Sistemi Hamiltoniani vincolati e teorie di gauge con applicazioni in relatività

The aim of this work is to study the mathematical structure of constrained Hamiltonian systems and their physical meaning. Such systems arise when there are constraint equations on the phase space. The presence of the constraints does not allow to have a one-to-one correspondence between a point on the phase space (q^\mu,p_\mu) and a point on velocities space (q^\mu,u^\mu). Since there are constraints, the entire phase space is no longer accessible, but only a sub-manifold of it, called constraint surface. It follows that the Hamiltonian function is not well-defined by the canonical Legendre transformation. Therefore, it is necessary a different reformulation of the Hamiltonian, which must include the presence of constraints. After imposing time invariance to all constraints, it is useful to define a distinction between first-class and second-class constraints. This depends on the property that their Poisson bracket with any other constraint vanishes or not on the constraint surface. The physical meaning of constraints is the presence of some gauge quantities that are not physical observables, since they do not affect the global system. Then, the degrees of freedom of the system are not just the number of canonical variables. So, following the method proposed by Dirac in 1964, one can find an invertible matrix, generated by the Poisson bracket of all second-class constraints, and use this matrix to define a new algebraic structure, the Dirac bracket, that allows to deal only with first-class constraints. One can also show that first-class constraints are generators of gauge transformations, which are transformations that do not change the global system. This implies that, after imposing the gauge, it is possible to obtain the physical degrees of freedom of the system. We also study the Hamiltonian formulation of Noether theorem in a constrained system. Starting from the invariance of the action under global transformations, it is found a conserved quantity Q, which vanishes identically when there is no gauge freedom. Nevertheless, it is proved that, in a constrained system, Q is the generator of canonical transformations. Finally, all these mathematical tools are involved in a physical application: the description of the spinless relativistic particle. Since the parameter tau, chosen to parametrize the world-line of a particle, is arbitrary, there must be some constraints in the system. It is found that the constraint is the mass shell equation so, using the Dirac bracket structure, it is possible to obtain an Hamiltonian that is function of physical observables such as velocities and momenta. The last step is the quantization of the system, which implies that all observables and constants of motion must be reformulated in a Hilbert space with the quantum-mechanical operator formalism. This leads to the local Klein-Gordon equation and the operator of boost along a chosen direction.