Loop quantum cosmology: I. Kinematics

May combine two claims.

For LRS models, the Higgs equation has a non-trivial solution only for k=1, in which case φ takes the form φ1 = 2^-1/2(aτ1+bτ2), φ2 = 2^-1/2(-bτ1+aτ2), φ3 = cτ3.

“This equation has a non-trivial solution only for k = 1, in which case φ can be written as φ 1 = 2 -1 2 (aτ 1 + bτ 2 ), φ 2 = 2 -1 2 (-bτ 1 + aτ 2 ), φ 3 = cτ 3”

May combine two claims.

For class A models, the diffeomorphism constraint (10) already vanishes if the Gauss constraint is solved; this follows from an interrelation of SU(2)-gauge transformations and diffeomorphisms due to the Higgs constraint.

“which already vanishes if the Gauß constraint is solved.This is a consequence of an interrelation of SU (2)-gauge transformations and diffeomorphisms due to the Higgs constraint”

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For transitive symmetry groups, B is a single point, U_B is finite-dimensional, and A_B makes no contribution; the reduced theory has finitely many degrees of freedom after quantum symmetry reduction.

“In the case of transitively acting symmetry groups, B = {x 0 } consists of a single point and the space U B is finite-dimensional, whereas A B certainly makes no contribution.The reduced theory will hence only have finitely many degrees of freedom after a quantum symmetry reduction”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

For Bianchi IX the group of inner automorphisms is isomorphic to SO(3) rotating the three edges.

“for Bianchi IX the group of inner automorphisms is isomorphic to SO(3) rotating the three edges.”

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The momenta p^I_i are represented as derivative operators on H_aux and are quantized to combinations of left and right invariant vector fields acting on the I-th copy of SU(2).

“the momenta get quantized to combinations of left and right invariant vector fields pI i = -iι l 2 P 3 J =1 ∂(h J ) A B ∂φ i I (x 0 ) ∂ ∂(h J ) A B = -iι l 2 P 3 J =1 1 2 (τ i h I + h I τ i ) A B ∂ ∂(h I ) A B = -1 2 iι l 2 P X (L) i (h I ) + X (R) i (h I ) (13”

• AssumptionThe authors state this as an assumption: The compactification of S/F can be chosen so that the three curves e_I are closed.

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The LRS Gauß operator contains only vector fields associated with h1; the axial holonomy h3 does not appear.

“Second, the operator contains only vector fields associated with the holonomy h 1 , whereas the axial holonomy h 3 does not appear at all.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The action of an inner automorphism generated by T_K on a point holonomy is Ad(exp(-δT_K)): h → exp(-δT_K) h exp(δT_K).

“we write the action of an inner automorphism generated by T K ∈ LS on a point holonomy as Ad(exp(-δT K )): S → S, h → exp(-δT K )h exp(δT K )”

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The configuration spaces for these models are finite-dimensional and are represented in quantum theory as spaces of point holonomies associated with the single point x0.

“The configuration spaces for our models are given by Higgs 'fields' in a single point, which shows that they are finite-dimensional.In quantum theory they will be represented as spaces of point holonomies [23] associated with a single point, the only point x 0 in the reduced manifold B.”

• cites [23] Kobayashi S, 1963Not checked yet

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The three labels j1,j2,j3 and the vertex contractor are still needed to specify a state; symmetrization only slightly decreases freedom.

“But there are still the three labels j 1 , j 2 and j 3 , and certainly the vertex contractor which are all needed to specify a state.The symmetrization implies only a minor decrease in the freedom.”

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After relaxing the partial gauge fixing, functions on Uτ3LRS can be extended to U[λ1]LRS by demanding conjugation invariance; only for those functions can the gauge-fixed Gauß constraint be used, while on arbitrary functions on U[λ1]LRS all three SU(2)-constraint components must be imposed.

“In this process, functions on the partially gauge-fixed configuration space U τ 3 LRS can be extended to functions on U [λ 1 ] LRS by demanding invariance under conjugation.Only for those functions can the gauge-fixed Gauß constraint be used.On an arbitrary function on U [λ 1 ] LRS we have to impose all three components of an SU (2)-constraint.”

May combine two claims.

For Bianchi I all inner automorphisms are trivial, there is no non-trivial action on the edges, and the diffeomorphism constraint vanishes.

“For example, for Bianchi I all inner automorphisms are trivial and there is no non-trivial action on the edges (the diffeomorphism constraint vanishes in this case)”

Probably not a substantial claim: may not pass S3 (load-bearing).

There are more gauge-invariant functions on the classical configuration space than naively expected.

“Thus there are more gauge-invariant functions on the classical configuration space than naively expected.”

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For LRS models the constraint (10) either vanishes identically when n1+n2=0 or is equivalent to the Gauß constraint, already solved by gauge-invariant states.

“For LRS models the situation is different: here, the constraint (10) either vanishes identically (n (1) + n (2) = 0) or is equivalent to the Gauß constraint, i.e. it is already solved by using gauge-invariant states.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For isotropic models U[λ1]iso is obtained by setting A=c.

“For isotropic models we can obtain U [λ 1 ] iso by setting A = c.”

May combine two claims.

Equation (14) implies all three holonomies are equal for an isotropic model, giving configuration space U[λ0]iso = diagonal SU(2) subgroup of SU(2)^3.

“Equation ( 14) then states that all three holonomies are the same for an isotropic model, i.e. in this case we have the configuration space U [λ 0 ] iso = {(h, h, h) : h ∈ SU (2)}, which is the diagonal SU (2)-subgroup of SU (2) 3 .”

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Spin network states with a reduced number of edges are some of the functions on U[λ1]; for isotropic models they come from anisotropic spin networks with two of three labellings zero.

“Of course, the spin network states with a reduced number of edges are some of those functions.For example, for isotropic models they can be obtained from spin networks in the anisotropic theory with two of the three labellings being zero.”

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In both LRS and isotropic cases there is a unique non-trivial sector, and no topological charge appears.

“Thus, we see that in both cases there is a unique non-trivial sector, and no topological charge appears.”

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These reduced-edge spin network functions do not exhaust all functions on U[λ1]iso: all gauge-invariant one-edge spin network functions are symmetric under c→-c when evaluated at h=exp(cτ3), but there exist gauge-invariant functions not symmetric under this reflection, e.g. tr[...] = 2cos^3(c/2)-2sin^3(c/2).

“But these do not comprise all functions on U [λ 1 ] iso : one can easily see that all such gauge-invariant spin network functions with one edge are symmetric under c → -c if they are evaluated in h = exp(c 3 ).But there are gaugeinvariant functions on U [λ] iso which are not symmetric under this reflection: one example is given by tr[exp(c 1 ) exp(c 2 ) exp(c 3 )] = 2 cos 3 (c/2) -2 sin 3 (c/2)”

May combine two claims.

Classically 3i is fixed, so quantizing the Gauß constraint with partial gauge fixing would involve only one component of invariant vector fields and force only that component to vanish.

“First, classically 3 i is fixed so that the quantization of the Gauß constraint consists of only one component of invariant vector fields (X 3 in the τ 3 -gauge), and it would force only this component to vanish if we would use this partial gauge fixing in quantum theory.”

May combine two claims.

The diffeomorphism constraint for isotropic models vanishes identically, consistent with symmetry reduction being stronger than gauge reduction, so an isotropic state is automatically invariant under inner automorphisms.

“The diffeomorphism constraint for isotropic models vanishes identically, which is consistent with the discussion above: symmetry reduction is stronger than gauge reduction, and therefore an isotropic state is automatically invariant with respect to inner automorphisms.”

May combine two claims.

For Bianchi IX, the isotropic metric is ds^2 = 4e^2[(dr)^2 + sin^2 r dΩ^2], an isotropic closed model of positive spatial curvature.

“In the isotropic case we obtain the metric ds 2 = e 2 3 I =1 ω I ⊗ ω I = 4e 2 [(dr) 2 + sin 2 r d 2 ] of an isotropic closed model of positive spatial curvature.”

May combine two claims.

In the classically reduced manifold (a single point) these transformations have no geometric meaning, but in quantization the auxiliary manifold is introduced and inner automorphisms act on it, moving edges depending on S.

“In the classically reduced manifold (consisting of a single point) these transformations have, of course, no geometric meaning.But in the course of quantization we introduced an auxiliary manifold when promoting point holonomies to holonomies associated with edges therein.The inner automorphisms act on this manifold and thereby move these edges depending on the symmetry group S.”

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Going from Uτ3 to U[λ1] introduces no new degrees of freedom if functions on U[λ1] are required to be gauge invariant under diagonal SU(2)-conjugation, restoring the original SU(2) gauge group.

“Going over from U τ 3 to U [λ 1 ] introduces no new degrees of freedom as long as we require functions on U [λ 1 ] to be gauge invariant under diagonal SU (2)-conjugation.This restores the original gauge group SU (2) of the non-symmetric theory.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Group averaging is simple because all graphs underlying cylindrical functions in the auxiliary Hilbert space have at most three edges.

“This is a simple procedure because all graphs underlying cylindrical functions in the auxiliary Hilbert space have at most three edges.”

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In that case the only inner automorphism with a non-vanishing diffeomorphism component is a rotation around the axial edge; due to the Higgs constraint, such a rotation on a transversal edge is equivalent to a gauge rotation of the associated holonomy.

“In this case the only inner automorphism with respect to which there is a non-vanishing component of the diffeomorphism constraint is a rotation around the axial edge.But owing to the Higgs constraint ( 14) such a rotation applied to a transversal edge is equivalent to a gauge rotation of the associated holonomy.”

May combine two claims.

For cosmological models there is just a single vertex; no new edges can be created, only spins can be changed.

“For cosmological models, there is just a single vertex.No new edges can be created; only the spins can be changed.”

May combine two claims.

For n1=0 there is no inner automorphism in the transversal plane and no diffeomorphism constraint is needed.

“For n (1) = 0, on the other hand, there is no inner automorphism in the transversal plane and no diffeomorphism constraint is needed.”

May combine two claims.

In loop quantum gravity, basic geometric quantities have been quantized and found to have discrete spectra.

“Basic geometric quantities have been quantized [10][11][12][13][14] and found to have discrete spectra.”

• cites [10] sourceNot checked yet

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• cites [11] sourceNot checked yet

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• cites [12] sourceNot checked yet

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May combine two claims.

The Wheeler-DeWitt equation is a hyperbolic differential equation involving the scale factor, and it remains hyperbolic after small perturbations of homogeneous metrics.

“This equation is a hyperbolic differential equation involving the scale factor of the universe, which plays a role analogous to a time variable.It remains hyperbolic after small perturbations of the homogeneous metrics [3].”

• cites [3] sourceNot checked yet

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This application allows loop quantum gravity to be tested in a large class of minisuperspaces and its features, such as the discrete volume spectrum, to be investigated in certain cosmological regimes.

“This allows us to test loop quantum gravity in a large class of minisuperspaces and to investigate its features-e.g. the discrete volume spectrum-in certain cosmological regimes.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The Wheeler-DeWitt equation is the quantized Hamiltonian constraint of general relativity.

“the Wheeler-DeWitt equation [1], which is the quantized Hamiltonian constraint of general relativity.”

• cites [1] sourceNot checked yet

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May combine two claims.

The Gauss and diffeomorphism constraints are similar: G generates conjugation in internal SU(2) space, while D generates conjugation in the homogeneous space S; their constants are the structure constants of SU(2) and S.

“We note here that G and D are very similar: G generates as gauge transformations conjugation in the internal SU (2) space, whereas D generates conjugation in the homogeneous space S. Therefore, the constants appearing in the constraints are the structure constants ij k of SU (2) and c I J K of S, respectively.”

May combine two claims.

Bianchi class A models constitute all homogeneous models with a freely acting symmetry group (F={1}) that can be treated in a Hamiltonian formulation; Bianchi class B models violate the principle of symmetric criticality.

“Bianchi class A models constitute all homogeneous models with a freely acting symmetry group (F = {1}) which can be treated in a Hamiltonian formulation (Bianchi class B models violate the principle of symmetric criticality [25,26]).”

• cites [25] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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• cites [26] Bojowald M, 2000, Class. Quantum Grav. sourceNot checked yet

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May combine two claims.

For LRS models the reduced gauge group is U(1), while for isotropic models the gauge is completely fixed and no gauge group remains.

“for LRS models the reduced gauge group is U(1), whereas for isotropic models it is fixed totally and there is no gauge group at all.”

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The reconstruction of invariant connections from the Higgs field pulls back a spin network function to a function on A_B × U_B; these functions are distributional states of the unreduced theory symmetric under [λ], and all symmetric states arise this way.

“The basic idea of [20] is to use the reconstruction of invariant connections from the Higgs field to pull back a spin network function, which is a function on the space A of generalized connections on modulo gauge transformations, to a function on the space A B × U B of generalized connections and Higgs fields on the reduced manifold B.”

• cites [20] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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Inner automorphisms of S are the remaining gauge freedom under the diffeomorphism group, and the diffeomorphism constraint enforces independence of the choice of base point.

“Inner automorphisms of S are the remaining gauge freedom under the diffeomorphism group, and the diffeomorphism constraint-which demands independence of the physical phase space under inner automorphisms-can be seen to enforce independence of the selection of a base point.”

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The space of point holonomies is the quantum configuration space only for freely acting symmetry groups; if F is non-trivial, condition (2) further constrains this space.

“The space of point holonomies will, however, be the quantum configuration space only for freely acting symmetry groups.If F is non-trivial, the condition (2) will further constrain this space.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The insertion can be viewed as a reduction of the vertex contractor in spin networks with three closed edges.

“This insertion can be viewed as a reduction of the vertex contractor in spin networks with three closed edges, which was not taken into account in the naive arguments presented above.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Bianchi models describe all possible types of metrics which are homogeneous in space.

“Bianchi models describe all possible types of metrics which are homogeneous in space [27].”

• cites [27] MacCallum M A H, 1972, Commun. Math. Phys. sourceNot checked yet

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Bianchi type I and IX can be constrained to isotropic metrics; the only other class A model for which this is possible is type VII0, which yields an isotropic model equivalent to type I.

“Bianchi type I and IX can even be constrained to isotropic metrics, i.e.F ∼ = SU (2).(The only other class A model for which this is possible is type VII 0 .It yields, however, an isotropic model equivalent to that of type I [27].)”

• cites [27] MacCallum M A H, 1972, Commun. Math. Phys. sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

The Wheeler-DeWitt operator in loop quantum gravity remains poorly understood.

“Unfortunately, this operator of the full theory remains poorly understood.”

• cites [17] Smolin L, 1993Not checked yet

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• cites [18] sourceNot checked yet

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• cites [19] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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Because the λ1 solution space is not a subgroup, the Peter-Weyl theorem no longer applies and all functions on U[λ1] must be determined by hand.

“The Peter-Weyl theorem no longer applies, and we have to determine all functions on U [λ 1 ] by hand.”

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There is gauge freedom in the directions of the edges e_I due to the diffeomorphism constraint acting by inner automorphisms of S; this freedom is fixed by solving the diffeomorphism constraint in section 5.

“There is some arbitrariness in the directions of the e I : depending on the special model there is gauge freedom due to the diffeomorphism constraint which acts by inner automorphisms of S. This freedom will be fixed by solving the diffeomorphism constraint in section 5”

• AssumptionThe authors state this as an assumption: The compactification of S/F can be chosen so that the three curves e_I are closed.

Probably not a substantial claim: may not pass S3 (load-bearing).

The Higgs constraint can be analysed more easily in the reduced model because there is only one Higgs vertex.

“In particular, the Higgs constraint ( [20] and section 2) can be analysed more easily because there is just one Higgs vertex in the reduced model.”

• cites [20] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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Using point holonomies with respect to G instead of Z_λ undoes the partial gauge fixing; this is mandatory because point holonomies transform under the adjoint representation of G, whereas Higgs fields need not transform under the adjoint representation of Z_λ.

“By using point holonomies with respect to the structure group G instead of Z λ the partial gauge fixing can be undone.This is even mandatory, because point holonomies necessarily transform under the adjoint representation of the structure group, whereas Higgs fields will not necessarily transform under the adjoint representation of Z λ .”

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The paper presents the basic setting for cosmological models within loop quantum gravity, and the kinematical level has been almost completely solved.

“In this paper we have presented the basic setting for the study of cosmological models within loop quantum gravity.The kinematical level has been almost completely solved.”