SU(2)loop quantum gravity seen from covariant theory

There is a quantization ambiguity in covariant loop gravity due to the existence of a family of possible Lorentz connections.

“Within this framework, we underline a quantization ambiguity due to the existence of a family of possible Lorentz connections.”

After solving the second class constraints, the natural configuration variables parametrising the system are the field χ and a generalization of the Ashtekar-Barbero connection of LQG.

“After solving the second class constraints, the natural configuration variables parametrising the system are the field χ a = -e ta /e t 0 ( a being an su (2) index and t the 0 space-time index), which is the space components of the time normal or internal time direction, and a generalization of the Ashtekar-Barbero connection of LQG ( i being a space index):”

At a given point x of intersection of the loop and a small surface, one needs to decompose the representation R(n,ρ) of SL(2,C) into representations V_jχ(x) of the subgroup SUχ(x)(2) which leaves the vector χ(x) invariant.

“Indeed, at a given point x of intersection of the loop and a small surface whose area we are computing, we need to decompose the representation R ( n,ρ ) of SL (2 , C ) into representations V j χ ( x ) of the subgroup SU χ ( x ) (2) , which leaves the vector χ ( x ) invariant.”

Despite the fact that the canonical connection A is non-commutative, A(ash) commutes with itself.

“Moreover, despite the fact that the canonical connection A is non-commutative, A (ash) commutes with itself (see appendix C).”

It is possible to solve explicitly the second class constraints at the quantum level using the generalized Ashtekar-Barbero connection.

“The study of this case is interesting because it can be a guide for the 'correct' diffeomorphism preserving quantization since it is possible to solve explicitly the second class constraints at the quantum level using this generalized Ashtekar-Barbero connection.”

The physical output of the resulting theory does not depend on the Immirzi parameter.

“Moreover, a nice feature of the resulting theory is that the physical output of the theory does not depend on the Immirzi parameter.”

The formalism that solves the second class constraints explicitly breaks the covariance of the theory.

“The disadvantage of this formalism is that it breaks the covariance of the theory when solving explicitly the second class constraints.”

There exists a true spacetime connection transforming properly under all diffeomorphisms.

“On the other hand, there exists a true spacetime connection, transforming properly under all diffeomorphisms.”

The canonical variable A(bar) is neither a Lorentz nor an SU(2) connection.

“Besides, the canonical variable A (bar) is in fact neither a Lorentz nor an SU (2) connection.”

The natural covariant generalization of the Ashtekar-Barbero connection is the only commutative one.

“Moreover, this connection is the only commutative one.”

The canonical analysis done preserving covariance leads to a two-parameter family of Lorentz connections with commutation relations such that the corresponding Wilson lines are eigenstates of the area operator.

“The canonical analysis done in preserving covariance, as described in appendix A, leads to a two-parameter family of Lorentz connections (58) with such commutation relations that the corresponding Wilson lines are eigenstates of the area operator.”

The addition of an extra term compared to the original Palatini action leads to the introduction of a new coupling constant β.

“Nevertheless, the addition of an extra term compared to the original Palatini action leads to the introduction of a new coupling constant β .”

Following the methods of LQG, one can derive the corresponding family of area spectra, which contain the Casimir operators of SL(2,C).

“Following the methods of LQG, one can derive the corresponding family of area spectra (66), which now contain the Casimir operators of SL (2 , C ) [10, 11].”

• cites [10] [10] S. Alexandrov and D. Vassilevich, Area spectrum in Lorentz covariant loop gravity , Phys. Rev. D 64 , (2001) 044023, [gr-qc/0103105]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There exists a Lorentz connection generalizing the Ashtekar-Barbero connection.

“We show the existence of a Lorentz connection generalizing the Ashtekar-Barbero connection and we loop-quantize the theory showing that it leads to the usual SU (2) Loop Quantum Gravity and to the area spectrum given by the SU (2) Casimir.”

There exists a natural covariant generalization of the Ashtekar-Barbero connection that makes it possible to derive LQG starting from the covariant quantization.

“It turns out there exists a natural covariant generalization of the AshtekarBarbero connection what makes possible to derive LQG starting from the covariant quantization.”

In the time gauge, one finds the exact set-up of LQG with Immirzi parameter β, reproducing Holst's result.

“In this gauge, one finds the exact set-up of LQG with Immirzi parameter β , reproducing Holst's result [12].”

• cites [12] [12] S. Holst, Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action , Phys. Rev. D 53 , 5966 (1996). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The SU(2) theory definitively breaks space-time diffeomorphism invariance.

“In particular, the scalar (Hamiltonian) constraint turns out hard to understand and the SU (2) theory definitively breaks the space-time diffeomorphism invariance, as it was foreseen in [6].”

• cites [6] [6] J. Samuel, Is Barbero's Hamiltonian formulation a gauge theory of Lorentzian gravity?, Class. Quantum Grav. 17 , L141 (2000), [gr-qc/0005095]; Phys. Rev. D 63 , 068501 (2001). - sourceNot checked yet

  Not checked in the free run.

The second class constraints can be properly imposed at the quantum level.

“In particular, the second class constraints can be properly imposed at the quantum level.”

A suitable projection of the canonical SL(2,C) connection gives an equivalent of the Ashtekar-Barbero connection.

“Indeed, a suitable projection of the canonical SL (2 , C ) connection A X i ( X is an sl (2 , C ) index) gives an equivalent of the Ashtekar-Barbero connection:”

Choosing the connection (5) as a basic variable, the covariant theory can be relatively easy quantized.

“It turns out that choosing the connection (5) as a basic variable, the covariant theory can be relatively easy quantized.”

In a particular gauge, one actually recovers the usual SU(2) spin network Hilbert space of LQG.

“Then, in a particular gauge, we actually recover the usual SU (2) spin network Hilbert space of LQG.”

Physical outputs of the quantization based on the generalized Ashtekar-Barbero connection depend non-trivially on the embedding of the canonical hypersurface into the space-time manifold.

“This covariant point of view allows to analyze closely the drawbacks of the SU (2) formalism: the quantization based on the (generalized) Ashtekar-Barbero connection breaks time diffeomorphisms and physical outputs depend non-trivially on the embedding of the canonical hypersurface into the space-time manifold.”

It is possible to write observables and a Hilbert space of quantum geometry states using functionals of the connection and of the foliation.

“However, it turns out possible to write observables and a Hilbert space of quantum geometry states using functionals of the connection and of the foliation.”

Taking A(ash) as a canonical variable, in any gauge χ=const, the phase space has the same structure as in the usual SU(2) approach.

“Thus, taking A (ash) as a canonical variable, in any gauge χ = const , the phase space has the same structure as in the usual SU (2) approach.”

Choosing the parameters a and b in equation (58) as specified yields a connection which reproduces the results of the SU(2) approach.

“As it had been noted in [11], if we choose the parameters of the connection, a and b in equation (58), to be as follows”

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The components A(ash) form a connection of the boosted subgroup SUχ(2) when χ is constant over the canonical hypersurface.

“Further, they form a connection of the 'boosted' subgroup SU χ (2) , constructed explicitly in Appendix B, when χ is constant over the canonical hypersurface.”

If one requires that the area operator be diagonal on the Wilson loops defined by the connection, one ends up with a two-parameter family of possible SL(2,C) connections.

“Then, following the philosophy of LQG, if we require that the area operator be diagonal on the Wilson loops defined by the connection, we end up with a two-parameter family of possible SL (2 , C ) connections.”

The quantization based on the generalized Ashtekar-Barbero connection breaks time diffeomorphisms.

“This covariant point of view allows to analyze closely the drawbacks of the SU (2) formalism: the quantization based on the (generalized) Ashtekar-Barbero connection breaks time diffeomorphisms and physical outputs depend non-trivially on the embedding of the canonical hypersurface into the space-time manifold.”

βA(bar) coincides with the usual Ashtekar-Barbero connection when expressing ω through the triad by means of the Gauss constraints.

“In particular, βA (bar) coincides with the usual Ashtekar-Barbero connection, when expressing ω ab i through the triad by means of (a half of) the Gauss constraints.”

Due to the commutator relation with the triad multiplet, the area spectrum given by this SL(2,C) connection coincides exactly with the one coming from Loop Quantum Gravity given by the Casimir operator of SU(2).

“Due to this last relation, the area spectrum given by this SL (2 , C ) connection coincides exactly with the one coming from Loop Quantum Gravity given by the Casimir operator of SU (2):”

The main technical difficulty of the covariant method is the resulting non-commutativity of the SL(2,C) connection.

“The main technical difficulty of this method is the resulting non-commutativity of the SL (2 , C ) connection.”

There exists a Lorentz connection which is a natural generalization of the Ashtekar-Barbero one.

“Remarkably, there exists such a Lorentz connection which is a natural generalization of the Ashtekar-Barbero one [11].”

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The chosen connection is commutative.

“As the Ashtekar-Barbero connection, it is commutative (see appendix C)”

One can use additional ambiguity to add to any quantity a combination of the second class constraints (57) in order to remove 6 more components, without modifying any commutation relations.

“However, we can use additional ambiguity to add to any quantity a combination of the second class constraints (57) in order to remove 6 more components, without modifying any commutation relations.”

Projected spin networks projected onto j=0 give the same Hilbert space of quantum states as obtained from the spin foam approach.

“and give the same Hilbert space of quantum states as obtained from the spin foam approach.”

The Immirzi parameter appears in the classical theory parametrizing different Ashtekar-Barbero connections of the SU(2) approach, which are all related by a canonical transformation.

“It appears in the classical theory parametrizing different AshtekarBarbero connections of the SU (2) approach, which all are related by a canonical transformation.”

In the time gauge, the infinite refinement changes a projected Wilson line (15) with j_s(e)=j_t(e)=j by an SU(2) Wilson line in the representation j dependent only on the SUχ0(2) components of the connection.

“Therefore, in the time gauge it has an effect to change a projected Wilson line (15) with j s ( e ) = j t ( e ) = j by an SU (2) Wilson line in the representation j dependent only of SU χ 0 (2) components of the connection.”

The Lorentz extension (5) of the Ashtekar-Barbero connection does not transform correctly under time diffeomorphisms.

“And indeed, it was shown that its Lorentz extension (5) does not tranform correctly under the time diffeomorphisms [11].”

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Provided each edge is associated with a simple representation R(0,ρ) of SL(2,C), the infinitely refined projected spin networks depend only on the solution of the second class constraints (19).

“Now we prove that provided each edge is associated with a simple representation R (0 ,ρ ) of SL (2 , C ), such infinitely refined projected spin networks depend only on the solution of the second class constraints (19).”

We are not able to carry out the second part of the program and solve the second class constraints on this Hilbert space as we had done for the connection A.

“However, we are not able to carry out the second part of the program and solve the second class constraints on this Hilbert space as we had done for the connection A .”

There does not exist any canonical transformation relating theories with different values of β within the covariant formalism.

“Nevertheless, let us note that there does not exist any canonical transformation, relating theories with different values of β , within our covariant formalism.”

The scalar product (14) does not depend on the choice of (χ1,...,χV) due to Lorentz invariance.

“This scalar product does not depend on the choice of ( χ 1 , . . . , χ V ) due to the Lorentz invariance.”

The constructed states are eigenvectors of all area operators.

“It is important to note that the constructed states are eigenvectors of all area operators [14].”

• cites [14] [14] S. Alexandrov, Hilbert Space for Covariant Loop Quantum Gravity , Phys. Rev. D 66 , (2002) 024028 [gr-qc/0201087]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The projected spin networks, evaluated in the time gauge, reduce to the usual SU(2) spin networks.

“Then, the projected spin networks, evaluated in the time gauge, reduce to the usual SU (2) spin networks, which are actually the natural SU (2) gauge invariant objects.”

The area operator following from the commutation relations (28) is expressed as a combination of two Casimir operators.

“The area operator following from the commutation relations (28) is expressed as a combination of two Casimir operators:”

The second class constraints now correspond to the constraints satisfied by the connection (5).

“The second class constraints now correspond to the constraints satisfied by the connection (5):”

In comparison with the Lorentz generalization of the Ashtekar-Barbero connection, the nontrivial part of the connection is contained in the boost rather than SU(2) components.

“The nontrivial part of the connection is contained in the boost rather than SU (2) components (see (28) and (33)).”

The covariant Ashtekar-Barbero connection (5) is only one among the two-parameter family of Lorentz connections found in the covariant approach.

“To answer this question, let us recall that the covariant Ashtekar-Barbero connection (5) is only one among the two-parameter family of Lorentz connections found in the covariant approach.”

Projected spin networks projected onto the trivial SU(2) representation j=0 solve the second class constraints at their vertices.

“it turns out that projected spin networks projected onto the trivial SU (2) representation j = 0 solve the second class constraints at their vertices”

If one imposes the additional restriction of transforming properly under time diffeomorphisms, there is only one connection satisfying it, possessing a genuine spacetime interpretation.

“Indeed, it turns out that if we impose this additional restriction, there is only one connection satisfying it [11], i.e. , possessing a genuine spacetime interpretation.”

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The Immirzi parameter enters the area spectrum.

“However, this parameter enters the area spectrum.”

The connection remains noncommutative.

“The connection remains noncommutative.”

For the Lorentz connection satisfying (19), the insertion of the projector on a representation j in the middle of an edge has a trivial effect: if j_s(e)=j_t(e)=j one gets identity, otherwise the result vanishes.

“One can note that for the Lorentz connection satisfying (19) the insertion of the projector on a representation j in the middle of an edge has a trivial effect: if j s ( e ) = j t ( e ) = j we get identity, otherwise the result vanishes.”

The full (kinematical) structure of SU(2) Loop Quantum Gravity has been recovered at the level of the Hilbert space.

“We have actually recovered the full (kinematical) structure of SU (2) Loop Quantum Gravity at the level of the Hilbert space.”

We end up with the same kinematical Hilbert space H0.

“In this way we end up with the same kinematical Hilbert space H 0 .”

The projected spin networks realize an orthonormal basis in H0.

“Thus, the projected spin networks realize an orthonormal basis in H 0 .”

The Ashtekar-Barbero connection cannot be extended in such a way that it transforms as a true space-time connection under 4-dimensional diffeomorphisms.

“In terms of transformation properties it means that the Ashtekar-Barbero connection cannot be extended in such a way that it transforms as a true space-time connection under 4-dimensional diffeomorphisms.”

For the simple representations β(j)=0, which implies that the projected boost generators vanish.

“For the simple representations β ( j ) = 0 which implies that the projected boost generators then vanish.”

The area attached to an edge at a vertex does not depend at all on the SL(2,C) representations.

“Interestingly, this does not depend at all on the SL (2 , C ) representations.”

The unique spacetime Lorentz connection diagonalizing the area operator corresponds to the choice a=b=0 in equation (58), leading to the shifted connection.

“This connection corresponds to the choice a = b = 0 in equation (58) what leads to the following shifted connection [10, 11]”

• cites [10] [10] S. Alexandrov and D. Vassilevich, Area spectrum in Lorentz covariant loop gravity , Phys. Rev. D 64 , (2001) 044023, [gr-qc/0103105]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The criterion to resolve the quantization ambiguity is that the connection transforms properly under time diffeomorphisms.

“The answer is affirmative and the corresponding criterion is actually simply that it transforms properly under the time diffeomorphisms.”

The reason the implementation fails is that loop operators are not mapped to the time translated loop operators after symmetry transformation.

“The reason is that loop operators are not mapped to the time translated loop operators after symmetry transformation.”

In the time gauge, A is reduced to its SUχ0(2) part and becomes simply an SU(2) connection.

“As a result A is reduced to its SU χ 0 (2) part and becomes simply an SU (2) connection.”

The connection (31) satisfies the constraints.

“It is clear then that the connection (31) satisfies the constraints:”

Such invariant functions are entirely given by the functions fχ0(A)=f(A,χ=χ0) taken for χ constant on the hypersurface equal to χ0.

“Such invariant functions are in fact entirely given by the functions f χ 0 ( A ) = f ( A , χ = χ 0 ) taken for χ constant on the hypersurface equal to χ 0 .”

For this choice of connection, the area spectrum does not depend on the Immirzi parameter.

“Moreover, for this choice of connection the area spectrum does not depend on the Immirzi parameter.”

The Dirac bracket is equal to the initial Poisson bracket when both the considered quantities commute with only the φ ij constraints.

“an interesting property of this Dirac bracket is that it is equal to the initial Poisson bracket when both the considered quantities commute with only the φ ij constraints:”

Due to Lorentz invariance, the infinitely refined projected spin networks still give a solution of the problem representing a basis of the physical Hilbert space.

“Due to the Lorentz invariance the infinitely refined projected spin networks still give a solution of our problem representing a basis of the physical Hilbert space.”

The nondynamical part of the connection given by Γ X i (∼Q) does not vanish in the time gauge.

“The nondynamical part of the connection given by Γ X i ( ∼ Q ) does not vanish in the time gauge.”

These constraints give raise to a Dirac bracket.

“These constraints give raise to a Dirac bracket.”

Computing the holonomies of A, one gets group elements belonging to the SUχ0(2) subgroup.

“Computing the holonomies of A , we get group elements belonging to the SU χ 0 (2) subgroup.”

Imposing that a spin network based on the initial SL(2,C) connection A commutes with φ leads to an equation on the Casimir operators of the representation living on the chosen link.

“Imposing that it commutes with φ leads to an equation on the Casimir operators of the representation living on the chosen link:”

A (ash) X i gives an embedding of A (bar) a i into the Lorentz algebra.

“Thus, A (ash) X i gives an embedding of A (bar) a i into the Lorentz algebra.”

For general β, the Poisson algebra is complicated by the R change of basis.

“The situation is complicated by the R change of basis.”

One can accomplish the most important step: to implement the second class constraints at the Hilbert space level.

“In particular, one can accomlish the most important step: to implement the second class constraints at the Hilbert space level.”

I(Q) ̃A commutes with ∼P.

“Then, as I ( Q ) ˜ A commutes with ∼ P , we get”

This equation is exactly the same as the one arising in the construction of spin foam models from the generalised Palatini action.

“This equation is exactly the same as the one arising in the construction of spin foam models from the generalised Palatini action [25].”

• cites [25] [25] E.R. Livine, Immirzi parameter in the Barrett-Crane model? , gr-qc/0103081. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The Dirac brackets change the commutation relations of the canonical variables so that the connection becomes non-commutative.

“They change the commutation relations of the canonical variables so that the connection becomes non-commutative.”

The commutation relations (61) and the area spectrum (66) found for arbitrary connection of the family (58) depend only on the parameters a and b from the definition of the connection but not on β.

“They depend only on the peremeters a and b from the definition of the connection but not on β .”

The covariant generalization of the Ashtekar-Barbero connection A given by (5) is also commuting.

“This allows to conclude that the covariant generalization of the Ashtekar-Barbero connection A given by (5) is also commuting.”

The second class constraints lead to Dirac brackets.

“which lead to Dirac brackets.”

A X i (0,0) is the only connection from the family (58) which transforms correctly under the time diffeomorphisms.

“More precisely, it is the only connection from the family (58), which transforms correctly under the time diffeomorphisms.”

There exists a true space-time connection, the unique Lorentz connection which transforms properly under space-time diffeomorphisms.

“We have also underlined the existence of a true space-time connection, the unique Lorentz connection which transforms properly under space-time diffeomorphisms.”

If χ = const, A (ash) a i transforms as a connection with respect to the transformations generated by G ( χ ) a.

“It turns out that if χ = const , A (ash) a i transforms as a connection with respect to the transformations generated by G ( χ ) a :”

It is the commutator of the connection and triad multiplet from which we derive the spectrum of area and other geometrical operators.

“It is this commutator from which we derive the spectrum of area and other geometrical operators.”

The Immirzi parameter does not appear in the commutation relations of the connection and triad multiplet (28).

“The main reason is that it does not appear in the commutation relations of the connection and triad mutliplet (28).”

The resulting area spectrum for generic (a,b) is given by the formula in the text.

“The resulting area spectrum for generic ( a, b ) is”

The Poisson bracket of two simple spin network functionals whose graphs intersect only at some common vertices vanishes.

“Therefore, the Poisson bracket of two such functionals whose graphs intersect only at some common vertices vanishes.”

This representation takes into account only the commutation relations (36) at the chosen points.

“This representation takes into account only the commutation relations (36) at the chosen points.”

The quantity (3) commutes with itself.

“it commutes with itself.”

We have a complete realisation of the Dirac brackets of the connection and the triad.

“Therefore, we have a complete realisation of the Dirac brackets of the connection and the triad.”

Simple spin networks based on graphs with a chosen initial set of points as vertices give a representation of the initial commutation relations and quantum structure that reproduces exactly the kinematical setting and boundary states of the Barrett-Crane model.

“we consider the simple spin networks based on such graphs and we obtain a representation of the initial commutation relations and quantum structure which reproduces exactly the kinematical setting and the boundary states of the Barrett-Crane model [16].”

• cites [16] [16] E.R. Livine, Projected Spin Networks for Lorentz connection: Linking Spin Foams and Loop Gravity , gr-qc/0207084. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The dependence on the Immirzi parameter in the SU(2) case appears only after the identification (a,b)=(-β,1).

“The dependence on the Immirzi parameter in the SU(2) case appears only after the identification ( a, b ) = ( -β, 1) (4).”

We can change A to ̃A without modifying anything in the Poisson brackets.

“First, we can change A to ˜ A without modifying anything in the Poisson brackets.”

One has the same structure as previously replacing I(P)A of the case β→∞ by I(P)R^{-1} ̃A.

“Then, one has the same structure as previously replacing I ( P ) A of the case β → ∞ by I ( P ) R -1 ˜ A .”

We have derived the framework of SU(2) Loop Quantum Gravity from the explicitly Lorentz covariant formalism.

“In the present work, we have derived the framework of SU (2) Loop Quantum Gravity from the explicitly Lorentz covariant formalism.”

This quantization procedure linking the canonical theory to the spin foam setting is based on solutions at a finite number of points to the second class constraints, relating the (space-like part of the) connection to the triad.

“Nevertheless, this quantization procedure linking the canonical theory to the spin foam setting is based on solutions at a finite number of points to the second class constraints, relating the (space-like part of the) connection to the triad.”

The operator ̂I(Q) ̃A can be deduced as the Christoffel symbol Γ( ̃Q) constructed with the ̂ ̃Q operator.

“The operator ̂ I ( Q ) ˜ A can be deduced as the Christofel symbol Γ( ˜ Q ) constructed with the ̂ ˜ Q operator.”

The action of the area operator on such a Wilson line depends on the details of embedding of the surface and line into 3d space.

“The action of the area operator on such a Wilson line depends on the details of embedding of the surface and line into 3d space [10].”

• cites [10] [10] S. Alexandrov and D. Vassilevich, Area spectrum in Lorentz covariant loop gravity , Phys. Rev. D 64 , (2001) 044023, [gr-qc/0103105]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

We were able to reproduce the Hilbert space structure of SU(2) LQG from the Hilbert space of covariant quantization.

“so that we were able to reproduce the Hilbert space structure of SU (2) LQG from the Hilbert space of covariant quantization.”

We can choose the operator ̂I(P)R^{-1} ̃A to be the multiplication by I(P) ̃A.

“Then, we can choose the operator ̂ I ( P ) R -1 ˜ A to be the multiplication by I ( P ) ˜ A .”

The simple spin network is an eigenvector of the area operator of a surface intersecting the graph at the end of an edge (at the vertex).

“The simple spin network is its eigenvector with eigenvalue:”

Λ X i (∼Q) depends only on the field χ.

“Using the explicit expressions (46), one can show that actually Λ X i ( ∼ Q ) depends only on the field χ :”

This procedure allows to create an SL(2,C) connection from an SU(2) one by introducing the vector field χ.

“Thus, this procedure allows to create an SL (2 , C ) connection from an SU (2) one by introducing the vector field χ .”

• AssumptionThe authors state this as an assumption: We have a hypersurface with an SU(2) connection a.

Loop quantum gravity with the Immirzi parameter was shown to come from a canonical analysis of the generalized Hilbert-Palatini action in the time gauge.

“Loop quantum gravity with the Immirzi parameter was shown to come from a canonical analysis of the generalized Hilbert-Palatini action in the so-called time gauge [12, 13].”

• cites [12] [12] S. Holst, Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action , Phys. Rev. D 53 , 5966 (1996). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [13] [13] N. Barros e Sa, Hamiltonian analysis of General relativity with the Immirzi parameter , Int. J. Mod. Phys. D 10 , (2001) 261, [gr-qc/0006013]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The existence of many Lorentz connections constructible from the canonical variables gives rise to a quantization ambiguity in the loop approach, since each of them can be used in the definition of loop variables.

“This gives rise to a quantization ambiguity in the loop approach, since each of them can be used in the definition of loop variables.”

The choice of a space triad and an SU(2) connection loses the explicit covariance of the theory and a space-time geometrical interpretation.

“This particular choice of variables loses the explicit covariance of the theory and a space-time geometrical interpretation [6].”

• cites [6] [6] J. Samuel, Is Barbero's Hamiltonian formulation a gauge theory of Lorentzian gravity?, Class. Quantum Grav. 17 , L141 (2000), [gr-qc/0005095]; Phys. Rev. D 63 , 068501 (2001). - sourceNot checked yet

  Not checked in the free run.

The SU(2) LQG can be derived from the quantized covariant formulation.

“Thus, we are able to derive the SU (2) LQG from the quantized covariant formulation.”

The derivation establishes an exact correspondence between the covariant formalism and the usual SU(2) formalism.

“This derivation establishes an exact correspondence between the covariant formalism and the usual one.”

As shown by Holst, in the time gauge, the additional term leads to Loop Quantum Gravity with β as Immirzi parameter in the quantum theory.

“As was shown by Holst [12], in the so-called time gauge , this additional term leads to Loop Quantum Gravity with β as Immirzi parameter in the quantum theory.”

• cites [12] [12] S. Holst, Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action , Phys. Rev. D 53 , 5966 (1996). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Reproducing the phase space of the SU(2) approach at the classical level is not equivalent to deriving the SU(2) setting from a covariant quantum theory.

“However, it simply amounts to break the covariance of the theory and translate the SU (2) connection variables into the new formalism: it is not equivalent to derive the SU (2) setting from a covariant quantum theory.”

The authors are not able to complete the quantization program for the true spacetime connection giving a full solution of the second class constraints at the Hilbert space level.

“However, we are still not able to complete the quantization program for this connection giving a full solution of the second class constraints at the Hilbert space level.”

The obtained results suggest that the Immirzi parameter should disappear from the physical output of a path integral formulation of quantum gravity and of its canonical quantization based on the covariant formulation.

“The obtained results suggest that the Immirzi parameter should disappear from the physical output of a path integral formulation of quantum gravity [9] as well as of its canonical quantization based on this covariant formulation [10, 11].”

• cites [9] [9] S. Alexandrov, SO(4,C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter , Class.Quant.Grav. 17 (2000) 4255, [gr-qc/0005085]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [10] [10] S. Alexandrov and D. Vassilevich, Area spectrum in Lorentz covariant loop gravity , Phys. Rev. D 64 , (2001) 044023, [gr-qc/0103105]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The second class constraints can be translated into simplicity constraints restricting the used SL(2,C) representations to simple ones, which have a vanishing quadratic Casimir operator.

“They can then be translated into the so-called simplicity constraints restricting the used SL (2 , C ) representations to simple ones, which have a vanishing quadratic Casimir operator [27, 25, 26].”

• cites [27] [27] R. De Pietri and L. Freidel, so (4) Plebanski Action and Relativistic Spin Foam Model , Class. Quant. Grav. 16 , (1999) 2187, [gr-qc/9804071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [25] [25] E.R. Livine, Immirzi parameter in the Barrett-Crane model? , gr-qc/0103081. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [26] [26] E.R. Livine and D. Oriti, Barrett-Crane spin foam model from generalized BF-type action for gravity , Phys. Rev. D 65 , (2002) 044025, [gr-qc/0104043]. - sourceNot checked yet

  Not checked in the free run.

Since the selected subspace depends on the field χ, this leads to considering gauge invariant functionals of both the connection A and the time normal field χ.

“Since this subspace depends on the field χ , this leads us to consider gauge invariant functionals of both the connection A and the time normal field χ .”

Thus the two-parameter family of connections represents a real quantization ambiguity of the loop approach.

“Thus, they represent a real quantization ambiguity of the loop approach.”

The loss of spacetime interpretation for the SU(2) connection is probably the reason why the quantum constraint algebra does not reproduce the classical one and contains an anomaly.

“In particular, this fact is probably the reason why the quantum constraint algebra does not reproduce the classical one and contains an anomaly [18, 19, 17].”

• cites [18] [18] J. Lewandowski and D. Marolf, Loop constraints: A habitat and their algebra , Int. J. Mod. Phys. D 7 , (1998) 299, [gr-qc/9710016]; R. Gambini, J. Lewandowski, D. Marolf, and J. Pullin, On the consistency of the constraint algebra in the spin network quantum gravity , Int. J. Mod. Phys. D 7 , (1998) 97, [gr-qc/9710018]. - sourceNot checked yet

  Not checked in the free run.

• cites [19] [19] L. Smolin, The classical limit and the form of the Hamiltonian constraint in nonperturbative quantum general relativity , gr-qc/9609034. - sourceNot checked yet

  Not checked in the free run.

• cites [17] [17] M. Reisenberger and C. Rovelli, Sum over surfaces form of Loop Quantum Gravity , Phys. Rev. D 56 , (1997) 3490, [gr-qc/9612035]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It is possible to sidestep the problem of solving the second class constraints for the spacetime connection at the Hilbert space level and impose the second class constraints at a finite number of points.

“Nevertheless, it is possible to sidestep this problem and impose the second class constraints at a finite number of points.”

When carrying the canonical analysis, we get second class constraints of two types φ ij = Π XY ̃ Q i X ̃ Q j Y and ψ ij.

“when carrying the canonical analysis [9], we get second class constraints of two types φ ij = Π XY ˜ Q i X ˜ Q j Y (56) and ψ ij”

• cites [9] [9] S. Alexandrov, SO(4,C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter , Class.Quant.Grav. 17 (2000) 4255, [gr-qc/0005085]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

This means that there is a unique loop quantization preserving all classical symmetries of general relativity.

“This means that there is a unique loop quantization preserving all classical symmetries of general relativity.”

The SU(2) LQG can be rigorously derived from the quantization of the covariant formalism based on the connection (5).

“Thus the SU(2) LQG can be rigorously derived from the quantization of the covariant formalism based on the connection (5).”

The present covariant canonical framework builds a bridge between spin foam models based on SL(2,C) and the canonical framework based on SU(2).

“The present covariant canonical framework builds a bridge between these two pictures”

• AssumptionRests on an assumption: There is a lack of an explicit link between existing spin foam models based on SL(2,C) and the canonical framework based on SU(2).

Hence the quantum diffeomorphism algebra contains an anomaly.

“Hence the quantum diffeomorphism algebra contains an anomaly.”

From the point of view of canonical quantization, the Barrett-Crane model can be interpreted as quantizing the system without imposing the second class constraints and imposing them only afterwards at the quantum level.

“From the point of view of canonical quantization, it can be interpreted as quantizing the system without imposing the second class constraints and imposing them only afterwards at the quantum level.”

As a result, the implementation of time diffeomorphism symmetry in quantum theory in the framework of loop quantization fails.

“As a result, the implementation of this symmetry in quantum theory in the framework of loop quantization fails.”

The results of section III say the Immirzi parameter does not become physical in the second quantization.

“The results of section III say it does not.”

The appearance of the Immirzi parameter in the results of SU(2) LQG is a consequence of the quantum anomaly in the four-dimensional diffeomorphism invariance.

“All this allows to conclude that the appearence of the Immirzi parameter in the results of the SU(2) LQG is a consequence of the quantum anomaly in the four-dimensional diffeomorphism invariance.”

The commutation relations of A with itself tell us that the operator for I(P)A will not be trivial.

“Nevertheless, the commutation relations A and itself tell us that the operator for I ( P ) A will then not be trivial.”

Even if the commutator of two connections contributes to some physical results, it is very likely that the Immirzi parameter will be cancelled.

“Therefore, even if this commutator contributes to some physical results, it is very likely that the Immirzi parameter will be cancelled.”

There is a two-parameter family of Lorentz connections transforming correctly under the Gauss and diffeomorphism constraints, such that the area operator is diagonal on Wilson lines defined with the new connection.

“It has been shown [11] that there is a two-parameter family of such Lorentz connections, i.e. , transforming correctly under the Gauss and diffeomorphism constraints.”

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

For β→∞, the theory corresponds to the usual Palatini action without the extra term introduced by the Immirzi parameter.

“This corresponds to the usual Palatini action without the extra term introduced by the Immirzi parameter”

The parameter β was identified as the Immirzi parameter of SU(2) LQG through the canonical analysis in the time gauge.

“It was identified as the Immirzi parameter of SU (2) LQG through the canonical analysis in the time gauge [12].”

• cites [12] [12] S. Holst, Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action , Phys. Rev. D 53 , 5966 (1996). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The condition of commuting with φ coincides exactly with the simplicity constraint used in spin foam models.

“Moreover, it coincides exactly with the simplicity constraint used in spin foam models.”

We suggested a quantization procedure which leads to the resulting picture of quantum geometry states similar to one of the Barrett-Crane model.

“In particular, we suggested a quantization procedure which leads to the resulting picture of quantum geometry states similar to one of a particular spin foam model - the Barrett-Crane model.”

The formal path integral constructed for the generalized Hilbert-Palatini action does not depend on β.

“It was shown that the formal path integral constructed for the generalized Hilbert-Palatini action does not depend on β [9].”

• cites [9] [9] S. Alexandrov, SO(4,C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter , Class.Quant.Grav. 17 (2000) 4255, [gr-qc/0005085]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

This hints toward a link between the space-time formalism given by spin foams and the canonical frame given by LQG and spin network states.

“This hints toward a link between the space-time formalism given by spin foams and the canonical frame given by LQG and spin network states.”

The solution to the problem is to use the second class constraint I X (Q) Y A Y i = Γ X i (∼Q).

“The solution to the problem is to use the (second class) constraint I X ( Q ) Y A Y i = Γ X i ( ∼ Q ).”

These two results point to the non-relevance of the Immirzi parameter in the space-time dynamics.

“These two results point to the non-relevance of the Immirzi parameter in the space-time dynamics.”

The second class constraints build a theory based on the coset SO(3,1)/SO(3) or SL(2,C)/SU(2) just as in the spin foam scheme.

“This shows that the second class constraints build a theory based on the coset SO (3 , 1) /SO (3) or SL (2 , C ) /SU (2) just as in the spin foam scheme.”

Once the quantization preserving four-dimensional diffeomorphism invariance is chosen, the Immirzi parameter remains unphysical as it was at the classical level.

“Instead, once the quantization preserving this symmetry is chosen, it remains unphysical as it was at the classical level.”

The physical spectra of geometrical operators, like the area spectrum (10), depend on the Immirzi parameter.

“The main reason is that the physical spectra of geometrical operators, like the area spectrum (10), depend on it.”

Considering only quantities that commute with the φ constraints allows to ignore the ψ constraints.

“This leads to think that considering only quantities that commute with the φ constraints allows to ignore the ψ constraints.”

The spin foam quantization of the generalized Hilbert-Palatini action was shown not to depend on the Immirzi parameter.

“Similarly, the spin foam quantization, which can be understood as a discrete path integral, of the generalized Hilbert-Palatini action was shown not to depend on the Immirzi parameter [25, 26].”

• cites [25] [25] E.R. Livine, Immirzi parameter in the Barrett-Crane model? , gr-qc/0103081. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [26] [26] E.R. Livine and D. Oriti, Barrett-Crane spin foam model from generalized BF-type action for gravity , Phys. Rev. D 65 , (2002) 044025, [gr-qc/0104043]. - sourceNot checked yet

  Not checked in the free run.

SU(2) LQG says the Immirzi parameter plays a role at the quantum level.

“The SU(2) LQG says it does.”

It seems possible to reproduce, within the canonical approach, the spin foam result that simplicity conditions impose a SO(D)/SO(D-1) coset structure to the quantum theory.

“it seems possible to reproduce, within the canonical approach, the spin foam result that simplicity conditions impose a SO ( D ) /SO ( D -1) coset structure to the quantum theory [28].”

• cites [28] [28] L. Freidel, K. Krasnov, and R. Puzio, BF Description of Higher-Dimensional Gravity Theories , Adv. Theor. Math. Phys. 3 , (1999) 1289, [hep-th/9901069]. -Not checked yet

  Not checked in the free run.

A X i ≡ A X i (0,0) is the unique true space-time connection diagonalizing the area operator.

“where A X i ≡ A X i (0 , 0) is the unique true space-time connection diagonalizing the area operator [11].”

• cites [11] [11] S. Alexandrov, On choice of connection in loop quantum gravity , Phys. Rev. D 65 , (2002) 024011, [gr-qc/0107071]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

This quantum ambiguity is fixed by the requirement to retain the classical symmetry.

“In any case, this quantum ambiguity is fixed by the requirement to retain the classical symmetry.”

• AssumptionThe authors state this as an assumption: The classical symmetry should be retained.

It can be used to construct the covariant generalization of the Ashtekar-Barbero connection (5).

“It can be used to construct the covariant generalization of the Ashtekar-Barbero connection (5).”