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Quantum Cosmology

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Book cover Non-minimal Higgs Inflation and Frame Dependence in Cosmology

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Abstract

In this chapter we will investigate the cosmos as a whole. First, we will try to approach the fundamental questions about the cosmos by resorting to various ideas and conceptional questions in order to understand the environment we live in. Then, we will investigate the tunnelling scenario in the framework of quantum cosmology. Finally, we will apply the results of the previous chapter to derive initial conditions for inflation.

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Notes

  1. 1.

    According to Karl Popper [51], a scientific theory has to make predictions that can be falsifiable.

  2. 2.

    This does not mean that the coupling constants in our patch of the cosmos must have the same values in different regions. Some authors even question if there is a fundamental meaning of physical laws at all or if they can also vary in different regions of the cosmos [61], but we will not follow these ideas.

  3. 3.

    There are approaches, such as string theory, which are even more ambitious and try to go beyond a quantization of gravity to find a “theory of everything”. See e.g. [49] for an overview and a summary of literature about string theory.

  4. 4.

    In cosmological minisuperspace models, the infinitely many degrees of freedom contained in the metric field \(g_{\mu \nu }(x)\) are usually reduced to a single degree of freedom, the homogeneous scale factor \(a(t)\).

  5. 5.

    Here, we use the terminology of DeWitt and Graham [28] who re-investigated Everett’s original work. Everett himself did not use the nomenclature “parallel worlds” but “relative states” [31].

  6. 6.

    In general, an arbitrary environmental degree of freedom cannot measure aspects of the system. Only those degrees of freedoms which can be modelled by local interaction between the environment and the system do participate in the process of measurement. For example a scattering process of photons with a molecule with centre of mass at a certain point will localize this object due to the electromagnetic interaction. The same setup with neutrinos instead of photons would not lead to such an efficient classical localization of that molecule, since the neutrinos will go through the molecule with almost no interaction.

  7. 7.

    The spectral theorem ensures that such an expansion is always possible for Hermitian operators.

  8. 8.

    Some authors, including Everett himself and Zurek, claim that the Born rule can be derived from the theory in terms of “relative frequencies” [31] or “envariance” [75] and is not an independent postulate. However, there are always some additional assumptions needed in the proofs of its derivation, so that it seems fair to say that the final status of this issue remains controversial.

  9. 9.

    The criterion of mutual orthogonality between two states \(|\! E_{i}\! \rangle \), \(|\! E_{j}\! \rangle \) is a measure of how well the environment can resolve the pointer positions.

  10. 10.

    They do not turn into an oscillating “cree” in the next moment, but we can be confident that the next morning our car still stands on the parking and the tree is still enrooted in the garden.

  11. 11.

    Although the process of continuous monitoring of the environment would take place just as well without our presence.

  12. 12.

    The continuous measurement of the system by the environment could perhaps also explain why we perceive time as continuous and uniform process. We do not mean the different clock rates due to relativistic effects, but rather the amount of continuous “information flow” carried away from the system into the environment and thereby increasing the entropy.

  13. 13.

    For example in a recollapsing universe or the “bounce” scenario of Loop Quantum Cosmology.

  14. 14.

    It could perhaps be interesting to relate the timescale of such a hypothetical “quantum re-coherence cycle” to a quantum version of the classical Poincaré recurrence theorem.

  15. 15.

    A discussion of the eternally ongoing reproduction of space-time can be found e.g. in [48].

  16. 16.

    This is a simplified presentation. In the context of string theory [60], the landscape is defined in a parameter-space of the dilaton fields, resulting from compactifications of extra dimensions.

  17. 17.

    In general, a factor ordering problem appears in the transition \(\mathcal{H}\rightarrow \hat{\mathcal{H}}\), leading to inequivalent quantum theories. We neglect this issue here.

  18. 18.

    See [45] for a systematical expansion of the Wheeler–DeWitt equation in powers of the Planck mass.

  19. 19.

    The formal analytic extension from \(N^0_\mathrm{{E}}=1\) to \(N^0_\mathrm{{E}}=-1\) should of course not be applied to \(a_\mathrm{{E}}(\tau )=\sin (N^0_\mathrm{{E}}\,H_\mathrm{{eff}}\tau )/H_\mathrm{{eff}}\) which would yield a negative \(a_\mathrm{{E}}(\tau )\) instead of (7.44) because in contrast to the sign-indefinite Lagrange multiplier \(N_\mathrm{{E}}\) the path integration over \(a_\mathrm{{E}}(\tau )\) in (7.35) semi-classically always runs in the vicinity of its positive geometrically meaningful value. For this reason, \(a_\mathrm{{E}}(\tau )\) never brings sign factors into the on-shell action even though it enters the action with odd powers.

  20. 20.

    Another interesting range of \(M_\mathrm{H}\) is below the instability threshold \(M_\mathrm{H}^\mathrm{inst}\) where \(\hat{V}_\mathrm{{RG}}\) becomes negative in the “true” high energy vacuum. As mentioned in the previous section, the tunnelling state rules out aperiodic solutions of effective equations with \(H^2<0\), which cannot contribute to the quantum ensemble of expanding Lorentzian signature models. Therefore, this range is semi-classically ruled out not only by the instability arguments, but also contradicts the tunnelling prescription.

  21. 21.

    This might seem to be equivalent to the tunnelling path integral of [64, 66], but the class of metrics integrated over is very different. We do not impose by hands \(a^{-}_\mathrm{{E}}=0\) as the boundary condition, but derive it from the saddle-point approximation for the integral over formally periodic configurations. The fact that periodicity gets violated by the boundary condition \(a^{-}_\mathrm{{E}}=0\) implies that the a priori postulated tunnelling statistical ensemble is exhausted at the dynamical level by the contribution of a pure vacuum state [7, 12, 13].

  22. 22.

    In the case of the vacuum no-boundary state when the vanishing thermal part of the effective action cannot present an obstacle to analytic continuation in the complex plane of \(N\), the situation stays the same. Indeed, any integration contour from \(-i\infty \) to \(+i\infty \) crosses the real axes an odd number of times, so that the contribution of only one such crossing survives, because any two (gauge-equivalent) saddle-points traversed in opposite directions give contributions cancelling one another.

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  1. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Christian Friedrich Steinwachs

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  1. Christian Friedrich Steinwachs
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Steinwachs, C.F. (2014). Quantum Cosmology. In: Non-minimal Higgs Inflation and Frame Dependence in Cosmology. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-01842-3_7

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