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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.
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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.
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47 No basis given · 15 Own interpretation · 9 Not checked yet · 3 Own result
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“We conclude that - All (convergent) observables of the form 3.9 are symmetric on H phys with the inner product 3.23”
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- The authors state this as an assumption: The integrals defining the observables converge for the results of section III.
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“Equation 3.14 is also sufficient for us to determine the physical inner product.”
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“and lead to a familiar Hilbert space for the relativistic free particle (IV B).”
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“and, if [ A ( t ) , ∂ ∂t θ ( Q ( t ) -τ )] = 0, we have [ A ] Q = τ = [ A ] L Q = τ = [ A ] R Q = τ .”
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“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 .”
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“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.”
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“Because its integrand is a one-form, expression 2.1 is in fact invariant under the more general reparametrizations (2.2).”
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“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.”
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“Furthermore, many of these observables have a clear interpretation in the classical limit”
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“that it may be applied to a large class of minisuperspace models - which includes the LRS Bianchi IX and Kantowski-Sachs cosmologies -”
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“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 .”
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“For example, the object N ( t ) commutes with the constraint but does not correspond to a gauge-invariant quantity.”
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“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).”
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“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.'”
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“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.”
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“and to provide reasonable results for simple test cases.”
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“Note that 3.9 does not depend on the choice of lapse function n ( t ).”
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“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.”
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- The authors state this as an assumption: The integrals defining the observables converge for the results of section III.
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“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.”
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“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
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“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.”
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“Result 3: O τ of 5.20 is densely defined and converges strongly to zero as τ →∞ .”
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“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.”
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“Note that 4.17 agrees with 3.23 since dE = 2 | p 0 | dp 0 .”
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“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 .”
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- Also stated in: § C. A Complete Set of Operators · p. 30 · plain
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“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 .”
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“Results for other systems remain formal, but a forthcoming paper [16] will address the full Bianchi IX model.”
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“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.'”
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“A physical inner product was identified and this approach was shown to predict the collapse of spacetimes for the appropriate Bianchi cosmologies.”
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“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:”
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“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.”
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“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.
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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.”
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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”
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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
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