling of the coherent state label and we restrict our calculation to the following special cases:

1) The edges of the graph are aligned with the orientation of the plaquettes (6-valent graph).

2) Two or more edges lie in a given plaquette (4-valent graph).

3) One edge is aligned with a given plaquette while a second edge lies in a given plaque (4-valent graph).

However, such combination of edges have measure zero in SO(3). For embeddings whose measure in SO(3) is non-trivial we do not obtain the correct semiclassical behavior for the volume operator for any valence of the graph.

This result suggests strongly that the area complexifier coherent states are not the correct states with which to analyse semiclassical properties in LQG. Moreover, as previously mentioned, if embedding inde-pendence (staircase problem) is to be eliminated, area complexifier coherent states should be ruled out as semiclassical states altogether.

Acknowledgments

I would like to thank very much Thomas Thiemann for illuminating discussions and useful comments on the manuscript. I am also grateful to the Perimeter Institute for Theoretical Physics for hospitality and financial support where parts of the present work were carried out. Research performed at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.

A Edge metric calculations

A.1 Edge metric components for the 4-valent graph In this Section we list the values for each of the terms √^t_t^eiej

eit_ej which comprise the matrix√

A⁻¹that appears in the computation of the expectation value of the volume operator. The method for computing such terms was described in Section 3.1 with the aid of a two-dimensional example. It is worth recalling that the term t_eiej = t^x_eiej+ t^y_eiej+ t^z_eiej represents the total number of plaquettes in each direction that are intersected by both the edges ei and ej. On the other hand, the terms tei = t^x_ei+ t^yei+ t^z_ei and tej = t^x_ej+ t^yej+ t^z_ej represent are the total number of plaquettes in each direction intersected by the edges e_i and e_j respectively.