Symmetry reduction for quantized diffeomorphism-invariant theories of connections

If a cylindrical function is not zero on the image of r_[λ], then some distribution in σ_[λ](B) does not annihilate it.

“There is no cylindrical function f ∈ that is non-trivial on Im(r [λ] ) and annihilated by all distributions in σ [λ] ( B ).”

Every state in the range of σ_[λ] is [λ]-symmetric.

“The image of σ [λ] contains only [λ]-symmetric states.”

The reconstruction map in the (2+1)-dimensional gravity example sends (A_i^a τ_i dx^a, φ^i τ_i) to A_i^a τ_i dx^a + φ^i τ_i dz.

“the reconstruction is r: (A i a τ i dx a , φ i τ i ) → A i a τ i dx a + φ i τ i dz.(15)”

For spherically symmetric electromagnetism, the centralizers Z_λn = U(1) coincide with the full group.

“For the Abelian group G the centralizers Z λ n = U(1) are equal to the full group.”

Acting on a gauge-invariant state (η_B)^K, p yields hK(η_B)^K.

“Acting on a gauge-invariant state (16) this yields p (η B ) K = hK(η B ) K .”

Gauge-invariant functions, namely cylindrical functions on A_B × U_B / G_B, provide a solution to the Gauß constraint.

“The Gauß constraint is solved by using gauge-invariant functions, i.e. cylindrical functions on A B × U B /G B .”

A projected spin-network state is gauge invariant if and only if the original spin-network state in the full space is gauge invariant.

“above calculations exhibit the fact that a projected spin-network state is gauge invariant if and only if the original spin-network state in is gauge invariant.”

The reduced spherically symmetric electromagnetic theory has only two canonical degrees of freedom.

“We see that the reduced theory is particularly simple, having only two canonical degrees of freedom.”

In that case all reduced bundles are trivial because B is contractible.

“Thus, a spherically symmetric principal fibre bundle with structure group U(1) is classified by an integer n, and, of course, all reduced bundles are trivial, because B is contractible, and they are not needed for the classification.”

A spherically symmetric principal fibre bundle with structure group U(1) is classified by an integer n.

“Thus, a spherically symmetric principal fibre bundle with structure group U(1) is classified by an integer n, and, of course, all reduced bundles are trivial, because B is contractible, and they are not needed for the classification.”

The physical Hilbert space can be identified with L2(U(1), dμ_H), where dμ_H is the Haar measure on U(1).

“The physical Hilbert space can be identified with L 2 (U (1), dµ H ), where dµ H is the Haar measure on U(1).”

The reduced theory for spherically symmetric electromagnetism is a theory of a U(1) connection only.

“This proves that the reduced theory is a theory of a U(1) connection only.”

Distributions in σ_[λ](B) distinguish cylindrical functions whose restrictions to [λ]-invariant connections are different.

“In particular, the space σ [λ] ( B ) separates the elements of that differ when restricted to [λ]-invariant connections.”

The generalized gauge-invariant [λ]-invariant connections in A/G are exactly the image of r_[λ].

“The subset of generalized gauge invariant, [λ]-invariant connections in A /G is given by Im(r [λ] ), the image of the map r [λ] .”

In the connection representation, p is quantized as -ih δ/δω.

“standard quantization rules lead in the connection representation to the quantization p = -ih δ δω .”

Gauge invariance in the full space forces the spin-network labelling to satisfy equality of sums of edge labels over intersections with any two S-orbits not containing a vertex.

“This follows from the fact that gauge invariance in forces the spin-network labelling to fulfil”

• AssumptionThe authors state this as an assumption: All edges of the spin network are oriented outwards.

The range of σ_[λ] is dense in the [λ]-symmetric state space.

“In conclusion, the map σ [λ] is injective (by construction of B , where the Higgs constraint (5) is assumed to be solved, and by the construction of σ [λ] ), and has a dense image in the space of [λ]-symmetric states.”

The holonomy of iω along an edge e equals h_{π(e)}(iω_x)(β_e)^n, where π(e) is the projected edge in B and β_e is a phase factor depending only on the geometry of e.

“Here π(e) := [x 1 , x 2 ] = [x(e(0)), x(e(1))] ⊂ B is an edge in B, and h π(e) (i ω x ) is the holonomy of the reduced connection i ω x along it.β e is a phase factor depending only on the geometry of e.”

For spherically symmetric electromagnetism, the physical Hilbert space is the space of gauge-invariant Ashtekar-Lewandowski square-integrable functions on generalized U(1) connections over B.

“According to theorem 3.2 the physical Hilbert space of spherically symmetric electromagnetism is given by L 2 (A B /G B , dµ AL ), the space of gauge invariant, Ashtekar-Lewandowski square-integrable functions on the space of generalized U(1) connections over B.”

The map σ_[λ] is injective.

“In conclusion, the map σ [λ] is injective (by construction of B , where the Higgs constraint (5) is assumed to be solved, and by the construction of σ [λ] ), and has a dense image in the space of [λ]-symmetric states.”

• AssumptionThe authors state this as an assumption: The Higgs constraint (5) is solved in the construction of B.

In the (2+1)-dimensional gravity example, the reduced structure group Z_λ equals the full gauge group G=SU(2).

“In this case the reduced group Z λ = G is identical to the full structure group.”

The image of B under σ_[λ] is dense among [λ]-symmetric states.

“This implies that σ [λ] ( B ) is dense in symm in the weak topology [40, chapter II, section 6.2, corollary 4], and, therefore, in the space of [λ]-symmetric states.”

• cites [40, chapter II, section 6.2, corollary 4]Not checked yet

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A gauge-invariant spin network projects to a spin network with labelling k_e^B = K for each edge of the projected graph.

“Therefore, a gauge-invariant spin network will project to a spin network with labelling k e B = K for each edge e B of the projected graph.”

For a principal bundle P, the collection of generalized connections that are both gauge-invariant and [λ]-invariant forms a closed subset of A/G.

“The set of generalized gauge-invariant [λ]-invariant connections on P is closed in A /G .”

For any spin network T, r_{[λ_n]}T is cylindrical on A_B.

“We can see that r [λ n ] T ∈ B is cylindrical on A B for any spin network T ∈ .”

The symmetry reduction map satisfies r_{[λ_n]} T_{γ,k} = (β_{γ,k})^n π(T_{γ,k}).

“This equation enables us to write r [λ n ] T γ, k = (β γ, k ) n π(T γ, k ).”

The curvature of that invariant connection gives the densitized magnetic field of a Dirac monopole with magnetic charge n, with B_x = -n sinθ and B_θ = B_φ = 0.

“We can calculate the curvature of this [λ n ]-invariant connection to obtain the (densitized) magnetic field of a Dirac monopole with magnetic charge n:”

The diffeomorphism constraint can be solved by group averaging.

“The diffeomorphism constraint can be solved by group averaging.”

If the underlying graph contains an edge neither entirely in an orbit nor entirely in B, then f depends on all components of a connection along that edge.

“If the underlying graph contains an edge that lies neither entirely in an orbit nor entirely in the manifold B, then f will depend on all components of a connection in points along the edge.”

Spin-network states are eigenstates of p.

“We see that spin-network states are eigenstates of p.”

For a compact Hausdorff topological group G and a subgroup H, the centralizer of H in G is a compact subgroup of G.

“Let G be a compact topological group which is Hausdorff and H be a subgroup of G.The centralizer Z G (H ) := {g ∈ G : gh = hg for all h ∈ H } is a compact subgroup of G.”

• cites [39]Not checked yet

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In the large-j limit, the spherically symmetric area spectrum is consistent with the horizon area spectrum (36).

“and in the large-j limit it is consistent with the spectrum (36) of the horizon area.”

For large j, the spherically symmetric area spectrum becomes equidistant rather than dense.

“In particular, for large j the spectrum becomes not dense but equidistant”

Spherically symmetric gravity has a topological charge taking values in N0 in the dreibein formulation.

“We see that spherically symmetric gravity has a topological charge taking values in N 0 (in the dreibein, not in the metric formulation, as we will see below).”

The term cosθ dφ τ3 in (25) leads automatically to the appearance of A3 as the combination A3-√2 in the connection form (26) of [24,25].

“We note that the term cos ϑ dϕ τ 3 in equation (25), which is the only term independent of the fields a, b and c, in the connection form A(x, ϑ, ϕ) derived above leads automatically to the appearance of A 3 as the combination A 3 -√ 2 in the connection form (26) of [24,25].”

• cites [24] sourceNot checked yet

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• cites [25] Barbero G J F, 1995, Phys. Rev. sourceNot checked yet

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In an appropriate topology, the area spectrum consists of all values of form (41) with j_v a nonnegative integer and j_+ a nonnegative half-integer arbitrary, while j_- is restricted by |j_+-j_v| ≤ j_- ≤ j_++j_v.

“Here we note that in an appropriate topology of the area spectrum is given by all values of the form (41) where j v ∈ N 0 and j + ∈ 1 2 N 0 are arbitrary, whereas j -is restricted by |j + -j v | j -j + + j v .”

• AssumptionThe authors state this as an assumption: Each edge containing a general point x starts at x.

In the λ-gauge-fixed quantization, choosing m=j yields the area spectrum (1/2)ι l_P^2 N0.

“In this way we obtain the spectrum 1 2 ι l 2 P N 0 for the operator, analogously to [43].”

• cites [43] Mäkelä J, 1998Not checked yet

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The calculation shows c is identified with A1 and (a,b) with (A2,A3) up to a rescaling by √2 used in [24].

“The calculation above shows that c has to be identified with A 1 and (up to a rescaling with √ 2 used in [24] in order to obtain the standard symplectic structure) (a, b) with (A 2 , A 3 ).”

• cites [24] sourceNot checked yet

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The reconstructed connection form (25) differs from the connection form (26) used in [24,25] only by a gauge rotation g1 := exp(-θ n_i φ τ_i) followed by g2 := exp(-φ n_i x τ_i).

“The two connection forms differ, however, only by a gauge rotation g 1 := exp(-ϑn i ϕ τ i ) followed by a second gauge rotation g 2 := exp(-ϕn i x τ i ).”

• cites [24] sourceNot checked yet

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• cites [25] Barbero G J F, 1995, Phys. Rev. sourceNot checked yet

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The basis states |K := (η_B)^K span an orthonormal basis with respect to the Haar measure on U(1)=A_B/G_B derived from the Ashtekar-Lewandowski measure on A_B.

“These states span an orthonormal basis with respect to the Haar measure on U(1) = A B /G B which derives from the Ashtekar-Lewandowski measure on A B .”

The electromagnetic model exhibits a simple reduction of degrees of freedom to finitely many ones corresponding to the classical theory.

“The electromagnetic model exhibits a simple reduction of the degrees of freedom to finitely many ones corresponding to the classical theory.”

The symplectic structure for rescaled variables (A_I,E_J) is {A_I(x), E_J(y)} = κι/4π δ^J_I δ(x,y), with κ=8πG.

“{A I (x), E J (y)} = κι 4π δ J I δ(x, y)(27)”

The area operator in the non-degenerate sector has eigenvalues (1/2)ι l_P^2 [2j_+(j_++1)+2j_-(j_-+1)-j_v(j_v+1)].

“with eigenvalues 1 2 ι l 2 P 2j + (j + + 1) + 2j -(j -+ 1) -j v (j v + 1).”

• AssumptionThe authors state this as an assumption: Each edge containing a general point x starts at x.

The partial gauge fixing can be undone in the quantum theory by extending the spin networks by SU(2) gauge invariance to spin-network functions on the space of SU(2) connections and appropriate Higgs fields over B.

“As described in the general framework we can undo the partial gauge fixing in the quantum theory by extending the spin networks by SU (2) gauge invariance to spin-network functions on the space of SU (2) connections and appropriate Higgs fields over B.”

The spherically symmetric three-dimensional volume element transverse to the orbits is dV = 4π√2 √(E^1 E) dx.

“and the spherically symmetric three-dimensional volume element dV 'transverse' to those orbits is”

The classical symmetry and constraint reduction of spherically symmetric quantum gravity reveals only one physical configuration degree of freedom, the mass (or a canonical pair mass and time).

“The classical symmetry and constraint reduction reveals that there is only one physical configuration degree of freedom, the mass (or a canonical pair: mass and time).”

Each edge holonomy in an SU(2) spin network can be projected to a U(1) holonomy by inserting the projector |m,n><m,n| in each edge with spin j, where m is arbitrary but fixed for each j.

“We can now project each edge holonomy in an SU (2) spin network to a U(1) holonomy by inserting the projector |m, n j m, n| j in each edge with spin j , where m, -j m j is arbitrary but fixed (for each j ).”

• AssumptionThe authors state this as an assumption: In the classification, only homomorphisms λ that are constant along B are used.

The transformed Gauß constraint in the gτ3g^-1 gauge is Ĝ3[Λ] = Ad^3i(g) Ĝi[Λ] =: Ĝ[Λ^i].

“The transformed Gauß constraint in the gτ 3 g -1 gauge is now”

The surface area A(x) of the two-dimensional spherical orbit at x is 4π E^1(x).

“The surface A(x) of the two-dimensional spherical orbit generated by the symmetry group S at x ∈ B is given by”

All conjugacy classes of homomorphisms λ: U(1)->SU(2) are represented by λ_k: z -> diag(z^k, z^-k) for k∈N0 after dividing by the Weyl group.

“All homomorphisms in Hom(U (1), T (SU (2))) are then given by λ k : z → diag(z k , z -k ) for any k ∈ Z, as in the electromagnetic example.However, here we have to divide out the action of the Weyl group, leaving only the maps λ k , k ∈ N 0 , as representatives of all conjugacy classes of homomorphisms.”

The k=1 sectors describe degenerate sectors different from the degenerate sector found in [24].

“Thus, the sectors with k = 1 describe degenerate sectors which are, however, different from that found in [24].”

• cites [24] sourceNot checked yet

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For a symmetry group acting on a principal fibre bundle, symmetric states of a diffeomorphism-invariant quantum theory of connections on that bundle can be defined.

“Given a symmetry group acting on a principal fibre bundle, symmetric states of the quantum theory of a diffeomorphism-invariant theory of connections on this fibre bundle are defined.”

Restricting to that Hilbert space gives a quantum symmetry reduction procedure inside the spin-network framework.

“Restriction to this Hilbert space yields a quantum symmetry reduction procedure within the framework of spin-network states, the structure of which is analysed in detail.”

The paper's main result, theorem 3.2, identifies spaces of symmetric states with certain spin-network spaces.

“Our main result is theorem 3.2, by means of which we can identify spaces of symmetric states with certain spin-network spaces.”

That area spectrum result clarifies and reconciles the relation between the general non-symmetric area operator spectrum and the older Bekenstein proposal that area eigenvalues scale linearly with n.

“This result clarifies (and reconciles) the relationship between the more complicated spectrum of the general (non-symmetric) area operator in loop quantum gravity and the old Bekenstein proposal that A n ∝ n.”

An S-symmetric principal fibre bundle with isotropy subgroup F is uniquely characterized by a conjugacy class [λ] of homomorphisms λ:F→G and a reduced bundle Q.

“Theorem 2.1.A S-symmetric principal fibre bundle P ( , G, π) with the isotropy subgroup F S of the action of S on is uniquely characterized by a conjugacy class [λ] of homomorphisms λ: F → G together with a reduced bundle Q( /S, Z G (λ(F )), π Q ).”

• cites [22] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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• AssumptionThe authors state this as an assumption: All orbits of the symmetry group S are of the same type.
• AssumptionThe authors state this as an assumption: The coset space S/F is reductive.
• AssumptionThe authors state this as an assumption: The action of S on the base manifold has orbits given by S/F for every point.

An S-invariant connection on P is uniquely classified by a reduced connection on Q and a Higgs field φ obeying equation (5).

“Then the connection ω is uniquely classified by the reduced connection ω on Q and the Higgs field φ: Q × LF ⊥ → LG obeying equation (5).”

• cites [22] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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• AssumptionThe authors state this as an assumption: All orbits of the symmetry group S are of the same type.
• AssumptionThe authors state this as an assumption: The coset space S/F is reductive.
• AssumptionThe authors state this as an assumption: The action of S on the base manifold has orbits given by S/F for every point.

An invariant connection on a manifold can be decomposed into a reduced connection on a submanifold B and scalar fields (a Higgs field) acted on by a representation of the reduced structure group.

“It is well known [20][21][22] that an invariant connection on a manifold can be decomposed into a reduced connection of a reduced gauge group on a submanifold B ⊂ plus some scalar fields on B acted on by a group action determined by the symmetry reduction (a representation of the reduced structure group).”

• cites [20][21][22]Not checked yet

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• AssumptionThe authors state this as an assumption: All orbits of the symmetry group S are of the same type.
• AssumptionThe authors state this as an assumption: The coset space S/F is reductive.

The approach to (2+1)-dimensional gravity yields the kinematical Hilbert space previously derived and used by Thiemann.

“As to (2+1)-dimensional gravity we shall show how our approach leads to the kinematical Hilbert space derived and used by Thiemann [19].”

• cites [19] sourceNot checked yet

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A classical multipole expansion for spherically symmetric electromagnetism has only electric and magnetic charge as parameters, matching the quantum symmetric-state result.

“This is illustrated by our example of spherically symmetric electromagnetism: classically a multipole expansion shows that there are only two parameters, electric and magnetic charge, corresponding exactly to the result for symmetric states of the quantum theory (see section 4.2).”

Those symmetric states, using a scalar product taken from the Ashtekar-Lewandowski measure for loop quantum gravity, constitute a Hilbert space.

“These symmetric states, equipped with a scalar product derived from the Ashtekar-Lewandowski measure for loop quantum gravity, form a Hilbert space of their own.”

Equation (6) includes as special cases the invariant connections found in [36], which are gauge equivalent through gauge transformations depending on angular coordinates.

“Equation ( 6) contains as special cases the invariant connections found in [36].These are gauge equivalent by gauge transformations depending on the angular coordinates (ϑ, ϕ), i.e. they correspond to homomorphisms λ which are not constant on the orbits of the symmetry group.”

• cites [36] sourceNot checked yet

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• AssumptionThe authors state this as an assumption: All orbits of the symmetry group S are of the same type.
• AssumptionThe authors state this as an assumption: The coset space S/F is reductive.
• AssumptionThe authors state this as an assumption: The action of S on the base manifold has orbits given by S/F for every point.

These black-hole-related models have only finitely many physical degrees of freedom.

“Similar to (2 + 1)-dimensional gravity they have only finitely many physical degrees of freedom”

Symmetric states must be distributions (generalized functions) on the function space over the quantum configuration space.

“A lesson, however, can nevertheless be drawn from group averaging.Symmetric states have to be distributions (generalized functions) on the function space over the quantum configuration space.”

For a non-compact space manifold, the symmetries can be interpreted as global asymptotic transformations at spatial infinity that are not part of the gauge group.

“For a non-compact space manifold one can interpret the symmetries as global asymptotic transformations at spatial infinity which are not regarded as elements of the gauge group [23].”

• cites [23] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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The canonical variables of the symmetry reduced theory are the radial components p and iω of the electric field and the U(1) connection form.

“The canonical variables of the symmetry reduced theory are given by the radial components p and iω of the electric field and the U(1) connection form, respectively.”

• cites [26] Thiemann T, 1996, Class. Quantum Grav. sourceNot checked yet

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In that example, the Higgs field supplies the z-component of the connection form.

“This shows that the Higgs field gives the z-component of the connection form.”

Spin-network functions with Higgs field vertices form an orthogonal basis of the auxiliary Hilbert space.

“An orthogonal basis is given by the set of spin-network functions with Higgs field vertices.”

The symmetry reduction can be imposed by adding a constraint to electrodynamics besides the Gauß constraint, yielding an Abelian BF-theory.

“In fact, the symmetry reduction considered here can be imposed by supplementing electrodynamics with a further constraint besides the Gauß constraint to arrive at an Abelian BF -theory [38].”

• cites [38] sourceNot checked yet

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Group averaging cannot give the desired result for spherically symmetric quantum gravity because one would need to average rotations of an edge while holding other edges fixed, which causes problems when imposing gauge invariance.

“We would have to average not only over rotated subgraphs of the graph underlying a given spin network which is to be averaged, but it would be necessary to average over all rotations of an edge while keeping the other edges fixed because these give the same holonomy when evaluated in an invariant connection.”

The emergence of the Higgs field reflects information contained in edges lying entirely in S-orbits.

“The emergence of the Higgs field can be understood as a reflection of the information which is contained in those edges associated with a cylindrical state which are located entirely in the orbits of S.”

In general, r_[λ] f involves an infinite number of Higgs vertices for a cylindrical function f.

“From this picture one can see that, in general, the function r [λ] f will involve an infinite number of Higgs vertices, where f ∈ is a cylindrical function.”

In a diffeomorphism-invariant quantum theory, defining symmetric states as invariant under the symmetry group's action on Hilbert space is inappropriate because the symmetry group is a subgroup of the diffeomorphism group.

“One could also imagine defining symmetric states as states which are invariant with respect to an action of the symmetry group on the Hilbert space, which would be the usual procedure applied to conventional quantum mechanical systems.In a diffeomorphism-invariant quantum theory, however, this is inappropriate because the symmetry group is a subgroup of the diffeomorphism group.”

Restricting the support of a spin-network function to symmetric connections produces a singular distribution.

“Constraining the support of a spin-network function to only symmetric connections yields a singular distribution.”

The conjugacy classes of homomorphisms correspond to the values of a topological charge.

“This confirms our claim that the conjugacy classes of homomorphisms correspond to the values of a topological charge.”

Degeneracy is crucial in the canonical partition function calculation of black hole entropy.

“The degeneracy plays a crucial role in the calculation [27,28] of black hole entropy using a canonical partition function approach.”

• cites [27] Torre C G, 1999, Int. J. Theor. Phys. sourceNot checked yet

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• cites [28] sourceNot checked yet

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For a compact space manifold, it is impossible to select translation-invariant states for cosmological models in a non-trivial way.

“It would then be impossible to select, for example, translation-invariant states for cosmological models in a non-trivial manner.”

The construction does more than quantize a reduced model because any quantum state of the reduced theory can be interpreted as a symmetric state of the full quantum theory.

“however, our construction provides more than just a quantization of a reduced model because we are able to interpret any quantum state of the reduced theory as a symmetric state of the full quantum theory.”

In a diffeomorphism-invariant quantum theory, there are no non-trivial states invariant under space transformations.

“However, analogously to the diffeomorphism group which does not act strongly continuously, there are no non-trivial states which are invariant with respect to space transformations.”

The λ-gauge-fixed area quantization is appropriate only in degenerate sectors.

“This quantization is appropriate only in the degenerate sectors.”

The topology with only one boundary component is B=R+∪{0}, Σ=R3, with boundary only at spacelike infinity.

“The topology with only one boundary component is given by B = R + ∪ {0}, = R 3 and has only the boundary at spacelike infinity.”

The partial gauge fixing used in the λ-gauge-fixed area quantization is inappropriate in the non-degenerate sector because of Higgs vertices.

“In doing so we have used a partial gauge fixing which, as discussed above, is inappropriate in the non-degenerate sector because of the Higgs vertices.”

Diffeomorphisms cannot deform an edge transversally.

“However, they cannot deform an edge transversally”

At a Higgs vertex, the spins of neighbouring edge holonomies can differ only by an integer, because j_v is integer-valued due to the Higgs loop.

“The crucial observation is that in a Higgs vertex the spins of the neighbouring edge holonomies can differ only by an integer value because the spin j v mentioned above is integer-valued due to the Higgs loop in the vertex.”

The presence of the Higgs field allows a non-trivial action of constraints.

“This property allows a non-trivial action of constraints.”

The appearance of 'Simon's subgraphs' of [51] is generic in the symmetry-reduced theory.

“Note in this context that the appearance of 'Simon's subgraphs' of [51] is generic in the symmetry-reduced theory.”

• cites [51] Bojowald M, 2000, Class. Quantum Grav. sourceNot checked yet

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The Gauß and diffeomorphism constraints are as easy to deal with in spherically symmetric gravity as in the full (3+1)-dimensional theory.

“The Gauß and diffeomorphism constraint are as easy to deal with as in the full (3 + 1)dimensional theory.”

The spin j_v can only be integer-valued because the Higgs field contributes left and right invariant vector fields leading to the loop.

“Because the Higgs field contributes by left and right invariant vector fields leading to the loop the spin j v can only be integer valued.”

Newly created vertices can always be shifted to their neighbouring vertices.

“The newly created vertices can always be shifted to their neighbouring vertices.”

The results reveal subtleties mainly due to the classical partial gauge fixing, which should be relaxed in the quantum theory.

“The results show that there are some subtleties, mainly due to the classical partial gauge fixing, which should be relaxed in the quantum theory.”

Holonomy variables are well suited to quantize the electromagnetic model and its observables.

“Holonomy variables turned out to be well suited in order to quantize the electromagnetic model and its observables.”

In general, topology can impose restrictions on which values of form (41) occur in the area spectrum.

“In general, however, the topology can impose restrictions on the possible values of the form (41) occurring in the area spectrum.”

Diffeomorphisms act only by dragging the ends of edges and the Higgs vertices attached to them.

“Diffeomorphisms act only by dragging the ends of edges and the Higgs vertices attached to them.”

The Hamiltonian constraint does not seem more easily solvable within loop quantum gravity with spin-network states than in the full theory.

“Unfortunately, the Hamiltonian constraint does not seem to be more easily solvable within the framework of loop quantum gravity with its spin-network states than in the full theory.”

The no-go theorem in [11, appendix C] is avoided in this setting because the graphs are one-dimensional.

“The no-go theorem of [11, appendix C] is evaded by the one-dimensional nature of our graphs.”

• cites [11, appendix C]Not checked yet

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The spectroscopy for spherically symmetric black holes is unaltered by the area spectrum (42) because it becomes uniformly spaced for large j, like spectrum (36).

“As a last remark we note that the spectroscopy for spherically symmetric black holes (cf, e.g., [46]) is unaltered by our area spectrum (42) because it becomes uniformly spaced for large j , like the spectrum (36).”

• cites [46] sourceNot checked yet

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A longitudinal deformation of an edge can always be absorbed by a reparametrization of the edge.

“and a longitudinal deformation can always be absorbed by a reparametrization of the edge.”

This allows only a subset of (41) as the area spectrum.

“This fact allows only a subset of (41) as the area spectrum.”