3 Quantum Cosmology and Gravitation
Our goal has been to obtain a consistent cosmological scenario starting from a a path to quantize gravity. This line of research got to be known as Quantum Cosmology. Let us remark that the usual interpretation of Quantum Mechanics assumes a partition of the world in a classic and a quantum domain. According to this interpretation, the collapse of the wave function occurs in the latter, due to classical measures. The classical domain is where the quantum potentialities turn into real facts. If we consider the universe as a whole a system, there is no place for a classic domain, and the orthodox interpretation is not useful. Hence, for Quantum Cosmology to make sense, we need a different interpretation of Quantum Mechanics. The aim is to give meaning to the solutions of the Schrodinger equation for Cosmology, the so-called Wheeler-De Witt equation). Among other things, this would allow us to test whether the initial singularity is avoided due to quantum effects, to get some information about the initial conditions of the universe, and to compare the theoretical spectrum of density perturbations with the precise meaures available today.
There are other technical problems to be solved.
First, the Wheeler-De Witt equation is a functional equation, defined on
the space of all the possible geometries, known as superspace.
There are no known exact solutions to this equation up to now. A possible
route is to decrease the infinite number of degrees of freedom of the problem,
to work with a reduced space, the minisuperspace. On the other hand, the
term that represents the reparametrization freedom, the lapse function,
is actually a lagrange multiplier of the hamiltonian. In the language of
Dirac, this means that the hamiltonian is weakly zero. Consequently, the
time variable ceases to exist in the quantization process, at least explicitly.
It might be still in the formalism, hidden in one of the dynamical variables,
or in related quantities.