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Quantum Cosmology and Quantum Gravity

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Primordial cosmology was considered some decades ago a high
especulative field in physics. This situation has changed radically in the last years. The
density of light chemical elements, like helium, permits to test cosmological models to
approximately t ~ 10 -4 s. Also, the analysis of the anisotropies of the cosmic
microwave background radiation (CMBR) can give us access to much more remote
epochs because they could lead to informations about the physical processes
responsible for for the fluctuations which have originated the large sacale structures.
However, there are still weak points in the standard cosmological model that require
the study of more fundamental theories for the firs instants in the history of the
Univesrse. They are, basically: The existence of an initial singualrity; The initial
conditions of the Universe. To deal with these problems it is necessary to formulate a
model aplicable to the Universe in his first moments. In particular, it is necessary to
determine if it is possible to construct a coherente scenario from a program of
quntization of the gravitational field aplied to cosmology, which we call Quantum
Cosmology. However, the usual interpretation of qquantum mechanics assumes the
division of the world in a quantum domain and a classical domain. It is in this last one
where the colapse of the wave function takes place by means of classical
measurements, where the quantum potentialities become actual facts. But if the
quantum system is the whole Universe and everything in it, there is no place for a
classical domain and obviously the usual Copenhaguen interpretation cannot be
applied. Hence, to give a sense to quantum cosmology, it is imperative to foemulate
alternative interpretations of quantum mechanics. The aim is to give a sense to the
solutions of the quantum gravitational analog of the Schrödinger equation (the
Wheeler-DeWitt equation) and verify of the cosmological singularities are eliminated
by quantum processes, extract informations about the initial conditions of the
Universe and compare the spectrum of density perturbations obtained with
astrophysical data of the CMBR and others. There are other technical difficulties to
be solved. Firstly, the Wheeler-DeWitt equation is a functional equation defined in the
space of all possible spacelike geometries, named superspace, and for which is very
hard to find exact solutions. Sometimes all but a finite number of degrees of freedom
are frozen to simplify the calculations and obtain qualitative answers in the so-called
minisuperspace. Also, the term that represents gauge freedom of time
reparametrization, the lapse function, is a Lagrange multiplier of the Hamiltonian
imposing that it is, in Dirac?s language, weakly zero. Hence, in the process of
quantization, the time variable does not appear, at least explicitly. It is possible that it is
present in the formalism but hidden in the dynamical variables or in quantities related
to them.