The $${\varvec{SL}}(2,\mathbb {R})$$ S L ( 2 , R ) totally constrained model: three quantization approaches

Gambini, Rodolfo; Olmedo, Javier

doi:10.1007/s10714-014-1768-1

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“…The Gambini-Porto-Pullin approach [117] is a competitor insofar as it is a timeless/semiclassical combination (their work with Torterolo [118] furthermore combines this with ideas about observables). Thus it is similarly motivated to Halliwell's approach and similarly applicable to Problem of Time and other issues in the foundations of Quantum Cosmology.…”

Section: More Minimalist Alternativesmentioning

confidence: 99%

Approaching the problem of time with a combined semiclassical-records-histories scheme

Anderson¹

2012

Class. Quantum Grav.

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I approach the Problem of Time and other foundations of Quantum Cosmology using a combined histories, timeless and semiclassical approach. This approach is along the lines pursued by Halliwell. It involves the timeless probabilities for dynamical trajectories entering regions of configuration space, which are computed within the semiclassical regime. Moreover, the objects that Halliwell uses in this approach commute with the Hamiltonian constraint, H. This approach has not hitherto been considered for models that also possess nontrivial linear constraints, Lin. This paper carries this out for some concrete relational particle models (RPM's). If there is also commutation with Lin -the Kuchař observables condition -the constructed objects are Dirac observables. Moreover, this paper shows that the problem of Kuchař observables is explicitly resolved for 1-and 2-d RPM's. Then as a first route to Halliwell's approach for nontrivial linear constraints that is also a construction of Dirac observables, I consider theories for which Kuchař observables are formally known, giving the relational triangle as an example. As a second route, I apply an indirect method that generalizes both group-averaging and Barbour's best matching. For conceptual clarity, my study involves the simpler case of Halliwell 2003 sharp-edged window function. I leave the elsewise-improved softened case of Halliwell 2009 for a subsequent Paper II. Finally, I provide comments on Halliwell's approach and how well it fares as regards the various facets of the Problem of Time and as an implementation of QM propositions. * eanderso@apc.univ-paris7.fr arXiv:1204.2868v2 [gr-qc] 31 Aug 20121 The spatial topology M is taken to be compact without boundary. hµν is a spatial 3-metric thereupon, with determinant h, covariant derivative Dµ, Ricci scalar Ric(h) and conjugate momentum π µν . Λ is the cosmological constant. Here, the GR configuration space metric M µνρσ = h µρ h νσ −h µν h ρσ is the undensitized inverse DeWitt supermetric with determinant M and inverse Nµνρσ itself the undensitized DeWitt supermetric hµρhνσ − hµν hρσ/2. In this paper, is a portmanteau of function dependence ( ) in finite theories such as particle mechanics or minisuperspace and functional dependence [ ] or mixed function-functional dependence ( ; ] in infinite theories such as field theories. I use S dS for the integral over space, including the finite case, for which this is taken to be a multiplicative 1.2 The inverted commas indicate that the Wheeler-DeWitt equation has, in addition to the Problem of Time, various technical problems, including A) regularization problems -not at all straightforward for an equation for a theory of an infinite number of degrees of freedom in the absense of background structure, while the mathematical meaningfulness of functional differential equations is open to question. N.B. this is not an issue in the specific examples in this paper as these are for a finite number of degrees of freedom. B) There are operator-ordering issues, which this paper's toy...

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Section: More Minimalist Alternativesmentioning

confidence: 99%

Approaching the problem of time with a combined semiclassical-records-histories scheme

Anderson¹

2012

Class. Quantum Grav.

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The $${\varvec{SL}}(2,\mathbb {R})$$ S L ( 2 , R ) totally constrained model: three quantization approaches

Cited by 1 publication

References 33 publications

Approaching the problem of time with a combined semiclassical-records-histories scheme

Approaching the problem of time with a combined semiclassical-records-histories scheme

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