Summary:
In this essay we attempt to present a review of the Hamiltonian formalism, not
only in the case of full pure gravity, but also for Bianchi cosmologies. In the
Lagrangian formalism we are allowed to solve the Euler Lagrange equations in
terms of the generalized accelerations whenever the determinant of the Hess
matrix is non vanishing, in which case the system is called regular. However,
if the Hess determinant vanishes we cannot solve the Euler equations in terms
of all the accelerations but only for some of them. Such systems are called
‘Singular Lagrangian Systems’ or ‘Constrained Hamiltonian Systems’ or
‘Degenerate Systems’. The handling of such systems according to the
Dirac-Bergmann algorithm is discussed in the first chapter along with an
attempt to outline the quantization in accordance with the Dirac’s proposal.
The quantization of the general relativity theory is a hard and complicated
process, and even nowadays we haven’t found any complete way to treat quantum
gravity. In order to gain appropriate knowledge and intuition, to treat the
complete problem, it is reasonable to make some simplifications bypassing some
of the difficulties encountered. The most logical and elegant way is to impose
to our problem symmetries and thus get specific solutions to the full problem.
Making the assumption that the four-dimentional space can be splitted into
three-dimentional with fixed time homogeneous spaces, we end up whith the 9
Bianchi models.
Next we determine the time dependent diffeomorfisms leading to matrix
transformations of the tables γ_αβ that are automorfisms of the Lie algebra
defined by the structure constants of each Bianchi model. In such a way we are
able to simplify the form of the line element and therefore the Einstein’s
equations without losing generality. So after all this process, in all Bianchi
models we come up with four arbitrary functions of time in the general solution
of all Einstein's equations, and this always gives us the right to set the
Shift vector Ν_α to zero (possibly we will have to pay the cost of a more
complicated γ_αβ ).
What is next is to follow the canonical analysis of the gravity Bianchi
cosmologies. Here we can see that the Bianchi types are divided into Class A
(Cττρ=0) and Class B (Cττρ=/0) models. The Dirac-Bergmann algorithm leads to
a closed algebra only for Class A models. In Class B we come up with further
constraints with no correlation to the Einstein’s field equations , imposing
our inability to write a valid Hamiltonian for Class B models.
Finally in the third chapter we concentrate on a diagonal Bianchi type III
model. As we see, although this ia a class B model, we can write a valid
Hamiltonian provided the model becomes axis-symetric. Then we proceed to
quantization in accordance to the Dirac’s proposal and solve the Wheeler DeWitt
equation obtained.
