The DOI matches its Crossref record; 17 of the 22 references it cites were found in a registry or an index, and 5 are not checked yet; none of the 9 sources its claims cite is checked for support yet.
Quantum observables and recollapsing dynamics
Open accessAbstract
Within a simple quantization scheme, observables for a large class of finite-dimensional time reparametrization invariant systems may be constructed by integration over the manifold of time labels. This procedure is shown to produce a complete set of densely defined operators on a physical Hilbert space, for which an inner product is identified, and to provide reasonable results for simple test cases. Furthermore, many of these observables have a clear interpretation in the classical limit and we use this to demonstrate that, for a class of minisuperspace models including LRS Bianchi IX and the Kantowski--Sachs model, this quantization agrees with classical physics in predicting that such spacetimes recollapse.
Checked 2026-10-06 · 5 references not checked yet
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
Marolf develops a Dirac-style quantization of time-reparametrization-invariant systems in which gauge-invariant observables are built by integrating one-forms over the time-label manifold, yielding densely defined operators on a physical Hilbert space with an identified inner product. For separable semi-bound minisuperspace models, including LRS Bianchi IX and Kantowski-Sachs, the construction predicts recollapse, in agreement with classical behavior.
Key findings
- Observables of the form \([A]_{Q=\tau}\), defined by time integration with a symmetric ordering, commute with the Hamiltonian constraint and are symmetric on the physical Hilbert space with the inner product 3.23.
- For the relativistic free particle, the paper verifies that \([\operatorname{sign}(P_0)]_{X_0=\tau}=1\), and that requiring Hermiticity of \([P_0\operatorname{sign}(P_0)]_{X_0=\tau}\), \([X_i]_{X_0=\tau}\), \([P_j]_{X_0=\tau}\), and \([\Theta]_{X_0=\tau}\) fixes the inner product up to overall normalization.
- For separable semi-bound models, Result 1 shows \([A]_{Q_0=\tau}\) is densely defined when \([\tilde A,H_1]\) is densely defined; Result 2 shows \([A]_{Q_0=\tau}\to 0\) strongly as \(\tau\to\infty\); Result 3 shows the operator \(O_\tau\) of 5.20 is densely defined and converges strongly to zero; Result 5 shows a complete set of bounded observables converges strongly to zero.
- The paper interprets this strong convergence to zero as the quantum analogue of classical recollapse for LRS Bianchi IX and Kantowski-Sachs, with \(q_0=\ln(\det g)\).
Methods
- Dirac constrained quantization is used with an auxiliary Hilbert space \(L^2(Q)\), an algebra of \(Q(t)\) and \(P(t)\), a lapse \(N(t)\), and physical states satisfying \(H|\psi\rangle=0\); observables are defined by integrals over time labels, e.g. 3.15 and 3.18–3.20.
- The formalism is tested on already deparametrized systems and the relativistic free particle, then applied to separable semi-bound systems with \(h=h_0-h_1=0\), \(h_0=p_0^2+V_0(q_0)\), \(V_0(q_0)\to\infty\) as \(q_0\to\infty\), and an exponential bound \(V_0(q_0)<W e^{-\alpha q_0}\) for large negative \(q_0\).
- The analysis proves commutation with the constraint, symmetry with respect to the physical inner product, convergence bounds for the defining integrals, and strong convergence of observables to zero as \(\tau\to\infty\).
Limitations
- The paper states that rigorous convergence and spectral results are obtained only for already deparametrized systems, the free relativistic particle, and separable semi-bound models; results for other systems remain formal, and full Bianchi IX is deferred to a forthcoming paper.
- The paper states technical difficulties: the spectrum of \(H\) may not contain zero, the defining integrals may diverge on physical states, discrete spectra or confining potentials can cause divergence, and issues remain around delta-functions in 3.14, self-adjointness versus symmetry, and uniqueness of the physical inner product.
- Evident from the text: the treatment is restricted to finite-dimensional models, does not address full quantum gravity, and the recollapse conclusion is based on strong convergence on a dense set rather than an exact unitary evolution in \(\tau\)-time; some states are excluded in a manner analogous to infinite-energy states in nonrelativistic quantum mechanics.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
47 No basis given · 15 Own interpretation · 9 Not checked yet · 3 Own result
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We conclude that - All (convergent) observables of the form 3.9 are symmetric on H phys with the inner product 3.23”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: The integrals defining the observables converge for the results of section III.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Equation 3.14 is also sufficient for us to determine the physical inner product.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“and lead to a familiar Hilbert space for the relativistic free particle (IV B).”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“and, if [ A ( t ) , ∂ ∂t θ ( Q ( t ) -τ )] = 0, we have [ A ] Q = τ = [ A ] L Q = τ = [ A ] R Q = τ .”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In a 'proper parametrization' for which N ( t ) > 0 and proper time always flows forward, this is just the number of times for which q ( t ) = τ .”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This object defines a well-behaved function on any part of the classical space of solutions for which the fields φ k vanish simultaneously only at a finite number of spacetime points and for which the metric is smooth whenever φ k = 0.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Because its integrand is a one-form, expression 2.1 is in fact invariant under the more general reparametrizations (2.2).”
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Within a simple quantization scheme, observables for a large class of finite dimensional time reparametrization invariant systems may be constructed by integration over the manifold of time labels.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Furthermore, many of these observables have a clear interpretation in the classical limit”
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“that it may be applied to a large class of minisuperspace models - which includes the LRS Bianchi IX and Kantowski-Sachs cosmologies -”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“This approach has the advantage that it is not necessary to identify an 'internal time function' [1]”
What the sources say
- Not checked yetcites [1] Kuchar K, 1992
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“it has the reassuring property (see IV B) that [sign( P 0 )] X 0 = τ is the identity operator for the free relativistic particle”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“However, we have seen that reparametrization invariant operators do commute with the Hamiltonian.”
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These test cases show that the methods of III give the usual results for 'already deparametrized systems' (IV A)”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“From 3.22 and 3.24, we see that any symmetric ordering of the factors ∂ ∂t Q ( t ), Q ( t ), and A ( t ) will produce a Hermitian gauge invariant that we may take to be a 'quantization' of [ a ] q = τ .”
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“and that certain properties of the resulting operators can be studied even though the models may not be exactly solvable.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This procedure is shown to produce a complete set of densely defined operators on a physical Hilbert space for which an inner product is identified and to provide reasonable results for simple test cases.”
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“In the examples we will see that our approach in fact provides an 'overcomplete set' of quantum observables. By this we mean a set of observables whose classical counterparts are overcomplete (see. for example [13]) on the (constrained) phase space.”
What the sources say
- Not checked yetcites [13] Higuchi A, 1991, Class. Quantum Grav. source
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“we will see that quantum analogues of these observables exist as densely defined operators on a space of physical states for interesting models”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We will also see that the inner product 3.23 is unique, provided that a simple set of observables are required to be symmetric.”
plainconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Thus, the case of A = X i does not correspond to a Newton-Wigner operator (which would leave the positive and negative frequency subspaces invariant).”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A similar calculation verifies that [sign( P 0 )] X 0 = τ = 1 1 as claimed.”
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It now follows that Hermiticity of a few interesting operators guarantees that the inner product 3.23 is unique.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“On the other hand, it may be checked that [ θ ( P 0 )signP 0 X i ] X 0 = τ is just the Newton-Wigner operator X i NW ( τ ) associated with the coordinate X i .”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The ˜ [ B ] Q 0 = τ form a complete set of bounded quantum observables.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: For each E0 and E1, the label k takes only a finite set of values.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Finally, we verify that we have constructed an overcomplete set of observables as densely defined operators on H phys .”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We have seen that the matrix elements are of the form 〈 E,k | [ A ] Q 0 = τ | E ′ , k ′ 〉 = δ ( E -E ′ ) f ( E,E ′ k, k ′ ) for eigenstates | E,k 〉 of H with eigenvalue E and some function f”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If we also ask that [ X i ] X 0 = τ and [ P j ] X 0 = τ ( i, j ∈ { 1 , 2 , 3 } ) be symmetric, then the inner product is fixed up to an overall normalization on the positive frequency subspace and a similar normalization on the negative frequency subspace.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If we require [ P 0 sign( P 0 )] X 0 = τ to be Hermitian on the physical space (spanned by | ǫ | p | , p ; t ′ 〉 ), it follows that the positive and negative frequency subspaces are orthogonal.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We will see that the approach of III gives the usual quantization of nonrelativistic particles when applied to systems of this form.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The most general rigorous results were obtained for already deparametrized systems (IV A), the free relativistic particle (IV B), and the separable semibound models of V.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We have seen that the construction of gauge invariants through integration over manifolds can be combined with Hamiltonian methods for constrained systems to build a complete set of gauge invariant operators that are densely defined in a physical Hilbert space for a number of time reparametrization invariant models.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Result 5: The ˜ [ B ] Q 0 = τ of Result 5 converge strongly to zero as τ →∞ .”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: For each E0 and E1, the label k takes only a finite set of values.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Result 2: For A as in Result 1, [ A ] Q 0 = τ → 0 as τ →∞ in the sense of convergence on a dense set of states; that is, [ A ] Q 0 = τ converges strongly to the zero operator.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Within our scheme, the canonical idea that an 'observable' is an object that commutes with the constraint (the Hamiltonian, H ) does not capture the entire notion of 'gauge invariant'”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We interpret this as a demonstration that our quantization of these models describes recollapsing cosmologies.”
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In particular, such observables can be used to show that, in correspondence with the classical result, our quantization of appropriate minisuperspace models describes spacetimes whose homogeneous slices expand to some maximal size and then recollapse.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“If the density ω is distributional, the resulting gauge invariant may be effectively local on M .”
What the sources say
- Not checked yetcites [2] DeWitt B, 1962
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“we use this to demonstrate that, for a class of minisuperspace models including LRS Bianchi IX and the KantowskiSachs model this quantization agrees with classical physics in predicting that such spacetimes recollapse.”
- Also stated in: § I. INTRODUCTION · p. 2 · plain
“for the models just mentioned it predicts that the corresponding spacetimes recollapse, in accord with the classical result.”
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § I. INTRODUCTION · p. 2 · plain
In the paper
“procedures based on a Klein-Gordon type inner product instead describe the universe as expanding 'forever' [4].”
What the sources say
- Not checked yetcites [4] Wald R, 1993, Phys. Rev. source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“Given any density ω on an n-manifold M , ∫ M ω is such an observable.”
What the sources say
- Not checked yetcites [2] DeWitt B, 1962
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Note that it is highly nontrivial for a quantization scheme to satisfy this last condition”
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: unclear — It is unclear whether it could be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“if zero lies in the point spectrum of H and a solution to the constraint is actually normalizable in the auxiliary space, the action of [ A ] Q 0 = τ on | E,q 0 〉 contains a divergent factor.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From this and the above conditions, it follows that H has a purely continuous spectrum with σ c ( H ) = ( -∞ , ∞ ).”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Since the spectrum of H includes zero, the set of physical states that are delta-function normalizable in the auxiliary inner product and satisfy H | ψ 〉 = 0 is not empty.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“They are of the form | E 0 = E 1 , E 1 , k ; t ′ 〉 where E 1 ranges over the spectrum of H 1 and k ranges over its full set of values.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“The LRS Bianchi IX and Kantowski-Sachs minisuperspaces fall into the class of models that we refer to as 'separable and semi-bound.'”
- Also stated in: § A. The System and Observables · p. 20 · hedged
“From [15], we see that when described by the appropriate variables and with a rescaled Hamiltonian, the LRS Bianchi IX and Kantowski-Sachs models may be placed in this form with q 0 = ln(det g ) where g is the 3-metric of a homogeneous slice.”
What the sources say
- Not checked yetcites [15] Ashtekar A, 1991
plainconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § A. The System and Observables · p. 20 · hedged
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For this reason, we feel justified in saying that when Q 0 has the interpretation of a scale factor for a cosmology, our quantization predicts that such spacetimes recollapse.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It now follows that quantization of 'separable semi-bound models' captures the classical notion of 'recollapse' expressed through 2.4 for a ( t ) independent of q 0 ( t ) and p 0 ( t ).”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“There is no exact unitary evolution in τ -time for semi-bound separable systems, just as there is no corresponding exact Hamiltonian evolution at the classical level.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“However, discrete spectra are associated with confining potentials as in [19,20]”
What the sources say
- Not checked yetcites [19] Haag R, 1992
- Not checked yetcites [20] Glimm J, 1987
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These conditions imply that, when viewed as an operator on the Hilbert space L 2 ( R ) in a ' q 0 - coordinate representation,' the spectrum of H 0 is purely continuous.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- It may state two separate claims as one. 16
- Probably not a substantial claim. 7
- The quote only partly states this claim. 2
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For example, an argument similar to that of 3.13 and 3.14 shows that any object of the form ∫ ∞ -∞ dtA ( t ), where A ( t ) is an algebraic combination of Q i ( t ) and P j ( t ), commutes with the Hamiltonian in a representation for which n ( t ) = 1. However, for more general choices of lapse, it does not commute with H .”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For example, the object N ( t ) commutes with the constraint but does not correspond to a gauge-invariant quantity.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The advantage of 2.1 is that it is invariant not just under reparametrizations ( t → T ( t ) with dT/dt > 0), but also under pull backs of the manifold of time labels through maps of degree 1 ( t → T ( t ) for any smooth T ( t ) of degree 1).”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Also, note that the commutator of an operator O with H is not in general gauge invariant so that this definition is itself 'gauge dependent.'”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The delta-function in E 1 -E 2 guarantees that our operator preserves the spectral subspaces of H , and therefore that - [Ω , H ] = 0 when 3.9 converges weakly.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: The spectrum of H is purely continuous, at least near the zero eigenvalue, for the applications considered.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“and to provide reasonable results for simple test cases.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that 3.9 does not depend on the choice of lapse function n ( t ).”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“When the quantum version of the integral 2.1 converges, we will define the [ A ] Q = τ through integration and show in III B that they are observables in the canonical sense; that is, that they commute with the Hamiltonian.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
- The authors state this as an assumption: The integrals defining the observables converge for the results of section III.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Of course, for some solutions s the integral in 2.1 may not converge, such as when q ( t ) takes the value τ an infinite number of times along s , and on such solutions [ a ] (2) q = τ is undefined.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“We introduce a space of physical states as in [3] and find a physical inner product in III C with respect to which our operators are symmetric.”
What the sources say
- Not checked yetcites [3] Dirac P A M, 1964
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In particular, [sign ( 1 N dq dt ) ] q = τ is the number of times for which q ( t ) = τ and N ( t ) > 0 minus the number of times for which q ( t ) = τ and N ( t ) < 0.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Result 3: O τ of 5.20 is densely defined and converges strongly to zero as τ →∞ .”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that the factor of ǫ ˜ p 0 + p 0 in 4.14 minimizes the amount by which this operator mixes positive and negative frequency states, though some mixing still occurs.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that 4.17 agrees with 3.23 since dE = 2 | p 0 | dp 0 .”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“given by 5.23 defines a bounded operator with domain H phys .”
- Also stated in: § C. A Complete Set of Operators · p. 30 · plain
“Furthermore, if B is self-adjoint, then so is ˜ [ B ] Q 0 = τ .”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: For each E0 and E1, the label k takes only a finite set of values.
- Also stated in: § C. A Complete Set of Operators · p. 30 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Then, under this identification, if | ψ 〉 ∈ H 1 lies in the domain of [ ˜ A, H 1 ], it also lies in the domain of [ A ] Q 0 = τ . In particular, if [ ˜ A, H 1 ] is densely defined, so is [ A ] Q 0 = τ .”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Results for other systems remain formal, but a forthcoming paper [16] will address the full Bianchi IX model.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It therefore follows from III A that all operators of the form [ A ] B = τ are symmetric (for Hermitian A ( t ) and B ( t )) and we need not have chosen x 0 as the 'intrinsic clock.'”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A physical inner product was identified and this approach was shown to predict the collapse of spacetimes for the appropriate Bianchi cosmologies.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Additionally, if we require [Θ] X 0 = τ to be Hermitian, where Θ is the time reversal operator (Θ | p ; t 〉 = | -p ; t 〉 ), these normalizations must be identical and the inner product is uniquely determined (up to an overal scale factor) to be:”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one. The quote only partly states this claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“if the dynamics is complete, such systems will generally have solutions for which q 0 ( t ) = τ at an infinite number of times in which case [ a ] q 0 = τ may diverge as well.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Note that convergence of these integrals guarantees that the matrix element in 5.5 (and correspondingly for [ A ] R Q 0 = τ vanishes in the large τ limit, verifying the recollapsing behavior of 2.4 in a weak sense.”
plaindirectn = 2- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The authors state this as an assumption.
The Hamiltonian is classically constrained to vanish (h=0) for the time reparametrization invariant systems considered.
“Because time reparametrization is to be a gauge symmetry of this system, the Hamiltonian is classically constrained to vanish: h = 0.”
The authors state this as an assumption.
The integrals defining the observables converge for the results of section III.
“The results of this section assume convergence of the integrals, which will be separately discussed for each of the examples of IV and V.”
The authors state this as an assumption.
The spectrum of H is purely continuous, at least near the zero eigenvalue, for the applications considered.
“Here, however, we will only be interested in applying these techniques to the case where the spectrum of H is purely continuous (at least near the zero eigenvalue).”
The authors state this as an assumption.
The one-form ωdt satisfies equation 3.10.
“Since ωdt is a one-form, we assume that it satisfies the equation”
The authors state this as an assumption.
The lapse function n(t) satisfies equation 3.7.
“From now on, we will assume that n ( t ) satisfies”
The authors state this as an assumption.
The parameter time evolution satisfies equation 3.6.
“In irreducible representations, N ( t ) is proportional to the identity: N ( t ) = n ( t ) 1 1 and we assume that 3.6 holds.”
The authors state this as an assumption.
For each E0 and E1, the label k takes only a finite set of values.
“We will also assume that, for each E 0 , E 1 , the label k takes only a finite set of values. This is the case for both LRS Bianchi IX and the Kantowski-Sachs Model.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“We need it in its manifestly real formulation as advocated, for the Lorentzian signature, first in [14] because [3] only applies when the gauge group is compact (in this case SU (2)).”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [14]
In the citing paper
“The gravitational sector can be described [14] in terms of 'real-valued Ashtekar variables', that is, by a canonical pair ( A i a , E a i /κ ) where a, b, c etc. are tensorial indices for Σ, i, j, k etc. are su (2) indices, A i a is an SU (2) connection and E a i is its conjugate momentum, an ad SU (2) transforming vector density of weight one.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [14]
In the citing paper
“the requirement that H has only a continuous spectrum near zero means (see [1,22]) that the almost local formalism can only be applied to homogeneous cosmological models by using a constraint H and lapse N which are nontrivially related to the ADM constraint and lapse.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“it provides [1,22] a positive-definite inner product on a physical Hilbert space H phys as well as a complete set of well-defined symmetric quantum observables on this space (perennials in the language of [18]) that, in a certain sense, capture the notion of 'the value of quantity A at the point in time at which quantity Q takes a specified value τ '.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“the almost local approach was shown in [1] to provide a complete quantization of Hamiltonian vacuum Bianchi models (see, e.g.[19][20][21]) and of similar finite-dimensional time reparametrization invariant systems.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“For such a system, it has already been shown in [1] that the external time formalism can be derived exactly.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“the approach of [1] (which we will refer to as the 'almost local formalism') contains no fundamental notion of time or evolution.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“Had the clock instead measured metric-defined proper time (assuming that it is coupled to the description of the gravitational field used in [1] or [22]), our system would be described (see the appendix) by a constraint of the form 0 = e 3α H 0 + H 1 (5.2) instead of by (3.2).”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“When H 1 = g ij P ⊥ i P ⊥ j with g ij a constant negative definite D Marolf metric, direct calculation as in section 4.2 of [1] shows that (4.10) holds exactly for A = Q i ⊥ as well.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“The associated physical operators describe a certain sort of average over 'the values of the quantity A(t) at the times for which B(t) = τ ', even when the corresponding classical quantity b(t) may take on the given value τ more than once along a trajectory.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“While it does not follow from the above argument, the results of section 4.2 of [1] show that when V (Q) = 0 and H 1 = ij g ij P ⊥ i P ⊥ h (for any negative definite constant metric g ij ) the (Q i ⊥ ) + τ and the (P ⊥ j ) + τ are just the usual positive frequency Newton-Wigner operators and so satisfy the canonical commutation relations exactly.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“Thus, the adiabatic approximation should hold when Ṗ / α P .Using h Ȧ ∝ [A, NH ] and the fact that H is typically of a form (see [1,22]) such that [α, H ] = hP α , this condition is V (Q) P α e 3α P (5.3) which we may expect to hold when e 3α is large.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“In addition, the complete formalism of [1] has allowed a more extensive treatment than was previously possible-we were able to address the issue of unitary evolution in the physical inner product and to derive the complete map from the physical Hilbert space to the corresponding Hilbert space of external time quantum mechanics ‡.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
In the citing paper
“From section 4.2 of [1], it follows that any corrections to (4.12) vanish when V (Q) = 0.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [1]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ A. The System and Observables · p. 19
“V 0 ( q 0 ) < We αq 0 for some positive W and α . This last condition is important only for large negative q 0 .”
§ A. The System and Observables · p. 20
“V ( q 0 ) < We -αq 0 for some W,α > 0”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ B. Rigorous Convergence and Recollapse Results · p. 25
“in the E ∗ → ∞ limit”
§ B. Rigorous Convergence and Recollapse Results · p. 25
“It follows that I ( E,E ∗ , τ ) → 0 as E ∗ → 0”
Citations
Cites
Quantum Theory of Gravity. I. The Canonical Theory
DeWitt, B. S. (1967) · Physical Review · doi:10.1103/physrev.160.1113 · Cited by 3,385
Basdevant, J.-L. (2007) · Springer eBooks · doi:10.1007/978-0-387-37748-3 · Cited by 3,105
Haag, R. (1996) · doi:10.1007/978-3-642-61458-3 · Cited by 2,443
Quantum Physics: A Functional Integral Point of View
Gllmm, J., Jaffe, A. M., & Wilczek, F. A. (1982) · Physics Today · doi:10.1063/1.2914804 · Cited by 1,062
Lectures on Non-Perturbative Canonical Gravity
Ashtekar, A. (1991) · Advanced Series in Astrophysics and Cosmology · doi:10.1142/1321 · Cited by 872
TIME AND INTERPRETATIONS OF QUANTUM GRAVITY
Kuchař, K. V. (2011) · International Journal of Modern Physics D · doi:10.1142/s0218271811019347 · Cited by 725
What is observable in classical and quantum gravity?
Rovelli, C. (1991) · Classical and Quantum Gravity · doi:10.1088/0264-9381/8/2/011 · Cited by 449
Time in quantum gravity: An hypothesis
Rovelli, C. (1991) · Physical Review D · doi:10.1103/physrevd.43.442 · Cited by 445
Quantum mechanics without time: A model
Rovelli, C. (1990) · Physical Review D · doi:10.1103/physrevd.42.2638 · Cited by 316
Relativistic quantum mechanics of supersymmetric particles
Marc Henneaux, A., & Teitelboim, C. (1982) · Annals of Physics · doi:10.1016/0003-4916(82)90216-0 · Cited by 189
Cited by
Sorted by citations
Loop quantum cosmology: a status report
Ashtekar, A., & Singh, P. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/21/213001 · Cited by 1,205
Quantum nature of the big bang: Improved dynamics
Ashtekar, A., Pawlowski, T., & Singh, P. (2006) · Physical Review D · doi:10.1103/physrevd.74.084003 · Cited by 1,147
(1998) · doi:10.12942/lrr-1998-1 · Cited by 917
Quantum nature of the big bang: An analytical and numerical investigation
Ashtekar, A., Pawlowski, T., & Singh, P. (2006) · Physical Review D · doi:10.1103/physrevd.73.124038 · Cited by 609
Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J., & Thiemann, T. (1995) · Journal of Mathematical Physics · doi:10.1063/1.531252 · Cited by 528
Self-citation
Metadata
- Authors
- Marolf, D.
- Affiliations
- University of California, Santa Barbara (US)
- Corresponding author
- Marolf, Donald
- ORCID
- Marolf, Donald
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1995-05-01
- Volume
- 12
- Issue
- 5
- Pages
- 1199-1220
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2021228295
- Semantic Scholar
- 9e2ab475cf168eb596c5f9fbdfbcb8c848307281
- Cited by
- 153
- References
- 22
- Citations by year
- 2026: 1 · 2025: 5 · 2024: 2 · 2023: 4 · 2022: 3 · 2021: 5 · 2020: 6 · 2019: 3 · 2018: 1 · 2017: 2 · 2016: 3 · 2015: 5 · 2014: 3 · 2013: 3 · 2012: 8
- FWCI
- 7.24
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Keywords
- quantum observables, time reparametrization invariance, minisuperspace model, Kantowski-Sachs model, recollapse, quantum cosmology, classical limit, finite-dimensional systems
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-06, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-06 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
