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Classical and Quantum Cosmology pp 407–465Cite as

Canonical Quantum Gravity

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Part of the book series: Graduate Texts in Physics ((GTP))

Abstract

Letting aside temporarily the possibility to unify all forces in a single consistent framework, a question one can ask oneself is whether quantum mechanics can resolve a particular singularity, such as the big bang or that inside a black hole.

Tirem-me daqui a metafísica!

     — Fernando Pessoa , Lisbon Revisited (1923)

Spare me metaphysics!

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Notes

  1. 1.

    Thanks to this, first-order quantization is fit for describing quantum gravitational systems at scales where the concept of metric and smooth spacetime can break down.

  2. 2.

    Spinors cannot be coupled directly to the metric but they are easily accommodated in the vielbein formulation [25, 26].

  3. 3.

    The scalar ϕ can be identified with the Barbero–Immirzi field, in which case \(\phi =\tilde{\beta }\) is a pseudo-scalar and V (ϕ) = 0. One can check that the Hamiltonian analysis of (9.24) is consistent with the one starting from the fundamental action (9.136) (Problems 9.1 and 9.2).

  4. 4.

    The context should be clear enough to avoid confusion with the Hubble parameter \(\mathcal{H}\) in conformal time.

  5. 5.

    A derivation of the York–Gibbons–Hawking boundary term can be found in [57]. Within first-order formalism, the boundary term is discussed in [58, 59].

  6. 6.

    This has nothing to do with the superspace of supersymmetry of Sect. 5.12

  7. 7.

    Very often in the literature, WKB wave-functions are defined as (9.98) (I ≡ 0), in which case “WKB state” and “semi-classical WKB state” are one and the same thing. Here we keep the distinction.

  8. 8.

    For the inquisitive reader, we notice that the profile (9.106) is periodic with period \(1/\mathcal{T}_{H} = 2\pi /H\). Its inverse is precisely the de Sitter temperature (5.132) giving the size of a field fluctuation. An explanation of this fact can be found in [115, 116].

  9. 9.

    In the high-j case, the Hamiltonian constraint is a difference equation of higher-than-second order. This may lead to an enlargement of the physical Hilbert space and, as a consequence, to the presence of solutions with incorrect large-volume limit [117]. Even if this were not the case, there is evidence (in 2 + 1 dimensions the proof is actually complete) that LQG has a well-defined continuum limit to quantum field theory only in the fundamental representation of the gauge group [118].

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    Gianluca Calcagni

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Calcagni, G. (2017). Canonical Quantum Gravity. In: Classical and Quantum Cosmology. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-41127-9_9

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