Quantum Cosmology in the Light of Quantum Mechanics
Abstract
:1. Introduction
2. Quantum Cosmology: A (Very) Brief Review
2.1. Classical Constraints
2.2. Canonical Quantisation
2.3. Minisuperspace
3. Classical Analogy: The Geometric Minisuperspace
4. Quantum Picture
4.1. Quantum Field Theory in the Spacetime
4.2. Quantum Field Theory in the Minisuperspace
5. Particles and Universes Propagating in Their Spaces
5.1. Semiclassical Universes: Classical Spacetime and Quantum Matter Fields
5.2. Semiclassical Particles: Geodesics and Uncertainties in the Position
6. Conclusions and Further Comments
Funding
Acknowledgments
Conflicts of Interest
References
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1  Time is created at the onset of the universe and thus the wave function of the universe cannot be a timedependent function, so we cannot apply an initial condition on the state of the universe. However, the universe may have a boundary where to impose the conditions that eventually would determine everything else in the whole history of the universe. 
2  These are essentially the relative states of Everett’s formulation of quantum mechanics [3]. However, Everett did not provide an explanation of why some states and no others are selected from the whole set of possible states. To explain it, Hartle needed to add, besides the boundary condition of the state of the universe and the equations of quantum mechanics, a new ingredient: the coarsegraining process that makes some states emerge from the decoherence process. These are the selected states of the Everett’s formulation. 
3  The outcomes of a classical experiment are exclusive, i.e., the cat is either dead or alive but not both. 
4  The existence of a semiclassical domain in the universe, and actually our own existence, can be seen as two possible outcomes of the cosmological experiment. As Hartle stated [5], we live in the middle of this particular experiment. 
5  
6  Note, however, that this space can still be infinite dimensional. 
7  It turns out that the quantum behaviour of a system is not a matter of its smallness, at least in principle. The analysis of the decoherence processes (see, for instance, Refs. [36,37]) shows that the quantumtoclassical transition essentially depends on the complexity of the system, i.e., on the number of constituents and the quantum coherence of their correlations. A macroscopic system typically contains a huge number of constituents so that the quantum correlations rapidly disappear. However, the application of the quantum theory to a model of the universe with a small number of degrees of freedom, e.g., a minisuperspace model of the universe, shows that the quantum effects might be important as well in the universe [38,39,40]. For instance, the quantum state of the universe could be given by a squeezed state, which is usually dubbed a quantum state without classical analogue [38]; or the composite quantum state of two otherwise semiclassical universes could be entangled [38,39,40]. Incidentally, it also reveals that the concept of classicality has different acceptations that must be taken with care in the context of quantum cosmology. As stated in Section 1, the application of the quantum theory to the universe challenges some of its fundamentals. 
8  Eventually, these inhomogeneous modes are interpreted as particles and gravitons propagating in the homogeneous and isotropic background spacetime. 
9  Spinorial and vector fields can be considered as well. 
10  For convenience, the scalar field has been rescaled according to $\phi \to \sqrt{2}\phi $. 
11  Unless otherwise indicated, we always consider cosmic time, i.e., $N=1$. 
12  Recall that the scalar field $\phi $ has been rescaled according to $\phi \to \sqrt{2}\phi $, see f.n. 10. 
13  
14  Let us bear in mind, however, that it is only an analogy. 
15  This way of obtaining the Klein–Gordon equation from the Hamiltonian constraint of a test particle that propagates in the spacetime has been wellknown for a long time. It can be seen, for instance, in Ref. [44]. However, it is not customarily used in quantum field theory. 
16  The exact meaning of “sufficiently localised” is specified in Section 5. 
17  By a future directed vector in the minisuperspace, we mean a vector positively oriented with respect to the scale factor component, which is the timelike variable of the minisuperspace. 
18  We do not call it second quantisation of the universe because in this formalism the field $\varphi $ can represent many universes. 
19  This can explicitly be seen in a very simplified cosmological model [38]. As we show below, it might have consequences in the quantum creation of universes. 
20  In the quantisation of a complex scalar field, it would be full of particle–antiparticle pairs. 
21  From now on, we omit the hats on top of the operators to ease the notation. 
22  Recall that the field $\phi $ was rescaled according to $\phi \to \sqrt{2}\phi $, see f.n. 10. 
23  The Euclidean gap also prevent matter and antimatter from collapse 
24  Classically, the changes, ${\tilde{g}}_{\mu \nu}={\Delta}^{1}{g}_{\mu \nu}$ and $d\lambda =m{\Delta}^{1}d\tau $, transform the action in Equation (21) [41] into
$$S=\int d\lambda \frac{1}{2}{\tilde{h}}_{ij}\frac{d{x}^{i}}{d\lambda}\frac{d{x}^{j}}{d\lambda}V\left(r\right),$$

25  For instance, in the case of a flat DeSitter spacetime, the modes with physical wavelengths ${\lambda}_{phys}$ much smaller than the cosmological horizon, ${L}_{h}=\frac{1}{{H}_{0}}$, can be interpreted in terms of particles with highly definite trajectories [41]. 
26  This will be done elsewhere. 
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RoblesPérez, S.J. Quantum Cosmology in the Light of Quantum Mechanics. Galaxies 2019, 7, 50. https://doi.org/10.3390/galaxies7020050
RoblesPérez SJ. Quantum Cosmology in the Light of Quantum Mechanics. Galaxies. 2019; 7(2):50. https://doi.org/10.3390/galaxies7020050
Chicago/Turabian StyleRoblesPérez, Salvador J. 2019. "Quantum Cosmology in the Light of Quantum Mechanics" Galaxies 7, no. 2: 50. https://doi.org/10.3390/galaxies7020050