Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity
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TL;DR
The paper constructs a regulated Wheeler-DeWitt constraint operator for four-dimensional non-perturbative Lorentzian vacuum quantum gravity in the continuum using real Ashtekar variables and loop/connection techniques. The operator is densely defined, requires no renormalization, is anomaly-free, and can be made symmetric.
Key findings
- A Wheeler-DeWitt constraint operator for 4D Lorentzian vacuum gravity is defined that is densely defined, finite on cylindrical functions, requires no renormalization, and is anomaly-free; symmetric ordering yields a symmetric operator.
- The regularization also produces well-defined operators including the Euclidean Wheeler-DeWitt operator, the generator of the Wick rotation transform, length operators, matter Hamiltonians, and generators of the asymptotic Poincaré group including quantum ADM energy.
- On spin-network states, the Hamiltonian constraint acts by annihilating, creating, and re-routing angular momentum quanta in units of ±ħ, ±ħ/2, 0; the spectrum at a given vertex is largely determined by the volume operator.
- Anomaly-freeness requires all loops α_ij(Δ) to be chosen as kinks with vertex at v(Δ); an arbitrary attachment of a_ij(Δ) is insufficient.
Methods
- Works in continuum canonical quantum gravity with real Ashtekar variables, SU(2) connections, densitized triads, and the Gauss, diffeomorphism, and Wheeler-DeWitt constraints; uses the Hilbert space L²(A/G, dμ₀) and gauge-invariant cylindrical functions.
- Regularizes the Euclidean Hamiltonian constraint by triangulating Σ into tetrahedra, using the volume operator V and holonomies, and replacing Poisson brackets by commutators; defines K via K = -{V, H_E[1]}, then defines the Lorentzian Wheeler-DeWitt operator using an identity that eliminates R_ab in favor of K.
- Checks anomaly freedom by evaluating commutators on diffeomorphism-invariant states; uses symmetric ordering and discusses group averaging for solutions and a physical inner product.
Limitations
- The paper states there is much freedom in the regularization step and that an important unresolved question is how to select the correct or physically relevant regularization of H.
- The paper states that no exact solution has been found in closed form except for cylindrical functions on two-valent graphs, which do not fully take the curvature term F_ab into account.
- The operator as written is not self-adjoint, though symmetric ordering is possible and self-adjoint extensions are expected; anomaly-freeness also requires the specific kink loop choice.
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Claims the paper makes
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- Findingreasoning§ Abstract · p. 1
A Wheeler-DeWitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum.
“AWheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum.”
- Also stated in: § Abstract · p. 7 · boosted
“Expressions (0.15), (0.16) are anomaly-free, densely defined operators corresponding to the original Wheeler-DeWitt operator in the continuum for non-perturbative four-dimensional Lorentzian canonical quantum vacuum gravity.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Also stated in: § Abstract · p. 7 · boosted
- Findingreasoning§ Abstract · p. 2
One can quantize the original Wheeler-DeWitt constraint in its non-polynomial, unrescaled form in a satisfactory way.
“One can quantize the original Wheeler-DeWitt constraint in its non-polynomial, unrescaled form in a satisfactory way.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingreasoning§ Abstract · p. 5
A solution to the anomaly freedom problem for the Euclidean Hamiltonian constraint is obtained if all loops αij(∆) are chosen to be kinks with vertex at v(∆), i.e., the arc aij(∆) joins the endpoints of ei(∆), ej(∆) in an at least C1 fashion.
“Specifically, a solution to the anomaly freedom problem is obtained 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 an at least C 1 fashion.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingreasoning§ Abstract · p. 6
The constraint algebra remains non-anomalous even after symmetric ordering.
“The constraint algebra remains non-anomalous even after symmetric ordering”
Quote in the paperSubstantial claim - Findingown data§ Abstract · p. 2
The unrescaled Euclidean Hamiltonian constraint operator and the generator of the Wick transform appear naturally in the regularization procedure and are constructed simultaneously as a side result.
“Quite surprisingly, however, the unrescaled Euclidean Hamiltonian constraint operator and the generator of the Wick transform appear very naturally in our regularization procedure so that we cannot avoid to construct them simultanously as a side result.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingreasoning§ Abstract · p. 1
The final Wheeler-DeWitt constraint operator is anomaly-free.
“The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric.”
- Also stated in: § Abstract · p. 7 · plain
“Finally, for the same reason that ˆ H E T [ N ] is anomaly-free, ˆ H T is anomaly free as well.”
- Also stated in: § Abstract · p. 7 · plain
“Expressions (0.15), (0.16) are anomaly-free, densely defined operators corresponding to the original Wheeler-DeWitt operator in the continuum for non-perturbative four-dimensional Lorentzian canonical quantum vacuum gravity.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Also stated in: § Abstract · p. 7 · plain
- Findingnone given§ Abstract · p. 1
The regulated Wheeler-DeWitt constraint operator is densely defined.
“The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric.”
- Also stated in: § Abstract · p. 7 · plain
“Expressions (0.15), (0.16) are anomaly-free, densely defined operators corresponding to the original Wheeler-DeWitt operator in the continuum for non-perturbative four-dimensional Lorentzian canonical quantum vacuum gravity.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Also stated in: § Abstract · p. 7 · plain
- Findingreasoning§ Abstract · p. 5
An arbitrary attachment of aij(∆) is insufficient to guarantee anomaly-freeness.
“An arbitrary attachment of a ij (∆) is insufficient to guarantee anomaly-freeness.”
Quote in the paperSubstantial claim - Findingnone given§ Abstract · p. 6
Upon taking a symmetrical ordering of (0.9), one arrives at a symmetric operator.
“Finally, upon taking a symmetrical ordering of (0.9) we manage to arrive at a symmetric operator 1 .”
Quote in the paperSubstantial claim - Findingreasoning§ Abstract · p. 1
The regulated Wheeler-DeWitt constraint operator does not require any renormalization.
“The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric.”
- Also stated in: § Abstract · p. 5 · plain
“no ill-defined products of distributions arise, we do not encounter any singularities, no renormalization of the operator is necessary.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Also stated in: § Abstract · p. 5 · plain
- Findingnone given§ Abstract · p. 1
The final Wheeler-DeWitt constraint operator is at least symmetric.
“The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingreasoning§ Abstract · p. 5
If triangulations are chosen such that for each point of Σ the number of edges of tetrahedra intersecting it is uniformly bounded from above, then only a finite number of terms are involved in the sum (0.10).
“Therefore, if we choose our triangulations T such that for each point of Σ, the number of edges of tetrahedra of T intersecting it, is uniformly bounded (uniformly in T ) from above by some integer then, no matter how fine the triangulations is, there are always only a finite number of terms involved in the sum (0.10).”
Quote in the paperSubstantial claim - Findingreasoning§ Abstract · p. 8
On spin-network states the Hamiltonian constraint acts by annihilating, creating and re-routing the quanta of angular momentum associated with the edges of the graph in units of ±ħ, ±ħ/2, 0.
“On such spin-network states the Hamiltonian constraint acts by annihilating, creating and re-routing the quanta of angular momentum associated with the edges of the graph in units of ± ¯ h, ± ¯ h/ 2 , 0.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingnone given§ Abstract · p. 9
The ADM Hamiltonian acts effectively by multiplying the states |γ, j> by certain algebraic factors depending on the spins j.
“the ADM Hamiltonian acts effectively by multiplying the states | γ, ⃗ j > by certain algebraic factors depending on the spins ⃗ j .”
Quote in the paperNot a substantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingnone given§ Abstract · p. 8
The final expression of the Wheeler-DeWitt constraint (0.15) is surprisingly simple: on each cylindrical function it is a low order polynomial in the volume operator and holonomy operators.
“The final expression of the Wheeler-DeWitt constraint (0.15) is surprisingly simple : on each cylindrical function it is a low order polynomial in the volume operator and holonomy operators”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingreasoning§ Abstract · p. 7
The regularized Wheeler-DeWitt operator can be built alone from the volume operator V and operators corresponding to holonomies along the edges of the tetrahedra of a triangulation.
“and we see that the Wheeler-DeWitt constraint operator can be built alone from the volume operator ˆ V and operators corresponding to holonomies along the edges of the tetrahedra of a triangulation.”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingnone given§ Abstract · p. 8
The Hamiltonian constraint operator on a given graph reduces to a finite number of mutually commuting (in the diffeomorphism invariant context), symmetric constraint operators, one for each vertex of γ.
“As the Hamiltonian constraint operator on a given graph γ reduces to a finite number of mutually commuting (in the diffeomorphism invariant context), symmetric constraint operators (which we expect to have self-adjoint extensions), one for each vertex of γ”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Findingnone given§ Abstract · p. 7
The procedure leads to the quantization of the original Wheeler-DeWitt constraint, rather than a modified version thereof.
“our procedure leads to the quantization of the original Wheeler-DeWitt constraint, rather than a modified version thereof.”
Quote in the paperSubstantial claim - Findingnone given§ Abstract · p. 8
The spectrum of the Hamiltonian constraint operator at a given vertex is largely determined by the spectrum of the volume operator.
“Remarkably, the spectrum of the Hamiltonian constraint operator at a given vertex is largely determined by the spectrum of the volume operator”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Premisecitation§ Abstract · p. 4
There is a well-defined, self-adjoint operator V on H corresponding to the total volume V whose action on cylindrical functions is perfectly finite.
“The first fact that we are going to use is that there is a well-defined, self-adjoint operator ˆ V on H corresponding to V [13] whose action on cylindrical functions is perfectly finite”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13]
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- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Premisereasoning§ Abstract · p. 1
To define the Lorentzian Wheeler-DeWitt constraint operator it would be sufficient to define the Euclidean operator and the generator of the transform.
“Given the developments in connection with the above mentioned Wick transform, in order to define then the Lorentzian WheelerDeWitt constraint operator it would be sufficient to define the Euclidean operator and the generator of the transform.”
Quote in the paperSubstantial claim - Premisecitation§ Abstract · p. 1
This framework has been successfully employed to arrive at the complete solution of the Gauss and diffeomorphism constraints.
“In particular, this framework has been successfully employed to arrive at the complete solution of the Gauss and diffeomorphism constraints [7].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [7]
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- Premisecitation§ Abstract · p. 1
Real-valued variables allow for the construction of a mathematically rigorous kinematical framework by means of which constraint operators can be regulated in a non-ambiguous fashion.
“The advantage of these real-valued variables is that they allow for the construction of a mathematically rigorous kinematical framework by means of which constraint operators can be regulated in a non-ambiguous fashion.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] source
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- Interpretationown data§ Abstract · p. 2
The requirement that the Hamiltonian constraint should be polynomial can be discarded.
“The requirement that the Hamiltonian constraint should be polynomial can be discarded.”
Quote in the paperSubstantial claim - Premisecitation§ Abstract · p. 2
The validity of those prior operators relies heavily on the assumption that one can exponentiate the generator of the Wick transform as outlined in [4].
“Their validity relies heavily on the assumption that one can actually exponentiate the generator of the Wick transform as outlined in [4].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] source
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- Interpretationnone given§ Abstract · p. 1
The technique introduced here can be used to produce completely well-defined regulated operators including the Euclidean Wheeler-DeWitt operator, the generator of the Wick rotation transform, length operators, Hamiltonian operators of the matter sector, and the generators of the asymptotic Poincaré group including the quantum ADM energy.
“The technique introduced here can also be used to produce a couple of completely well-defined regulated operators including but not exhausting a) the Euclidean Wheeler-DeWitt operator, b) the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, c) length operators, d) Hamiltonian operators of the”
Quote in the paperNot a substantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- Premisecitation§ Abstract · p. 2
All versions of Hamiltonian constraint operators for canonical quantum gravity in the continuum constructed so far in the literature, whether in the connection representation or in the loop representation, provide quantizations only of the Euclidean theory.
“It should be stressed at this point that all versions of Hamiltonian constraint operators for canonical quantum gravity in the continuum that have been constructed so far in the literature, whether in the connection representation [9] or in the loop representation [8, 10], suffer from the fact that they provide quantizations only of the Euclidean theory.”
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- Interpretationreasoning§ Abstract · p. 7
It is expected that Euclidean solutions found already in the literature for operators defined in [10,8] are far from being annihilated by our operator.
“It is therefore to be expected that the Euclidean solutions that have been found already in the literature for the operators defined in [10, 8] are far from being annihilated by our operator.”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [10] source
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- Interpretationreasoning§ Abstract · p. 9
The spin-network representation can be interpreted as the 'non-linear Fock-' (or occupation number) representation for quantum gravity.
“In other words, the spin-network representation can be interpreted as the 'non-linear Fock-' (or occupation number) representation for quantum gravity.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
- May combine two claims. 20
- Classified as probably not a substantial claim. 7
May combine two claims.
Findingreasoning§ Abstract · p. 5The regularized Euclidean Hamiltonian constraint operator (0.9) has the correct classical limit and its action on cylindrical functions is finite.
“is a regularized operator with the correct classical limit and whose action on cylindrical functions is indeed finite !”
Quote in the paperSubstantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
May combine two claims.
Findingreasoning§ Abstract · p. 5A tetrahedron contributes to the action of the regularized Euclidean Hamiltonian constraint on a cylindrical function only if it intersects the graph, and only if it intersects the graph in one of its vertices.
“that is, a tetrahedron contributes to the action on f only if it intersects the graph. Moreover, as one can show in more detail, it only contributes if it intersects the graph in one of its vertices.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Abstract · p. 5No ill-defined products of distributions arise, no singularities are encountered, and no renormalization of the operator is necessary.
“no ill-defined products of distributions arise, we do not encounter any singularities, no renormalization of the operator is necessary.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Abstract · p. 6If H_E T is ordered appropriately, both operators on the right hand side are symmetric and finite, and a finite and symmetric expression for K_T is produced.
“If we order ˆ H E T appropriately then both operators on the right hand side are symmetric and finite and we have produced a finite and symmetric expression for ˆ K T .”
Quote in the paperSubstantial claimMay combine two claims.
Premisecitation§ Abstract · p. 1The Hamiltonian constraint in Ashtekar variables is polynomial only after rescaling it by a non-polynomial function, a power of the square root of the determinant of the three-dimensional metric, so the original Wheeler-DeWitt constraint is altered.
“2) the Hamiltonian constraint is polynomial only after rescaling it by a non-polynomial function, namely a power of the square root of the determinant of the three-dimensional metric, that is, the original Wheeler-DeWitt constraint has actually been altered.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] source
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Interpretationnone given§ Abstract · p. 2The formulation never introduces complex variables and never needs to deal with Euclidean gravity; a Wick rotation transform is completely unnecessary, so problem 1 disappears.
“We never introduce complex variables and we never need to deal with Euclidean gravity. A Wick rotation transform is completely unnecessary : problem 1) also disappears.”
Quote in the paperSubstantial claimMay combine two claims.
Premisereasoning§ Abstract · p. 2Taking the square root of the rescaled Euclidean Hamiltonian constraint is a required technical step to preserve diffeomorphism invariance, but there remain problems with taking the square root of an infinite number of non-commuting, non-positive, not self-adjoint operators.
“While this seems to be a required technical step to do in order to preserve diffeomorphism invariance there remain problems with it that have to do with taking the square root of an infinite number of non-commuting, non-positive, not self-adjoint operators.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S2 (checkable). May combine two claims.
Interpretationreasoning§ Abstract · p. 2The resulting operator is surprisingly simple and the problem of computing its kernel is conceivable.
“The resulting operator is surprisingly simple and the problem of computing its kernel is conceivable.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisereasoning§ Abstract · p. 2The operator obtained is not to be confused with the operator defined in [11] because the present work is in the continuum rather than on a lattice and the present operator is completely well-defined on the whole Hilbert space while the one in [11] suffers from singularities on a large subspace of it.
“The operator we obtain is not to be confused with the operator defined in [11] (which is also formulated in terms of real variables) because a) we work in the continuum rather than on a lattice and b) our operator is completely well-defined on the whole Hilbert space while the one in [11] suffers from singularities on a large subspace of it.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [11] source
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Premisenone given§ Abstract · p. 2The mathematical difficulties associated with exponentiating the Wick transform are highly non-trivial and it is even possible that the Wick transform does not exist.
“The mathematical difficulties associated with this step are highly non-trivial and it is even possible that the Wick transform simply does not exist.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisenone given§ Abstract · p. 2On general grounds, an operator carrying density weight two would need to be renormalized, introducing a length scale, which breaks diffeomorphism invariance.
“On general grounds, such an operator would need to be renormalized (thus introducing a length scale) which breaks diffeomorphism invariance.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ Abstract · p. 2Since the Wheeler-DeWitt constraint carries density weight one, problem 2 disappears and one never has to take ill-defined square roots.
“Since the Wheeler-DeWitt constraint carries density weight one, problem 2) disappears, we never have to take ill-defined square roots.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisecitation§ Abstract · p. 2The new Hamiltonian constraint carries density weight two.
“what is worse is that the new Hamiltonian constraint carries density weight two.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] source
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Premisecitation§ Abstract · p. 2A solution to problem 2 was first suggested: take the square root of the rescaled Euclidean Hamiltonian constraint.
“A solution to this problem was first suggested in [8] : the idea is to take the square root of the rescaled Euclidean Hamiltonian constraint.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [8]
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Premisecitation§ Abstract · p. 1Ashtekar connection variables for Lorentzian gravity are complex valued and are subject to algebraically highly complicated reality conditions, difficult to impose at the quantum level.
“There are two, a priori, problems with these Ashtekar connection variables for Lorentzian gravity : 1) they are complex valued and are therefore subject to algebraically highly complicated reality conditions, difficult to impose on the quantum level, which make sure that we are still dealing with real general relativity”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] source
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Premisecitation§ Abstract · p. 1A solution to problem 1 has been suggested: one can define real Ashtekar variables which simplify the rescaled Hamiltonian constraint of Euclidean gravity and then construct a Wick rotation transform from the Euclidean to the Lorentzian regime where complex Ashtekar variables are needed to simplify the Lorentzian constraint.
“A solution to problem 1) has been suggested in [4] (see also [5]) : namely, one can define real Ashtekar variables [6] which simplify the rescaled Hamiltonian constraint of Euclidean gravity and then construct a Wick rotation transform from the Euclidean to the Lorentzian regime where the complex Ashtekar variables are needed to simplify the Lorentzian constraint.”
Quote in the paperSubstantial claim0 of 3 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone given§ Abstract · p. 6The integrated densitized trace of the extrinsic curvature is up to a constant factor the time derivative of the total volume with respect to the integrated Hamiltonian constraint.
“Recall that the integrated (densitized) trace of the extrinsic curvature is up to a constant factor just the time derivative of the total volume with respect to the integrated Hamiltonian constraint (which is a signature invariant statement)”
Quote in the paperNot a substantial claimMay combine two claims.
Premisereasoning§ Abstract · p. 7None of the operators in [8,9,10] can be used to define K along the lines proposed here because it was crucial that the classical identity K = -{V, H_E[1]} holds.
“none of the operators in [8, 9, 10] can be used to define ˆ K along the lines proposed here because it was crucial that the classical identity K = -{ V, H E [1] } holds”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [8]
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Interpretationcitation§ Abstract · p. 6The apparent contradiction with general arguments [15] is resolved because the infinitesimal generator of U(φt) cannot be defined since U(φt) does not act strongly continuously on H [7], so the question of whether the diffeomorphism constraint operator appears to the right in the commutator cannot even be asked.
“The resolution of the apparent contradiction is related to the fact that the infinitesimal generator of ˆ U ( φ t ), where φ t is a one parameter subgroup of Diff(Σ), cannot be defined since ˆ U ( φ t ) does not act strongly continuously on H [7]. Therefore the question of whether the diffeomorphism constraint operator appears to the right in the expression of the commutator of two Hamiltonian”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [15] 15. 351 T. Thiemann, 'Quantum Spin Dynamics (QSD) II', Harvard Preprint HUTMP96/B-352 - [13] C. Rovelli, L. Smolin, Nucl. Phys. B 442 (1995) 593, Erratum : Nucl. Phys. B 456 - (1995) 734 A. Ashtekar, J. Lewandowski, 'Quantum Theory of Geometry I : Area Operators', CGPG-96/2-4, gr-qc/9602046 - [14] A. Ashtekar, J. Lewandowski, J. Math. Phys. 36 (1995) 2170, J. Geo. and Phys. 17 (1995) 191 - [15] P. Haj ´ iˇ cek, K. Kuchaˇ r, Phys. Rev D41 (1990) 1091, Journ. Math. Phys. 31 (1990) 1723 - [16] A. Higuchi Class. Quant. Grav. 8 , 1983 (1991), Class. Quant. Grav. 8 , 2023 (1991). D. Marolf, 'The spe
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- cites [7]
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Interpretationreasoning§ Abstract · p. 8It seems feasible to determine the space of diffeomorphism invariant solutions to the Wheeler-DeWitt constraint as well as a physical inner product thereon by the group averaging method.
“it seems feasable to determine the space of diffeomorphism invariant solutions to the Wheeler-DeWitt constraint as well as a physical inner product thereon by the group averaging method [16, 7].”
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Premisereasoning§ Abstract · p. 7The Euclidean operator (0.9) and the ones proposed in [10] and [8,9] are entirely different; they only share that their classical limits or the square thereof are proportional to each other.
“Not even our Euclidean operator (0.9) and the ones proposed in [10] and [8, 9] have anything to do with each other, they are entirely different . The only thing they share is that their classical limits or the square thereof are proportional to each other.”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [10] source
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Interpretationnone given§ Abstract · p. 7The Wheeler-DeWitt constraint operator as defined is not self-adjoint yet, but a symmetrical ordering can be performed without picking up singularities to render it symmetric, and it is expected to possess self-adjoint extensions.
“As it stands it is not self-adjoint yet but it is clear that a symmetrical ordering can be performed (without picking up singularities) to render it symmetric and we expect it to possess self-adjoint extensions.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Interpretationnone given§ Abstract · p. 8There is a lot of freedom involved in the regularization step reflecting the fact that the quantum theory of a given classical field theory is not unique.
“There is a lot of freedom involved in the regularization step reflecting the fact that the quantum theory of a given classical field theory is not unique.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ Abstract · p. 8The method proposed opens access to other well-defined and finite operators including the length of a curve, matter Hamiltonians for canonical Yang-Mills theory, and the ADM Hamiltonian.
“It is clear that the method proposed here opens access to other well-defined and finite operators which were so far out of reach in a representation in which the intrinsic metric is not diagonal because we are able to make sense out of an operator corresponding to q ab . Examples are operators corresponding to the length of a curve [19], matter Hamiltonians for canonical Yang-Mills theory [20] or”
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- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
Probably not a substantial claim: may not pass S3 (load-bearing).
Interpretationreasoning§ Abstract · p. 9Roughly, the spins associated with the edges of a graph indicate the 'number' of elementary string-like excitations of the gravitational field along the edges of the graph.
“Roughly, the spins associated with the edges of a graph indicate the 'number' of elementary string-like excitations of the gravitational field along the edges of the graph”
Quote in the paperNot a substantial claim- The authors state this as an assumption: There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
The authors state this as an assumption.
There exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than by test functions of rapid decrease.
“if, as we do in the sequel, one makes the assumption that there exists a phase for quantum gravity in which the excitations of the gravitational fields can be probed by loops rather than, say, test functions of rapid decrease”
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The Wheeler-DeWitt operator in loop quantum gravity remains poorly understood.
“Unfortunately, this operator of the full theory remains poorly understood.”
From Loop quantum cosmology: I. Kinematics (2000) · cites this paper as [18]
Regularization choices for the scalar constraint have to be made and the result is not unique.
“Regularization choices have to be made and the result is not unique.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [352]
Thiemann's proposed formula for the Hamiltonian constraint lacks crossing symmetry.
“(Unfortunately, Thiemann's formula for the Hamiltonian constraint lacks crossing symmetry.)”
From Spin foam models (1998) · cites this paper as [59]
Ashtekar's canonical transformation makes the Hamiltonian constraint polynomial.
“The famous discovery due to Ashtekar [13] is that this is indeed possible by performing a certain canonical transformation on the gravitational phase space.”
From Quantum spin dynamics (QSD): V. Quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories (1998) · cites this paper as [13]
The other solutions of the constraints that exist for D=4 are absent for D>4.
“We note that the other solutions of the constraints present for D = 4 (see equation ( 61)) are instead absent for D > 4.”
From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [73]
In any dimension D>4, a non-degenerate B field satisfies constraints (69) if and only if there exists a frame field e such that B^{IJ}_{μν}=± e e^I_{[μ} e^J_{ν]}.
“It is then proven [73] that, in any dimension D > 4, a non-degenerate B field satisfies the constraints (69) if and only if there exists a frame field e µ I such that B µν I J = ± e e [µ I e ν] J (70 )”
From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [73]
The formulation of general relativity as a constrained BF theory holds in any dimension and in both Euclidean and Lorentzian signatures.
“that this formulation of general relativity as a constrained BF theory holds true in any dimension (and in both Euclidean and Lorentzian signatures) [73].”
From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [73]
The second constraint is implemented by requiring representation functions to be invariant under SO(3), realizing them as harmonic functions on SO(4)/SO(3)≅S^3, which was proven to completely characterize simple representations of SO(D) for any D.
“We can implement the second constraint at this level by requiring that all the representation functions have to be invariant under the subgroup SO(3) of SO(4), so realizing these representations in the space of harmonic functions over the coset SO(4)/SO(3) S 3 , which was proven in [65,73] to be a complete characterization of the simple representations of SO(D) for any dimension D”
From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [73]
If one does not work with half densities, the faithful implementation of the reality conditions on scalar fermion fields at the quantum level is inconsistent with the implementation of the canonical Anti-Poisson brackets, as observed in [21].
“b) The faithful implementation of the reality conditions on scalar fermion fields at the quantum level is inconsistent with the implementation of the canonical Anti-Poisson brackets as observed in [21].”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [21]
The result that the inner product is uniquely selected by reality conditions can be established at the non-gauge-invariant level, without dealing with complicated path observables defined in [15,21].
“We do not need to deal with the complicated path observables defined in [15, 21] in order to establish this result because we can prove everything at the non-gauge-invariant level.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [21]
Difficulties with the awkward reality structure of the fermion field make the framework of [21] incomplete: most interesting operators defined in [15] do not have an adjoint on the Hilbert space defined, and this prevented proving that their inner product is selected by the classical reality conditions.
“However, certain difficulties having to do with the awkward reality structure of the fermion field make their framework incomplete in the sense that most of the interesting operators defined in [15] do not have an adjoint on the Hilbert space defined. In particular, this prevented the authors from proving that their inner product is selected by the classical reality conditions.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [21]
With fermionic half-densities, the natural Berezin integral can be used to define the inner product, and adjoints can be rigorously computed for any operator constructed from fermion variables, proving that the inner product is uniquely selected by the reality conditions.
“With the fermionic half-densities on the other hand we can then basically use the natural Berezin integral also employed in [21] to define the inner product. However, now we can rigorously compute adjoints for any, not even gauge-invariant operator constructed from fermion variables and then prove that the inner product is uniquely selected by the reality conditions.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [21]
The authors of [21] formulated a kinematical framework for diffeomorphism invariant theories of fermions coupling to real-valued Ashtekar gravity and other gauge fields, defining a natural inner product by the Berezin integral, showing a class of operators are well-defined, and indicating how to define diffeomorphism invariant distributions involving fermions.
“The authors of [21] have followed this trend and formulated a kinematical framework for diffeomorphism invariant theories of fermions coupling to real-valued Ashtekar gravity and other gauge fields. That is, they define a natural inner product by means of the Berezin integral, show that a certain class of the operators defined in [15] are well-defined on it and indicate how to define diffeomorphis”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [21]
The kinematical framework developed in this paper is applied in [22], which extends the dynamical framework of [16,17,18,19,20] to the standard model coupled to gravity and other kind of matter.
“The kinematical framework developed in this paper is applied in [22] which extends the dynamical framework of [16, 17, 18, 19, 20] to the standard model coupled to gravity and other kind of matter.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [22]
In [21], the authors partly solve the problem by restricting the adjoint operation to a certain sub-algebra of 'real elements' of the gauge-invariant observable algebra, but the action of the Hamiltonian constraint derived in section 3 does not preserve any cyclic subspace generated by that subalgebra.
“In [21] the authors partly solve the problem by restricting the adjoint operation to a certain sub-algebra of 'real elements' of the (gauge-invariant) observable algebra. However, as one can show, the action of the Hamiltonian constraint derived in section 3 does not preserve any cyclic subspace generated by that subalgebra.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [21]
For the Einstein-Dirac theory in variables (A^i_a, E^a_i), the symplectic structure for each fermionic degree of freedom is Θ = i/2 ∫ d^3x √det(q)[η ̇η - ̇ηη] with scalar-valued η.
“The second reason becomes evident when recalling that for the Einstein-Dirac theory, formulated in terms of the variables ( A i a , E a i ) [22] which have proved to be successful for quantizing the Wheeler-DeWitt constraint [16, 17], the symplectic structure for each of the fermionic degrees of freedom is given by Θ = i 2 ∫ Σ d 3 x √ det( q )[ η ˙ η -˙ ηη ] with scalar-valued η .”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [22]
If one uses scalar-valued Grassman variables, the momentum conjugate to ξ would be the density π = i√det(q) ξ; the density weight comes from √det(q), which in quantum gravity becomes an operator, leading to inconsistencies if one wants an inner product incorporating the reality condition π = -i√det(q) ξ.
“The first reason comes from the fact that in the latter case the momentum conjugate to ξ would be the density π = i √ det( q ) ξ [22]. The density weight itself is not troublesome, what is troublesome is that the density weight comes from the function √ det( q ) which in quantum gravity becomes itself an operator. This immediately leads to inconsistencies as already stated in [21] if one wants to”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [22]
There is no minus sign missing in this canonical transformation because the fermion fields are anti-commuting and the action is invariant under this transformation.
“Notice that there is no minus sign missing in this canonical transformation because we take the fermion fields to be anti-commuting, the action is invariant under this transformation [23, 22].”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [22]
If one uses scalar-valued Grassman variables, the momentum conjugate to ξ would be the density π = i√det(q) ξ; the density weight comes from √det(q), which in quantum gravity becomes an operator, leading to inconsistencies if one wants an inner product incorporating the reality condition π = -i√det(q) ξ.
“The first reason comes from the fact that in the latter case the momentum conjugate to ξ would be the density π = i √ det( q ) ξ [22]. The density weight itself is not troublesome, what is troublesome is that the density weight comes from the function √ det( q ) which in quantum gravity becomes itself an operator. This immediately leads to inconsistencies as already stated in [21] if one wants to”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [21]
Cites
Discreteness of area and volume in quantum gravity
Rovelli, C., & Smolin, L. (1995) · Nuclear Physics B · doi:10.1016/0550-3213(95)00150-q · Cited by 1,365 · see citing works
Quantum observables and recollapsing dynamics
Marolf, D. (1995) · Classical and Quantum Gravity · doi:10.1088/0264-9381/12/5/011 · Cited by 153 · see citing works
Knot invariants as nondegenerate quantum geometries
Brügmann, B., Gambini, R., & Pullin, J. (1992) · Physical Review Letters · doi:10.1103/physrevlett.68.431 · Cited by 94 · see citing works
Flagged first, then by citations
Research works printed but not linked: 24
- DeWitt, 1967, Phys. Rev., 160, 1113, 10.1103/PhysRev.160.1113 · DOI · Paper page
- DeWitt, 1967, Phys. Rev., 162, 1195, 10.1103/PhysRev.162.1195 · DOI · Paper page
- DeWitt, 1967, Phys. Rev., 162, 1239, 10.1103/PhysRev.162.1239 · DOI · Paper page
- Ashtekar, 1986, Phys. Rev. Lett., 57, 2244, 10.1103/PhysRevLett.57.2244 · DOI · Paper page
- Ashtekar, 1987, Phys. Rev. D, 36, 1587, 10.1103/PhysRevD.36.1587 · DOI · Paper page
- Barbero, 1995, Phys. Rev. D, 51, 5507, 10.1103/PhysRevD.51.5507 · DOI · Paper page
- Ashtekar, 1995, J. Math. Phys., 36, 6456, 10.1063/1.531252 · DOI · Paper page
- Rovelli, 1994, Phys. Rev. Lett., 72, 446, 10.1103/PhysRevLett.72.446 · DOI · Paper page
- B. Brügmann and J. Pullin, Nucl. Phys. B 363 221 · DOI
- Brügmann, 1992, Nucl. Phys. B, 385, 587, 10.1016/0550-3213(92)90060-O · DOI · Paper page
- T. Thiemann, Quantum Spin Dynamics (QSD), Harvard University Preprint.
- Rovelli, 1995, Nucl. Phys. B, 456, 734
- Ashtekar, 1995, J. Math. Phys., 36, 2170, 10.1063/1.531037 · DOI · Paper page
- Ashtekar, 1995, J. Geom. Phys., 17, 191, 10.1016/0393-0440(95)00028-G · DOI · Paper page
- Hajíček, 1990, Phys. Rev D, 41, 1091, 10.1103/PhysRevD.41.1091 · DOI
- Hajíček, 1990, J. Math. Phys., 31, 1723, 10.1063/1.529015 · DOI
- Higuchi, 1991, Class. Quant. Grav., 8, 1983, 10.1088/0264-9381/8/11/010 · DOI
- Higuchi, 1991, Class. Quant. Grav., 8, 2023, 10.1088/0264-9381/8/11/012 · DOI
- C. Rovelli and L. Smolin. Spin networks and quantum gravity. preprint CGPG-95/4-1;
- J. Baez, Spin network states in gauge theory, Adv. Math. (in press)
- T. Thiemann, The matrix elements of the volume operator for canonical quantum gravity, Harvard University Preprint.
- T. Thiemann, The length operator in canonical quantum gravity, Harvard University Preprint.
- T. Thiemann, A regularization of canonical Einstein-Yang-Mills quantum theory, Harvard University Preprint.
- T. Thiemann, A Hamiltonian for canonical quantum gravity, Harvard University Preprint.
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Sorted by citations
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Quantum nature of the big bang: Improved dynamics
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“As in the full theory [9, 15], the term involving triads can be written as”
BackgroundMethodQuantum geometry and black hole entropy
Ashtekar, A., Baez, J., Corichi, A., & Krasnov, K. (1998) · Physical Review Letters · doi:10.1103/physrevlett.80.904 · Cited by 1,105 · see citing works
Mathematical structure of loop quantum cosmology
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The Spin-Foam Approach to Quantum Gravity
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Quantum geometry of isolated horizons and black hole entropy
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Self-citationAre the spectra of geometrical operators in loop quantum gravity really discrete?
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Oko w, A., & Lewandowski, J. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/16/302 · Cited by 58 · see citing works
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Bojowald, M., & Perez, A. (2010) · General Relativity and Gravitation · doi:10.1007/s10714-009-0892-9 · Cited by 35 · see citing works
A combinatorial approach to diffeomorphism invariant quantum gauge theories
Zapata, J. A. (1997) · Journal of Mathematical Physics · doi:10.1063/1.532159 · Cited by 32 · see citing works
Combinatorial Space from Loop Quantum Gravity
Zapata, J. A. (1998) · General Relativity and Gravitation · doi:10.1023/a:1026699012787 · Cited by 31 · see citing works
Simplifying the spectral analysis of the volume operator
Loll, R. (1997) · Nuclear Physics B · doi:10.1016/s0550-3213(97)00334-9 · Cited by 27 · see citing works
Guedes, T. L. M., Mena Marugán, G. A., Müller, M., & Vidotto, F. (2025) · Physical Review D · doi:10.1103/hjdk-kdhk · Cited by 5 · see citing works
- Authors
- Thiemann, T.
- Affiliations
- Harvard University (US)
- Corresponding author
- Thiemann, T.
- Venue
- Physics Letters B
- Publisher
- Elsevier BV
- Published
- 1996-07
- Volume
- 380
- Issue
- 3-4
- Pages
- 257-264
- ISSN
- 0370-2693 (ISSN-L)
- OpenAlex
- W1998335592
- Cited by
- 284
- References
- 3
- Citations by year
- 2026: 1 · 2025: 36 · 2024: 10 · 2023: 7 · 2022: 7 · 2021: 4 · 2020: 3 · 2019: 4 · 2018: 6 · 2017: 4 · 2016: 9 · 2015: 9 · 2014: 12 · 2013: 12 · 2012: 5
- FWCI
- 14.32
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physics Letters B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- Wheeler-DeWitt equation, Lorentzian quantum gravity, non-perturbative quantum gravity, Hamiltonian constraint operator, Wick rotation, Euclidean quantum gravity, Poincaré group generators
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