Anisotropies in Kaluza-Klein quantum cosmology

By using the classically solvable minimal massless scalar minisuperspace model we study two examples of wave functions of the universe in the semiclassical limit. The first one consists of a Lorentzian component and a Euclidean component and admits a clear semiclassical interpretation as a superposition of universes. The second one consists of a Lorentzian (or Euclidean) component and another oscillatory component which corresponds to (neither Lorentzian nor Euclidean) complex classical solutions. This example has some resemblance to Hawking's minimal massive scalar minisuperspace model. We suggest a possible way of recovering the classical interpretation in such a case.

The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.

We consider a proposal to define the wave function of the Universe as a sum over spacetimes that eventually inflate. In the minisuperspace model, we explicitly show that a simple family of initial conditions, parametrized by a positive real number $a_0$, can be imposed to realise this prescription. The resulting wave function is found to be proportional to the Hartle-Hawking wave function and its dependence on $a_0$ is only through an overall phase factor. Motivated by this observation, we ask whether it is possible to analytically extend $a_0$ to an extended region $\bar{\mathcal{D}}$ in complex $a_0-$plane, while retaining the Hartle-Hawking form of the wave function. We use the condition for convergence of path integral and a recent theorem due to Kontsevich and Segal, further extended by Witten, to explicitly find $\bar{\mathcal{D}}$. Interestingly, a special point on the boundary of $\bar{\mathcal{D}}$ recovers the exact no-boundary wave function. Following that, we show that our prescription leads to a family of quantum states for the perturbations, which give rise to scale-invariant power spectra. If we demand, as an extra ingredient to our prescription, a matching condition at the "no-boundary point" in $\bar{\mathcal{D}}$, we converge on a unique quantum state for the perturbations.

We implement the no-boundary proposal for the wave function of the universe in an exactly solvable Bianchi IX minisuperspace model with two scale factors. We extend our earlier work (Phys. Rev. Lett. 121, 081302, 2018 / arXiv:1804.01102) to include the contribution from the $\mathbb{C}\text{P}^2 \setminus B^4$ topology. The resulting wave function yields normalizable probabilities and thus fits into a predictive framework for semiclassical quantum cosmology. We find that the amplitude is low for large anisotropies. In the isotropic limit the usual Hartle-Hawking wave function for the de Sitter minisuperspace model is recovered. Inhomogeneous perturbations in an extended minisuperspace are shown to be initially in their ground state. We also demonstrate that the precise mathematical implementation of the no-boundary proposal as a functional integral in minisuperspace depends on detailed aspects of the model, including the choice of gauge-fixing. This shows in particular that the choice of contour cannot be fundamental, adding weight to the recent proposal that the semiclassical no-boundary wave function should be defined solely in terms of a collection of saddle points. We adopt this approach in most of this paper. Finally we show that the semiclassical tunneling wave function of the universe is essentially equal to the no-boundary state in this particular minisuperspace model, at least in the subset of the classical domain where the former is known.

In quantum cosmology the DeWitt boundary condition is a proposal to set the wave function of the universe to vanish at the classical big-bang singularity. In this Letter, we show that in many gravitational theories including general relativity, the DeWitt wave function does not take a desired form once tensor perturbations around a homogeneous and isotropic closed universe are taken into account: anisotropies and inhomogeneities due to the perturbations are not suppressed near the classical singularity. We then show that Hořava-Lifshitz gravity provides a satisfactory DeWitt wave function. In particular, in the limit of z=3 anisotropic scaling, we find an exact analytic expression for the DeWitt wave function of the universe with scale-invariant perturbations. In general cases with relevant deformations, we show that the DeWitt wave function can be systematically expanded around the classical big-bang singularity with perturbations under control.

We calculate the four point correlation function for scalar perturbations in the canonical model of slow-roll inflation. We work in the leading slow-roll approximation where the calculation can be done in de Sitter space. Our calculation uses techniques drawn from the AdS/CFT correspondence to find the wave function at late times and then calculate the four point function from it. The answer we get agrees with an earlier result in the literature, obtained using different methods. Our analysis reveals a subtlety with regard to the Ward identities for conformal invariance, which arises in de Sitter space and has no analogue in AdS space. This subtlety arises because in de Sitter space the metric at late times is a genuine degree of freedom, and hence to calculate correlation functions from the wave function of the Universe at late times, one must fix gauge completely. The resulting correlators are then invariant under a conformal transformation accompanied by a compensating coordinate transformation which restores the gauge.

A Casimir piston for massless scalar fields obeying Dirichlet boundary conditions in high-dimensional spacetimes within the frame of Kaluza–Klein theory is analyzed. We derive and calculate the exact expression for the Casimir force on the piston. We also compute the Casimir force in the limit that one outer plate is moved to the extremely distant place to show that the reduced force is associated with the properties of additional spatial dimensions. The more dimensionality the spacetime has, the stronger the extra-dimension influence is. The Casimir force for the piston in the model including a third plate under the background with extra compactified dimensions always keeps attractive. Further we find that when the limit is taken the Casimir force between one plate and the piston will change to be the same form as the corresponding force for the standard system consisting of two parallel plates in the four-dimensional spacetimes if the ratio of the plate-piston distance and extra dimensions size is large enough.

In Kaluza-Klein models, we investigate soliton solutions of Einstein equation. We obtain the formulas for perihelion shift, deflection of light, time delay of radar echoes and PPN parameters. We find that the solitonic parameter k should be very big: |k|\geq 2.3\times10^4. We define a soliton solution which corresponds to a point-like mass source. In this case the soliton parameter k=2, which is clearly contrary to this restriction. Similar problem with the observations takes place for static spherically symmetric perfect fluid with the dust-like equation of state in all dimensions. The common for both of these models is the same equations of state in our three dimensions and in the extra dimensions. All dimensions are treated at equal footing. To be in agreement with observations, it is necessary to break the symmetry between the external/our and internal spaces. It takes place for black strings which are particular examples of solitons with k\to \infty. For such k, black strings are in concordance with the observations. Moreover, we show that they are the only solitons which are at the same level of agreement with the observations as in general relativity. Black strings can be treated as perfect fluid with dust-like equation of state p_0=0 in the external/our space and very specific equation of state p_1=-(1/2)\epsilon in the internal space. The latter equation is due to negative tension in the extra dimension. We also demonstrate that dimension 3 for the external space is a special one. Only in this case we get the latter equation of state. We show that the black string equations of state satisfy the necessary condition of the internal space stabilization. Therefore, black strings are good candidates for a viable model of astrophysical objects (e.g., Sun) if we can provide a satisfactory explanation of negative tension for particles constituting these objects.

A process of idea filtration in two distinct streams of physics i.e., 1) The dimensionality perspective of spacetime, and 2) The quantum perspective leads us to an understanding of what might be a true reality of all that we perceive. The conclusions arrived at in this paper are a bit perplexing in the sense that our perceived reality could be a manifestation of a combination of 4D + n (n > 0) flat space-time, universal wave function, and cognizance. The work is based on a review and analysis of the main concepts in quantum theory, relativistic physics, and cosmology. Key ideas and conclusions are filtered and logically connected to arrive at what might be a view of the true reality. A significant part of the paper is dedicated to the concept of the “observer” and the “ability of cognizance” that should accompany the “observer”. Though the “observer” is central to modern physics, it is not known what constitutes observation, and the term observer, often open to interpretations, does not have a standard definition and hence, is lacking in clarity. In our analysis, we have argued that the environment, in which the observer-observed system is embedded, emerges as an all-knowing, cognizant, and ideal observer that has the knowledge of the observer-observed system. At a philosophical level, we link to the fundamentals of physics, “consciousness” or “ability of cognizance” as an unavoidable and key element in not only carrying out the observation but perhaps, as believed by many, having a role in shaping the reality perceived. In the review and analysis of another stream of physics, that of General Theory of Relativity (GTR) and cosmology, we examine the question of reality from a cosmological and dimensionality perspective. Research on the 4D and 5D constructs of the universe indicates that the “reality” perceived in 4D spacetime as matter, distance, time, etc., is a manifestation of a higher dimensional 4D + n (n > 0) reality. Theoretical research on this front points towards a 4D-spacetime embedded in a 5D or higher dimensional flat space with matter and energy being a manifestation of the higher dimensions. The flow of logic in this paper leans towards a view of an ultimate true reality that is flat 4D + n (n > 0) space combined with cognizance, universal wave function, and the environment.

(abbreviated) We study quantized solutions of WdW equation describing a closed FRW universe with a $\Lambda $ term and a set of massless scalar fields. We show that when $\Lambda \ll 1$ in the natural units and the standard $in$-vacuum state is considered, either wavefunction of the universe, $\Psi$, or its derivative with respect to the scale factor, $a$, behave as random quasi-classical fields at sufficiently large values of $a$, when $1 \ll a \ll e^{{2\over 3\Lambda}}$ or $a \gg e^{{2\over 3\Lambda}}$, respectively. Statistical r.m.s value of the wavefunction is proportional to the Hartle-Hawking wavefunction for a closed universe with a $\Lambda $ term. Alternatively, the behaviour of our system at large values of $a$ can be described in terms of a density matrix corresponding to a mixed state, which is directly determined by statistical properties of $\Psi$. It gives a non-trivial probability distribution over field velocities. We suppose that a similar behaviour of $\Psi$ can be found in all models exhibiting copious production of excitations with respect to $out$-vacuum state associated with classical trajectories at large values of $a$. Thus, the third quantization procedure may provide a 'boundary condition' for classical solutions of WdW equation.

Why two is more attractive than one… or Bianchi class-A models and Reissner-Nordström black holes in quantum N=2 supergravity

If a cat, a cannonball, and an economics textbook are all dropped from the same height, they fall to the ground with exactly the same acceleration under the influence of gravity. This equality of gravitational accelerations of different things is one of the most accurately tested laws of physics. That law, however, tells us little about cats, cannonballs, or economics. This lecture expands on this theme to address the question of what features of our world are predicted by a fundamental ``theory of everything'' governing the regularities exhibited universally by all physical systems. This may consist of two parts: a dynamical law governing regularities in time (e.g superstring theory) and a law of cosmological initial condition governing mostly regularities in space (e.g. Hawking's no-boundary initial condition). The lecture concludes that: (1) ``A theory of everything'' is not a theory of everything in a quantum mechanical universe. (2) If the laws are short enough to be discoverable then they are probably too short to predict everything. (3) The regularities of human history, economics, biology, geology, etc are consistent with the fundamental laws of physics but do not follow from them. (Public lecture given at The Future of Theoretical Physics and Cosmology: Stephen Hawking 60th Birthday Symposium.)

4a-H-5 The Wave Function of the Universe by the New Euclidean Path-integral Method

We investigate the quantum evolution of the metric operators for Bianchi-Type I model universes in the Heisenberg picture in order to remove the need to consider the wave function of the universe and interpret its “spin” variables. The calculation is analogous to that of the Zitterbewegung of the Dirac electron. We consider the behavior of the metric near the classical singularity, and consider the curvature there. Although factor ordering questions preclude the presentation of an unambiguous result for the curvature invariants, it does seem that the classical t −4 divergence of the Kretschmann scalar is not removed by quantization.

The wave function of the Universe is calculated for an inflationary cosmological model with a variable gravitational "constant." The resulting probability distribution for the initial states of the Universe is peaked at the highest maximum of the effective potential, $V={V}_{max}$, with the initial value of the Planck mass ${{m}_{P}}^{(0)}\ensuremath{\sim}{V}_{max}^{\frac{1}{4}}$.