Quantum spin dynamics (QSD)

The volume operator is densely defined on thrice-differentiable cylindrical functions.

“This demonstrates that V is a densely defined operator on thricedifferentiable cylindrical functions.”

This disappearance of regulator dependence before the limit is not generally true if one uses a larger space than Diff as in [24] or a different regularization scheme.

“This is, in general, not true if one works with a larger space than Diff as in [24] or chooses another regularization scheme than the one chosen in the present paper.”

• cites [24] sourceNot checked yet

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The paper constructs, in the continuum, an anomaly-free operator for the Wheeler-DeWitt constraint of Lorentzian four-dimensional canonical non-perturbative vacuum gravity.

“An anomaly-free operator corresponding to the Wheeler-DeWitt constraint of Lorentzian, four-dimensional, canonical, non-perturbative vacuum gravity is constructed in the continuum.”

The Wheeler-DeWitt constraint acts on spin-network states by annihilating, creating, and rerouting angular-momentum quanta associated with the edges of the underlying graph.

“The action of the Wheeler-DeWitt constraint on spin-network states is by annihilating, creating and rerouting the quanta of angular momentum associated with the edges of the underlying graph”

In all our constructions the regulator dependence disappears even before taking the limit.

“It is important to notice that in all our constructions the regulator dependence actually disappears even before taking the limit.”

The constructed Wheeler-DeWitt operator has no factor-ordering singularities.

“This operator is entirely free of factor-ordering singularities and can be defined in symmetric and non-symmetric form.”

The commutator [Ô1,Ô2] appears meaningless on Diff because Ô has domain in Diff but range in H.

“The trouble is that Ô has domain in Diff but range in and so the commutator [ Ô 1 , Ô 2 ] seems to be meaningless on Diff .”

Spin-network states essentially diagonalize the ADM energy.

“while the ADM energy is essentially diagonalized by the spin-network states.”

The same technique gives rigorous meaning to other operators, including the length of a curve, Poincaré generators for asymptotically flat topologies, and matter contributions to the Hamiltonian constraint.

“Using the same technique one can give rigorous meaning to a whole bunch of other operators in a representation where the metric is not diagonal, including but not exhausting: (i) the operator corresponding to the length of a curve [19]; (ii) the generators of the Poincaré group for asymptotically flat topologies [22] and (iii) matter contributions to the Hamiltonian constraint [23].”

• cites [19] Ashtekar A, 1995, J. Math. Phys.Not checked yet

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• cites [22] sourceNot checked yet

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• cites [23] sourceNot checked yet

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Restricting to normalizable solutions, the physical Hilbert space is the one already given in prior work.

“If we restrict ourselves to normalizable solutions then the physical Hilbert space turns out to be the one already given in [14].”

• cites [14] Baez J, 1994, Lett. Math. Phys. sourceNot checked yet

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The Wheeler-DeWitt operator can be defined in both symmetric and non-symmetric form.

“This operator is entirely free of factor-ordering singularities and can be defined in symmetric and non-symmetric form.”

The Hamiltonian constraint acts by annihilating, creating, and rerouting angular-momentum quanta in units of ±h, ±h/2, and 0.

“the Hamiltonian constraint acts by annihilating, creating and re-routing the quanta of angular momentum (with which the graphs of the so-called spin-network [28][29][30] states are 'coloured') in units of ±h, ±h/2, 0.”

• cites [28] sourceNot checked yet

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• cites [29] sourceNot checked yet

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• cites [30]Not checked yet

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This number is independent of ε.

“Moreover, this number turns out to also be independent of .”

For any finite regularization parameter, a consistent operator family on cylindrical functions does not exist.

“It turns out that the answer is negative: for any finite such an operator does not exist.”

The number ψ(Ô1(ε1,γ)Ô2(ε2,γ)fγ) is well defined for any graph γ with respect to which Ô(ε,γ)fγ is cylindrical and for diffeomorphism-invariant ψ.

“ψ( Ô1 ( , γ ) Ô2 ( , γ )f γ ) is well defined where γ is any graph with respect to which Ô( , γ )f γ is cylindrical and where ψ is diffeomorphism invariant.”

For the non-symmetric operator, the complete kernel of the Wheeler-DeWitt constraint operator is found in the companion paper.

“For the non-symmetric operator we are able to find the complete kernel of the Wheeler-DeWitt constraint operator in [1].”

• cites [1] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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The non-polynomiality of the theory, determined by 1/sqrt(det q), can be hidden in a Poisson bracket.

“So what we have achieved is to hide the non-polynomiality of the theory as determined by 1/ √ det(q) in a Poisson bracket.”

• AssumptionThe authors state this as an assumption: The quantizations of the total volume and the integrated trace of extrinsic curvature can be defined densely on suitable subspaces of cylindrical functions, and can be made self-adjoint if a symmetric operator is desired.

Using the real connection representation yields a well-defined quantum theory.

“We work in the real connection representation and obtain a well defined quantum theory.”

It is possible to construct a finite symmetric operator for the original non-rescaled Lorentzian Wheeler-DeWitt constraint whose quantum constraint algebra is non-anomalous.

“We show that it is indeed possible to construct a finite (symmetric) operator corresponding to the original, non-rescaled, Lorentzian Wheeler-DeWitt constraint whose quantum constraint algebra is non-anomalous.”

• AssumptionThe authors state this as an assumption: There exists a phase of quantum gravity in which gravitational-field excitations can be probed by loops rather than by rapidly decreasing test functions.

Linear combinations of spin-network states diagonalize the ADM energy operator similarly to how Fock states diagonalize the Maxwell Hamiltonian, with spins playing the role of occupation numbers.

“On the other hand, linear combinations of such states diagonalize the ADM energy operator very much in the same way as Fock states diagonalize the Maxwell Hamiltonian, the role of the occupation number being played by the spins of the spin-network state.”

Corollary 3.1: For three mutually distinct analytic curves intersecting only at v, there exist parametrizations, δ>0, and an analytic diffeomorphism such that s1 and s2 take the standard form of Lemma 3.1 and the third component of s3(t) vanishes except possibly at t=0.

“Let s 1 , s 2 , s 3 be three mutually distinct analytic curves which intersect only at their starting point v.There exist parametrizations of these curves, a number δ > 0 and an analytic diffeomorphism such that s 1 , s 2 take their standard form as described in the above lemma and (s 3 (t)) 3 = 0 except, possibly, at t = 0.”

The routing can be achieved for an arbitrary number of edges with the same α(s1,s2) and β(s1,s2).

“This can be achieved for an arbitrary number of edges e with the same α(s 1 , s 2 ), β(s 1 , s 2 ).”

Functions on graphs with only divalent vertices are annihilated by Ĥ_E(N).

“functions on graphs with only divalent vertices are annihilated by Ĥ E (N ).”

In the collinear case, if e does not point into the wedge y ≥ x^n, x ≥ 0, then an arc a chosen close enough to the vertex cannot intersect e.

“If e does not point into the wedge y x n , x 0 then again the arc a cannot possibly intersect e if we choose it close enough to the vertex.”

Functions cylindrical on graphs with only such degenerate vertices are annihilated by Ĥ^E(N).

“So we see already that functions cylindrical on graphs with only such degenerate vertices are annihilated by Ĥ E (N ).”

Lemma 3.1: For two distinct analytic curves intersecting only at their starting point v, there exist parametrizations, δ>0, and an analytic diffeomorphism such that in the adapted frame the curves take standard form (i) s1(t)=(t,0,0), s2(t)=(0,t,0) if tangents are linearly independent at v, and (ii) s1(t)=(t,0,0), s2(t)=(t,t^n,0) for some n≥2 if tangents are collinear at v.

“Let s 1 , s 2 be two distinct analytic curves which intersect only in their starting point v.There exist parametrizations of these curves, a number δ > 0 and an analytic diffeomorphism such that in the corresponding frame the curves are given by (i) s 1 (t) = (t, 0, 0), s 2 (t) = (0, t, 0), t ∈ [0, δ] if their tangents are linearly independent at v and (ii) s 1 (t) = (t, 0, 0), s 2 (t) = (t, t n ,”

H_E^T[N] has the correct limit (3.1) as all Δ shrink to their basepoints.

“is an expression which has the correct limit (3.1) as all shrink to their basepoints”

In the case s_k(Δ) ∩ γ = ∅, [h_s_k(Δ), V]f = 0.

“it follows from ( 2.8 ) that [h s k ( ) , V ]f = v∈V (γ ) h s k ( ) Vv f -v∈V (γ ∪s k ( )) Vv h s k ( ) f = -f v∈V (s k ( )) Vv h s k ( ) = 0”

The argument that divalent vertices do not contribute extends to vertices of arbitrary valence whose incident edges all have co-planar tangents.

“The argument actually extends to the case of vertices of arbitrary valence but such that all incident edges have co-planar tangents.”

As it stands, the projections of Ĥ^E(N) given in (3.7) are not symmetric operators on H.

“As it stands, not even the projections of Ĥ E (N ) given in (3.7) are symmetric operators on H.”

The operator only annihilates Diff but not the extensions * discussed in [24].

“in view of [25] it is clear that it only annihilates Diff but not any of the extensions * discussed in [24].”

• cites [25] Gambini R, 1995, Int. J. Mod. Phys. sourceNot checked yet

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• cites [24] sourceNot checked yet

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Expression (3.7) is finite, i.e. a cylindrical function, independently of T because E(v) is determined by the graph and not by the triangulation.

“In particular, the expression (3.7) is finite (i.e. a cylindrical function) independently of T because the number E(v) is determined by the graph and not by the triangulation and so does not change as we make the triangulation finer and finer.”

In Situation (ii), if e does not point into an octant where both x and y are positive, then an arc a chosen in the x-y plane with positive x-y components cannot intersect e.

“If e does not point into an octant of the frame where both x/y are positive then again there is no danger that a intersects e since we choose a to lie in the x-y plane with positive x-y components as mentioned above.”

The sum of Poisson brackets is of order ε^3 and therefore vanishes in the limit ε→0 regardless of the finite-ε brackets, even if the continuum brackets do not vanish on the diffeomorphism constraint surface.

“Thus the sum of Poisson brackets is of order 3 and therefore trivially vanishes in the limit → 0 whatever the brackets might be at finite , that is, even if the continuum brackets would not vanish on the constraint surface defined by the diffeomorphism constraint.”

• AssumptionRests on an assumption: The classical Poisson brackets {O1,O2} vanish identically on the constraint surface defined by the diffeomorphism constraint.

The end result of the regularized operator is independent of the additional edge added at a 2-valent vertex.

“the end result of the regularized operator is independent of that additional edge”

In the classical expression, for each Δ one can replace by 8 because all eight tetrahedra shrink to their basepoints and dependence on mirror images drops out.

“In the classical expression (3.3) we can replace, for each , ∈D( ) by 8 as all the tetrahedra shrink to their basepoints so that the dependence on the mirror images of drops out as promised †.”

H_E^Δ[N] tends to the correct classical value as Δ shrinks to v(Δ).

“tends to the correct value 2 [N tr(F ∧ {A, V })] as we shrink to the point v( ),”

Choosing a frame adapted to s1,s2 partially fixes the gauge freedom associated with analyticity-preserving diffeomorphisms.

“By choosing a frame adapted to s 1 , s 2 we have partially fixed the gauge freedom associated with analyticity-preserving diffeomorphisms.”

If s3 is not transversal to s1 and s2 at v, then (3.8) vanishes trivially because the volume operator annihilates vertices whose incident edges have co-planar tangents.

“Notice that if s 3 was not transversal to s 1 , s 2 at v then (3.8) would vanish trivially because the volume operator annihilates vertices which are such that all incident edges have co-planar tangents.”

There is no problem ordering V to the left or right of the holonomies because V has finite action on holonomies of A.

“Note that we have no problems in ordering V to the left or to the right of the holonomies involved as V has a finite action on holonomies of A as is clear from (2.8).”

Terms involving α23 and α31 drop out because the volume operator annihilates divalent vertices.

“The terms involving α 23 , α 31 drop out which follows from the fact that the volume operator annihilates divalent vertices as is obvious from the expression (2.8).”

The topology of the routing is diffeomorphism-invariantly defined because n, m, and m are diffeomorphism invariant.

“Notice that the topology of the routing is diffeomorphism-invariantly defined as the numbers n, m, m are diffeomorphism invariant.”

In the latter case of c14, the first non-vanishing derivative of e_z at t = 0 is again z,z e_z^(m)(0), because all derivatives of z(e(t)) not involving a z-partial vanish.

“In the latter case, the first non-vanishing derivative of e z at t = 0 would again be z ,z e (m ) z (0), because by definition of a diffeomorphism preserving the adaptedness of the frame, all derivatives of z (e(t)) = z t, t n , e z (t) at t = 0 which do not involve at least one partial derivative with respect to z must vanish.”

Expression (3.5) is finite no matter how fine the triangulation is, provided the number of tetrahedra intersecting the vertices of γ stays bounded as the triangulation becomes finer.

“we see that this expression is finite no matter how 'fine' the triangulation is, provided the number of tetrahedra intersecting the vertices of γ stays bounded as we go to finer and finer triangulations.”

The sign of ė_z(0) is preserved under change of adapted frames.

“We show that sgn( ėz (0)) is preserved under change of adapted frames”

On the cylindrical subspace labelled by graph γ, the action of the Euclidean Hamiltonian constraint reduces to equation (3.7).

“it follows from the considerations in section 3.1.1that on the cylindrical subspace labelled by the graph γ the action of the Euclidean Hamiltonian constraint reduces to”

This does not guarantee that Ĥ^E(N) itself is symmetric.

“Notice that this is far from guaranteeing that Ĥ E (N ) itself is symmetric (see [1]).”

• cites [1] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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With the symmetric ordering in (3.9), each ĥ^E_γ[N] separately becomes a symmetric operator on H.

“With this choice, each ĥE γ [N ] in (3.9) separately becomes a symmetric operator on H.”

Divalent vertices do not contribute to the Euclidean Hamiltonian constraint.

“Let us now show that divalent vertices do not contribute: if v is a divalent vertex of γ , if s 3 is the segment of the edge added in requirement (i) and if α 12 is the corresponding loop along segments s 1 , s 2 of the two edges e 1 , e 2 of γ incident at v then”

By choosing β(s1,s2) sufficiently large, one can make e_z^(n+1)(0) positive for an arbitrary finite number of edges.

“upon choosing one and the same β(s 1 , s 2 ) large enough we can beat the terms z ,z e (m+1) z (0) for an arbitrary (but finite) number of edges e as to make e (n+1) z (0) > 0.”

The action of (3.4) on cylindrical functions is finite no matter how fine the triangulation T is, provided a certain criterion is satisfied.

“its action on cylindrical functions is indeed finite no matter how fine the triangulation T is, provided that a certain criterion is satisfied”

Checking the commutator on cylindrical functions yields a finite sum of commutators whose classical limit is a finite number of Poisson brackets integrated over regions of coordinate volume of order ε^3.

“when checking the commutator on cylindrical functions we get again only a finite sum of commutators between terms from that Riemann sum, the classical limit of which is a finite number of Poisson brackets between functions which are integrated over regions of which have coordinate volume of order 3 .”

V annihilates any cylindrical function whose underlying graph is not at least 3-valent.

“V annihilates any cylindrical function, whether gauge invariant or not, whose underlying graph is not at least 3-valent”

If γ' is larger than γ and e1,e2 are edges of γ' that are parts of edges e1,e2 of γ incident at the same vertex v, then the loops made from corresponding segments are diffeomorphic, guaranteeing cylindrical consistency up to a diffeomorphism.

“if γ is larger than γ and if e 1 , e 2 are edges of γ , which are the parts of the edges e 1 , e 2 of γ incident at the same vertex v at which e 1 , e 2 are incident, then the loop made from the corresponding segments s 1 , s 2 and s 1 , s 2 are diffeomorphic thus guaranteeing cylindrical consistency up to a diffeomorphism.”

A solution to anomaly freedom is obtained for the non-symmetric operator when edges a_ij are attached irrespective of differentiability at endpoints with respect to s_i and s_j as prescribed in section 3.1.3.

“We now show that a solution to the anomaly freedom problem is obtained: (i) for the non-symmetric operator in the case where we attach the edges a ij ( ) irrespective of their differentiability at their endpoints with respect to s i ( ), s j ( ) as prescribed in section 3.1.3;”

The requirement that expression (3.18) vanish on diffeomorphism-invariant states is satisfied.

“(b) The fact that the classical Poisson bracket vanishes on the constraint surface of the phase space defined by the diffeomorphism constraint translates in the quantum theory into the requirement that (3.18) should vanish on diffeomorphism-invariant states and this we checked to be the case.”

If one takes the limit T → v(T), the result vanishes trivially.

“One might wonder what happens if one actually takes the limit and sends → v( ).It is easy to see that the result vanishes trivially which is not what we want.”

The theory does not have a faithful representation of the Dirac algebra.

“We conclude that for both the symmetric and the non-symmetric constraint operator defined here the constraint algebra is consistent (and so we do not have a strong anomaly), but we do not have a faithful representation of the Dirac algebra (and so have a weak anomaly).”

Ê(v) reduces to the correct value on f.

“Moreover, Ê(v) reduces to the correct value on f .”

There exists a diffeomorphism φ such that φ(γ) = γ and φ(T(γ)) = φ(T(γ)) for the specific T(γ).

“Such a diffeomorphism clearly exists: there are diffeomorphisms which leave the image of the graph invariant, while moving its points and off the graph it can put the arc a ij ( ) into any diffeomorphic shape.”

The Euclidean Hamiltonian operator defined in equation (3.10) is non-anomalous.

“Theorem 3.1. The Euclidean Hamiltonian operator Ĥ E (N ) = Ĥ E γ (N ), D γ as defined by (3.10) is non-anomalous.”

 and B commute after removing the diffeomorphisms φ and φ'.

“This, together with the fact that the volume operator at one vertex does not act on the tetrahedra incident at another vertex shows that Â, B commute upon removing the diffeomorphisms ϕ, ϕ .”

The volume operator [32] does not annihilate co-planar vertices.

“Namely, the volume operator [32] does not annihilate co-planar vertices.”

• cites [32] Thiemann T, 1998, Class. Quantum Grav.Not checked yet

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Up to higher order in h, the commutator Ĥ_E(M), Ĥ_E(N)/(ih) reproduces (MN,a - NM,a) q^ab V_b.

“The answer is, trivially, in the affirmative: up to higher order in h the commutator Ĥ E (M), Ĥ E (N) /(ih) reproduces (MN ,a -NM ,a )q ab V b so that we can just define (3.18) using that commutator because by construction it is well defined.”

Commutators with holonomies along edges incident at v are non-vanishing only for the parts K_v and V_v of V and K.

“In order to arrive at this expression we use the same arguments as before to conclude that only tetrahedra intersecting a vertex of γ give a contribution and we realize that all the commutators with holonomies along edges incident at v are non-vanishing only for the parts Kv and Vv of V and K, respectively.”

Equation (5.3) is cylindrically consistently defined up to a diffeomorphism when the projectors p are introduced.

“However, K is consistently defined up to a diffeomorphism and γ ∪ e i ( (γ )) = γ so that (5.3) is cylindrically consistently defined up to a diffeomorphism, provided we introduce again the projectors p .”

Neither Ĥ_E nor T acts at the additional vertices introduced by acting with Ĥ_E or T.

“To see this we use the fact that neither Ĥ E nor T act at the additional vertices introduced by acting with Ĥ E , T (in the symmetric case this requires lemmas 3.1 and 3.2) and find that for a spin-network function f cylindrical with respect to γ we have”

For different vertices, the commutators vanish on a diffeomorphism-invariant state because the volume operator parts at different vertices do not act on edges incident at the other vertex.

“while for v = v the commutators vanish when evaluated on a diffeomorphism-invariant state as in (3.16) after getting rid of some diffeomorphism operators, again because all operators involved in the commutators only contain the parts Vv , Vv of the volume operator which do not act on edges incident at different vertices.”

If the symmetric version of Ĥ_E(N) is used, then K is symmetric.

“Moreover, consider the case where we are using the symmetric version of Ĥ E (N).Then K is also symmetric and since Ĥ E [1] is an imaginary-valued operator, K and V are both real-valued operators.”

In [17,18] the assigned loop has the topology of a triangle, not a kink.

“In [17,18] the loop assigned does not have the topology of a kink but the topology of a triangle.”

• cites [17] sourceNot checked yet

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• cites [18] sourceNot checked yet

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The operator T[N] applied to a function cylindrical with respect to γ equals the expression in equation (5.3).

“The result, when applied to a function cylindrical with respect to a graph γ , reads”

Both families defined in (3.10) are self-consistent up to a diffeomorphism.

“The self-consistency up to a diffeomorphism of both families defined in (3.10) can be checked as follows.”

Only tetrahedra intersecting a vertex of γ contribute to T[N].

“In order to arrive at this expression we use the same arguments as before to conclude that only tetrahedra intersecting a vertex of γ give a contribution and we realize that all the commutators with holonomies along edges incident at v are non-vanishing only for the parts Kv and Vv of V and K, respectively.”

The triangulation adapted to a graph was designed to be diffeomorphism covariant for each vertex separately, meaning each operator Ĥ^E_v, p, Ê(v), Ĥ^E, and ĥ^E is covariantly defined.

“Our triangulation adapted to a graph was geared at being diffeomorphism covariant for each of its vertices separately meaning that each of the operators Ĥ E v and even each of the operators p , Ê(v), Ĥ E , ĥE separately is covariantly defined.”

The expected constraint algebra has the commutator on a state f as a sum of terms Û(φ) Â B f - Û(φ') B Â f, with [Â,B]f=0 and Â,B symmetric.

“Given the fact that we can only define an operator corresponding to finite diffeomorphisms the structure of the constraint algebra that we do expect is precisely the one displayed in (3.16), namely that the commutator on a state f is a sum of terms of the form Û(ϕ) Â Bf -Û(ϕ ) B Âf where [ Â, B]f = 0 and Â, B are symmetric operators.”

The dependence of H_E on momenta is through {A_a, V}, which is not an analytic function of E^a_i.

“The dependence of H E on the momenta is through {A a , V } which is not even an analytic function of E a i .”

The commutator in equation (3.16) vanishes identically because the volume operator part at one vertex does not act on holonomies along edges incident at the other vertex.

“Now we just need to use corollary 3.1 of [1] (the non-symmetric operator is treated in a comment after this corollary) to see that the commutator in (3.16) vanishes identically because the part Vv of the volume operator involved in Ĥ E (γ ) does not act on the holonomies along edges incident at v involved in Ĥ E (γ ) and vice versa since the vertices v, v are different.”

• cites [1] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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Operator (3.10) still depends on the diffeomorphism class [T] of the triangulation assignment.

“the operator (3.10) still carries a sign of the regularization procedure: it depends on the diffeomorphism class [T ] of the triangulation assignment which labels the freedom that we have in our regularization scheme.”

The complete symmetric or non-symmetric Lorentzian Wheeler-DeWitt operator defined in (5.5) is anomaly-free.

“Theorem 5.1.The complete (symmetric or non-symmetric) Lorentzian Wheeler-DeWitt operator Ĥγ [N] := Tγ [N] -Ĥ E γ [N] = v∈V (γ ) N v Tv -Ĥ E v (5.5) is anomaly-free.”

The volume operator at one vertex does not act on tetrahedra incident at another vertex.

“This, together with the fact that the volume operator at one vertex does not act on the tetrahedra incident at another vertex shows that Â, B commute upon removing the diffeomorphisms ϕ, ϕ .”

No loop assignments different from those for Ĥ_E are involved in defining T.

“There are no loop assignments different from the ones made for Ĥ E involved in the definition of T .”

For both the symmetric and non-symmetric versions of the constraint operator, no anomaly is found.

“while one of the versions of the constraint is actually not symmetric as one would expect from this result, another version actually is ordered Ĥ E (N ) symmetrically and for both versions we find that there is no anomaly!”

The number Ĥ^E_v f_γ depends only on the diffeomorphism class of the loop assignments α_ij(T).

“Then, obviously, the number Ĥ E v f γ depends only on the diffeomorphism class of the loop assignments α ij ( ) (this was first observed in [17]).”

• cites [17] sourceNot checked yet

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For both symmetric and non-symmetric constraint operators defined here, the constraint algebra is consistent.

“We conclude that for both the symmetric and the non-symmetric constraint operator defined here the constraint algebra is consistent (and so we do not have a strong anomaly), but we do not have a faithful representation of the Dirac algebra (and so have a weak anomaly).”

For every separate choice of the vertices v and v', the commutator expression (3.15) vanishes on a diffeomorphism-invariant state.

“ψ Ĥ E (γ ∪ (γ )) Ĥ E (γ ) -Ĥ E (γ ∪ (γ )) Ĥ E (γ ) f = 0 (3.15) for each choice of v( (γ )) = v( (γ ∪ (γ ))) = v, v( (γ )) = v( (γ ∪ (γ ))) = v prime separately.”

For the symmetric operator, anomaly freedom is obtained only if all loops α_ij are chosen to be kinks with vertex at v(T), i.e. the arc joins endpoints in at least C^1 fashion; arbitrary attachment is insufficient.

“(ii) for the symmetric operator only if all the loops α ij ( ) are chosen to be kinks with vertex at v( )!That is, the arc a ij ( ) joins the endpoints of e i ( ), e j ( ) in at least a C 1 fashion (see [1] for the details of the attachment).An arbitrary attachment of a ij ( ) is insufficient to guarantee anomaly-freeness.”

• cites [1] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

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Each term in sum (5.2) tends to 8/κ^3 N tr({A,K}∧{A,K}∧{A,V}) as tetrahedra shrink to basepoints, giving the correct continuum limit.

“Each term in the sum (5.2) labelled by a tetrahedron tends to 8/κ 3 N tr({A, K} ∧ {A, K} ∧ {A, V }) as we shrink the tetrahedra to their basepoints, that is, we get the correct continuum limit.”

Each term K_v contains only the part V_v of the full volume operator.

“So, each term Kv in the sum over vertices contains only the part Vv of the full volume operator.”

Under the other option, computing such commutators is impossible because the regularized operator does not exist.

“Given option (b) this is impossible because the operator Ô( ) does not exist.”

Because the original Wheeler-DeWitt constraint has density weight one, no renormalization is necessary.

“Since the original Wheeler-DeWitt constraint has density weight one, no renormalization is necessary.”

The limit ε→0 shrinking the loop cannot have meaning on H.

“Therefore the limit → 0 (shrinking of the loop) just cannot have any meaning on H.”

Spin-network states form an orthonormal basis on the Hilbert space.

“An orthonormal basis on H is given by the so-called spin-network states [28][29][30]”

• cites [28] sourceNot checked yet

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• cites [29] sourceNot checked yet

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• cites [30]Not checked yet

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Solutions of the diffeomorphism constraint are elements of the dual space.

“It turns out that the solutions of the diffeomorphism constraint are elements of [14].”

• cites [14] Baez J, 1994, Lett. Math. Phys. sourceNot checked yet

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The pair (A_i^a, E^a_i) is canonical on the phase space of Lorentzian gravity subject to the SU(2) Gauss, diffeomorphism, and Wheeler-DeWitt constraints, neglecting a Gauss term.

“Then one can show that A i a := i a +K i a , E a i is a canonical pair † on the phase space of Lorentzian gravity subject to the SU (2) Gauss constraint, the diffeomorphism constraint and the Wheeler-DeWitt constraint”

The spin-network representation is the nonlinear Fock representation of quantum gravity.

“We argue that the spin-network representation is the 'nonlinear Fock representation' of quantum gravity, thus justifying the term 'quantum spin dynamics (QSD)'.”

Smooth cylindrical functions are dense in the Hilbert space.

“Consider the set of smooth cylindrical functions := Cyl ∞ (A/G) which can be shown to be dense in H.”

The resulting theory can be recognized as Euclidean four-dimensional gravity written in real-valued Ashtekar variables.

“This theory can be, not surprisingly, recognized as Euclidean four-dimensional gravity written in terms of real-valued Ashtekar variables.”

The only way to give meaning to that limit is to change the operator topology.

“The only way to give meaning to it is to change the operator topology.”

If a state is diffeomorphism invariant and the Hamiltonian constraint is diffeomorphism covariantly defined, then solving the dual constraint equation is meaningful.

“On the other hand, we will see that if ψ is diffeomorphism invariant and Ĥ (N) is diffeomorphism covariantly defined, then solving ψ( Ĥ (N)f ) = 0 is meaningful.”

A well-defined self-adjoint operator corresponding to the total volume has already been constructed, with finite action on cylindrical functions.

“first, it is a fact that a well defined, self-adjoint operator V on H corresponding to V [32,33], whose action on cylindrical functions is perfectly finite (we follow [33]), has already been constructed”

• cites [32] Thiemann T, 1998, Class. Quantum Grav.Not checked yet

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• cites [33] Thiemann T, 1998, Class. Quantum Grav.Not checked yet

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The calculus on generalized connections modulo gauge transformations for compact gauge groups provides a rigorous kinematic framework for regularizing constraint operators unambiguously.

“This calculus provides a rigorous kinematic framework by means of which constraint operators can be regularized in a well defined and unambiguous manner.”

• cites [9] sourceNot checked yet

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• cites [10] Ashtekar A, 1987Not checked yet

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• cites [11] Barbero F, 1995, Phys. Rev. sourceNot checked yet

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• AssumptionThe authors state this as an assumption: There exists a phase of quantum gravity in which gravitational-field excitations can be probed by loops rather than by rapidly decreasing test functions.

The framework has already been used to obtain the general solution of the Gauss and diffeomorphism constraints for either signature.

“In particular, this framework has already been employed successfully to arrive at the general solution of the Gauss and diffeomorphism constraints [14] for either signature.”

• cites [14] Baez J, 1994, Lett. Math. Phys. sourceNot checked yet

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Ô must have range in H rather than Diff.

“there is nothing wrong with Ô having a range in rather than Diff -actually this must happen.”

The canonical constraints of general relativity can be put into polynomial form by a complex canonical transformation on the gravitational phase space.

“The celebrated observation due to Ashtekar [4] is that the canonical constraints of general relativity can indeed be cast into polynomial form if one performs a certain complex canonical transformation on the gravitational phase space.”

• cites [4] sourceNot checked yet

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The spin-network representation is the nonlinear Fock representation for quantum gravity.

“Thus, the spin-network representation is the 'nonlinear Fock representation for quantum gravity' and this motivates us to call the quantum theory we obtain 'quantum spin dynamics (QSD)' in analogy with QED or QCD.”

The continuum operator has a domain only on diffeomorphism-invariant distributions.

“Therefore, for any , Ô ( ) Diff =: Ô so the continuum operator actually has a domain only on diffeomorphism-invariant distributions”

The Gauss and diffeomorphism constraints can be solved.

“Moreover, Gauss and diffeomorphism constraints can be solved (see [11] and references therein for an introduction to these concepts).”

• cites [11] Barbero F, 1995, Phys. Rev. sourceNot checked yet

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To include more generalized solutions, a new scalar product must be constructed.

“However, in order to incorporate more (generalized) solutions one has to construct a new scalar product [20].”

• cites [20] sourceNot checked yet

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Choosing the arc a_ij as straight as possible so that α_ij looks like a triangle justifies the approximation h_{α_ij} ≈ 1 + 1/2 (δt)^2 ṡ_i^a(0) ṡ_j^b(0) F_ab(v(T)).

“This will then justify the approximation h α ij ( ) ≈ 1 + 1 2 (δt) 2 ṡi ( ) a (0)ṡ j ( ) b (0) F ab (v( )) where [0, δt] is the subinterval [0, 1] corresponding to s i ( ) as compared to the whole edge of γ of which it is a segment.”

Exact cylindrical consistency is not required because the triangulation assignment is defined only up to a diffeomorphism when evaluating a diffeomorphism-invariant state ψ on Ĥ_E f.

“The reason why we do not need to require exact cylindrical consistency is because the assignment of the triangulation is only defined up to a diffeomorphism if we care only about the evaluation of a diffeomorphism-invariant state ψ on the states Ĥ E f”

This result invalidates the main conclusion of [24].

“This surprising result invalidates the main conclusion of [24].”

• cites [24] sourceNot checked yet

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If the commutator vanishes only in the limit ε→0, then the theory still has an anomaly, in the sense that its classical limit has non-vanishing Poisson brackets on the diffeomorphism constraint surface.

“if the commutator only vanishes in the limit → 0 then one still has an anomaly in the sense that the classical limit of that theory has non-vanishing Poisson brackets on the constraint surface defined by the diffeomorphism constraint.”

• cites [25] Gambini R, 1995, Int. J. Mod. Phys. sourceNot checked yet

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• AssumptionRests on an assumption: The classical Poisson brackets {O1,O2} vanish identically on the constraint surface defined by the diffeomorphism constraint.

If ė_z(0) ≠ 0, then by analyticity e is curved away from the x-y plane near v and does not intersect a for any adapted frame, hence for any frame.

“By analyticity it then follows that e is curved away from the x-y plane in a neighbourhood of v and so does not intersect a for any choice of adapted frame and therefore for any frame.”

For a diffeomorphism-invariant state ψ, the number ψ Ĥ_E f depends only on diffeomorphism-invariant properties of the triangulation assignment.

“provided the state ψ is diffeomorphism invariant, the number ψ Ĥ E f would only depend on the diffeomorphism-invariant properties of the triangulation assignment (this was first observed in [17]).”

• cites [17] sourceNot checked yet

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• AssumptionRests on an assumption: The measure µ0 is diffeomorphism invariant.

The volume operator defined in [32] does not vanish for vertices involving only edges with co-planar tangents.

“according to the second reference in [33], that operator does not vanish for vertices which involve only edges with co-planar tangents.”

• cites [33] Thiemann T, 1998, Class. Quantum Grav.Not checked yet

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T is diffeomorphism covariant if and only if Ĥ_E is.

“Diffeomorphism covariance is immediate because T is covariantly defined if and only if Ĥ E is.”

The method of obtaining K from V and Ĥ_E[1] via their commutator cannot be applied to other operators for H_E[1] in the literature.

“Notice that the method of getting K from V , Ĥ E [1] simply through the commutator of these two operators is not possible to apply using any of the other operators corresponding to H E [1] as defined so far in the literature:”

The limit T → v(T) is inappropriate.

“This shows that the limit → v( ) is, in fact, inappropriate.”

The question of whether a factor ordering can place V_a to the right in (3.18) cannot be asked in this case.

“To conclude, in our case the question of whether a factor ordering can be found such that Va stands to the right in (3.18) cannot even be asked.”

For those other approaches, defining the operator for the Wick rotation generator is far from easy.

“and thus for those approaches the operator corresponding to the generator of the Wick rotation transform is far from easy to define.”

An operator for q^ab(x) is meaningless in this representation because both numerator and denominator in its expression are meaningless.

“Also an operator corresponding to q ab (x) is entirely meaningless in our representation: one could write it as q ab = E a i E b i / det(q), however, both nominator and denominator are meaningless [14].”

• cites [14] Baez J, 1994, Lett. Math. Phys. sourceNot checked yet

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By Stone's theorem, no self-adjoint generator exists for that representation.

“Therefore, by Stone's theorem, there does not exist a self-adjoint generator.”

The requirement of anomaly-freeness provides a selection rule among the operators [32,33], which a priori are both bona fide quantizations of the classical volume functional.

“It is very appealing that the requirement of anomaly-freeness provides us with a selection rule among the operators [32,33] which a priori from a purely kinematical point of view are both bona fide quantizations of the classical volume functional.”

• cites [32] Thiemann T, 1998, Class. Quantum Grav.Not checked yet

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• cites [33] Thiemann T, 1998, Class. Quantum Grav.Not checked yet

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The quantum constraint algebra cannot closely resemble the classical constraint algebra.

“Therefore the quantum constraint algebra cannot have any close analogy with the classical constraint algebra.”

The representation of one-parameter subgroups of the diffeomorphism group on the Hilbert space is not strongly continuous.

“As was shown in [10] the representation of (one-parameter subgroups of) the diffeomorphism group on H is not strongly continuous.”

• cites [10] Ashtekar A, 1987Not checked yet

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The considerations of [34] do not apply to this case.

“We conclude that the considerations of [34] are not applicable to our case.”

• cites [34] Thiemann T, 1998, Class. Quantum Grav.Not checked yet

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Diffeomorphism covariance is a sufficient requirement for the quantum theory to correspond to a continuum theory.

“Diffeomorphism covariance is therefore a sufficient requirement for our quantum theory to correspond to a continuum theory.”

Quantization of a classical theory does not require making sense of every classical function as an operator.

“The answer is that the quantization of a classical theory does not force us to make sense out of every classical function as an operator.”

Because those operators do not involve the volume operator but depend on an operator corresponding to abc ijk E^a_i E^b_j that does not vanish when only two independent tangent directions exist, the authors expect them to be anomalous.

“But because these operators do not involve the volume operator, rather they depend on an operator corresponding to abc ij k E a i E b j which does not vanish if there are only two independent tangent directions in the problem, we expect these operators to be anomalous.”

What must be avoided is that the Hamiltonian has non-trivial action at v1 and v2.

“One sees that what has to be avoided is that the Hamiltonian has non-trivial action at v 1 , v 2 .”