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Cosmology of Quantum Gravities

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Classical and Quantum Cosmology

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

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Abstract

A theory of everything explaining all the cosmological puzzles is an attractive utopia. It should explain how an expanding universe can arise from the chaotic foam of quantum geometry; whether the big-bang singularity of classical general relativity is replaced by some configuration without infinities; what is the fate of other gravitational singularities such as those met in black holes; what caused the early inflationary expansion, and what the late one; how the degrees of freedom of the Standard Model of particles emerged; what is the cosmological constant and why it acquired such a small value as observed today; and so on.

Magna et spatiosa res est sapientia; vacuo illi loco opus est; de divinis humanisque discendum est, de praeteritis de futuris, de caducis de aeternis, de tempore. De quo uno vide quam multa quaerantur: primum an per se sit aliquid; deinde an aliquid ante tempus sit sine tempore; cum mundo coeperit an etiam ante mundum quia fuerit aliquid, fuerit et tempus. […] Quamcumque partem rerum humanarum divinarumque conprenderis, ingenti copia quaerendorum ac discendorum fatigaberis. Haec tam multa, tam magna ut habere possint liberum hospitium, supervacua ex animo tollenda sunt. Non dabit se in has angustias virtus; laxum spatium res magna desiderat. Expellantur omnia, totum pectus illi vacet.

— Seneca, Ad Lucilium Epistularum Moralium, XI, 88, 33–35

Wisdom is a great and spacious thing; it needs plenty of free space. It teaches us about the divine and the human, the past and the future, the transient and the eternal, and about time. See how many issues arise just about the latter: First, whether time is anything by itself; then, if anything existed prior to time and without time; if time began with the world or, since something must have existed before the world, if also time existed before the world. […] Whatever part you embrace of human and divine sciences, you will have to make a great effort in studying a vast number of things. These are so many and so important that, in order for them to have free shelter in your soul, you will have to remove all superfluous things. Virtue does not confine itself to narrow quarters; great things wish large space. Let us expel everything else from our breast and make it empty to make room for virtue.

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Notes

  1. 1.

    In order to make sense of d S as an indicator of the physical geometry, one should ensure that the diffusion process or resolution-dependent probing be well defined. If the solution P of (11.2) was regarded as a probability density function, as in transport theory, then it should be positive semi-definite and normalized to 1. However, in many approaches to quantum gravity P becomes negative for certain values of ℓ and x, in which case the interpretation of (11.2) becomes problematic, even if the return probability \(\mathcal{P}\) is positive-definite. In fact, if there were no operationally sensible procedure by which the test particle could be found “somewhere” with a certain probability, then no clear geometric and physical meaning could be attached to (11.3). The negative probabilities problem can be fixed either by modifying (11.2) [41] or by adopting a quantum-field-theory viewpoint [189], where P is an amplitude rather than a probability.

  2. 2.

    For instance, in the diffusion interpretation on a sphere the particle can come back to x′ more easily than on a plane and, if we wait too long (T → +∞), the return probability tends to a constant. In the resolution interpretation, waiting too long means taking too low a resolution (1∕ℓ → 0), so that the sphere cannot be distinguished from a point. In both cases, d S → 0 instead of d S → D.

  3. 3.

    In this section, we reserve the symbol k for this cut-off.

  4. 4.

    An automorphism of a graph is an edge-preserving permutation of vertices. The set of automorphisms of a graph forms a group called automorphism group.

  5. 5.

    In [67, 68], phase D had not been discovered yet and it was indicated as “phase C.”

  6. 6.

    Restriction of the integration interval to the positive semi-axis α ∈ [0, +∞) leads to the definition of the Feynman Green’s function G F. While two-point functions such as G H define canonical inner products, Green’s functions such as G F are not solutions of the Hamiltonian constraint equation but they are true transition amplitudes (propagators), in the sense that they take into account the relative ordering in the “time” variable ϕ labelling the states (and thus defining a background-independent notion of “in” and “out”) [105]. In other words, true transition amplitudes propagate solutions of the constraint equation into other solutions.

  7. 7.

    Contrary to LQG spin networks, the combinatorial structure of GFT states is not embedded in an abstract space; no cylindrical equivalence conditions are imposed; states associated to different graphs have a different scalar product.

  8. 8.

    Starting from the fundamental level, this can be realized by grouping a different number \(\mathcal{N}\) of fundamental quanta in the effective building blocks.

  9. 9.

    The reader should consider this scale hierarchy cum grano salis, since the techniques and interpretations in GFC are still under intense study at the time of writing.

  10. 10.

    For WDW quantum cosmology with non-commutative mini-superspace coordinates or non-commutative phase-space variables, see [268] and [269], respectively. In the second case, where not only mini-superspace coordinates but also their conjugate momenta obey a non-commutative algebra, also black-hole backgrounds have been studied and the wave-function found to vanish at the central singularity [270–272].

  11. 11.

    For other types of non-locality which include infrared modifications to gravity, see [407–429].

  12. 12.

    Notice that \(\mathcal{F}_{2} = 0\) in [349, 362, 403] and \(\mathcal{F}_{0}\neq 0\neq \mathcal{F}_{2}\) in [377, 405].

  13. 13.

    The alternative name of “dimensional reduction” is often employed in the quantum-gravity literature, despite the fact that it is already in use in Kaluza–Klein and string scenarios, where spacetime has D > 4 topological dimensions and compactification to four observable dimensions is performed. For this reason, and to include also all scenarios where the dimension in the UV is not smaller than in the IR, we prefer the naming “flow.”

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    Gianluca Calcagni

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Calcagni, G. (2017). Cosmology of Quantum Gravities. In: Classical and Quantum Cosmology. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-41127-9_11

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