Timeless Quantum Mechanics and the Early Universe

Chataignier, Leonardo

doi:10.1007/978-3-030-94448-3

Cited by 4 publications

(25 citation statements)

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“…This formalism is an extension to general mechanical models of what was found in [54] for the particular case of a quasi-de Sitter universe with cosmological perturbations, which in turn was a continuation of the work done in [19]. Moreover, the results presented here were previously reported in the author's thesis [33], to which the reader is referred for further details. Throughout the article, we adopt units in which c = = 1 and we omit the summation sign over repeated indices.…”

Section: Introductionsupporting

confidence: 57%

“…6 The definition of the inner product for solutions to the WDW equation by a gauge-fixing procedure has been considered before [40,[58][59][60]] but its connection with WKB time and the weak-coupling expansion was only elucidated in [19,33,54] and the current article to the best of our knowledge. In [61], the question of equivalence of the Born-Oppenheimer approach and the gauge-fixing method was posed and analyzed for a simple toy model.…”

Section: Conditional Probabilitiesmentioning

confidence: 99%

“…1 If Ĥ involves a non-trivial metric for the q-configuration space that also depends parametrically on Q, then the determinant G in ( 23) may be replaced by G → Gh, where h is the determinant of the q-sector metric [19,33].…”

Section: Quantum Theorymentioning

confidence: 99%

“…2 In [33,54] it is explained that this expansion of the wave function is indeed equivalent to the traditional BO factorization Ψ = ψ0(Q)ψ(Q; q). Moreover, the justification of a single phase factor multiplying ψ(Q; q) in ( 24) instead of a superposition of the type exp (iM W0(Q)) + exp (−iM W0(Q)) may be a consequence of decoherence [56].…”

Section: Phase Transformation Of the Quantum Constraintmentioning

confidence: 99%

“…Several different approaches have been devised to extract a meaningful notion of dynamics from (1). Noteworthy are the 'Born-Oppenheimer' (BO) [6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the 'relational observables' approaches [20][21][22][23][24][25][26][27][28][29][30][31][32][33]. The former approach was inspired by the Born-Oppenheimer approximation scheme in molecular physics [34][35][36][37], which is well suited for models in which the dynamical fields can be separated according to characteristic energy scales m ≪ √ M , such that a formal perturbative expansion of (1) in powers of m 2 /M or, more schematically, 1/M can be performed.…”

Section: Introductionmentioning

confidence: 99%

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Beyond semiclassical time

Chataignier

2022

Preprint

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We show that the usual Born-Oppenheimer type of approximation used in quantum gravity, in which a semiclassical time parameter emerges from a weak-coupling expansion of the Wheeler-DeWitt constraint, leads to a unitary theory at least up to the next-to-leading order in minisuperspace models. As there are no unitarityviolating terms, this settles the issue of unitarity at this order, which has been much debated in the literature. Furthermore, we also show that the conserved inner product is gauge-fixed in the sense that the measure is related to the Faddeev-Popov determinant associated with the choice of semiclassical time as a reparametrization gauge. This implies that the Born-Oppenheimer approach to the problem of time is, in fact, an instance of a relational quantum theory, in which transition amplitudes can be related to conditional probabilities.

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Section: Introductionsupporting

confidence: 57%

Section: Conditional Probabilitiesmentioning

confidence: 99%

Section: Quantum Theorymentioning

confidence: 99%

Section: Phase Transformation Of the Quantum Constraintmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Beyond semiclassical time

Chataignier

2022

Preprint

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Beyond semiclassical time: dynamics in quantum cosmology

Chataignier

2023

J. Phys.: Conf. Ser.

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We review two approaches to the definition of the Hilbert space and evolution in mechanical theories with local time-reparametrization invariance, which are often used as toy models of quantum gravity. The first approach is based on the definition of invariant relational observables, whereas the second formalism consists of a perturbative construction of the Hilbert space and a weak-coupling expansion of the Hamiltonian constraint, which is frequently performed as part of the Born-Oppenheimer treatment in quantum cosmology. We discuss in which sense both approaches exhibit an inner product that is gauge-fixed via an operator version of the usual Faddeev-Popov procedure, and, in the second approach, how the unitarity of the effective Schrödinger evolution is established perturbatively. We note that a conditional probability interpretation of the physical states is possible, so that both formalisms are examples of quantum mechanics with a relational dynamics.

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Analyzing quantum gravity spillover in the semiclassical regime

Sahota,

Lochan

2023

Eur. Phys. J. C

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One of the standard approaches of incorporating the quantum gravity (QG) effects into the semiclassical analysis is to adopt the notion of a quantum-corrected spacetime arising from the QG model. This procedure assumes that the expectation value of the metric variable effectively captures the relevant QG subtleties in the semiclassical regime. We investigate the viability of this effective geometry approach for the case of dust dominated and a dark energy dominated universe. We write the phase space expressions for the geometric observables and construct corresponding Hermitian operators. A general class of operator ordering of these observables is considered, and their expectation values are calculated for a unitarily evolving wave packet. In the case of dust dominated universe, the expectation value of the Hubble parameter matches the “semiclassical” expression, the expression computed from the scale factor expectation value. In the case of the Ricci scalar, the relative difference between the semiclassical expression and quantum expectation is maximum at singularity and decays for late time. For a cosmological constant driven universe, the difference between the semiclassical expressions and the expectation value is most pronounced far away from the bounce point, hinting at the persistent quantum effect at the late time. The parameter related to the shape of the distribution appears as a control parameter in these models. In the limit of a sharply peaked distribution, the expectation value of the observables matches with their semiclassical counterpart, and the usage of effective geometry approach is justified.

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Timeless Quantum Mechanics and the Early Universe

Cited by 4 publications

References 0 publications

Beyond semiclassical time

Beyond semiclassical time

Beyond semiclassical time: dynamics in quantum cosmology

Analyzing quantum gravity spillover in the semiclassical regime

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