Classical Covariance

Classical mechanics, from point particles through rigid objects and continuum mechanics is reviewed based on the notions of tensors, transformations, and the metric, as developed in the first two chapters. The geodesic equation on flat and curved spaces is introduced and solved in a classical setting. Motion in a potential, particularly a gravitational potential, is discussed. Galilean covariance and transformations are introduced. Time as a fourth dimension is shown to arise even in a classical setting, even if not as rigorous as it would be in relativity theory.

Galilean covariance and the spin-orbit interaction

We have used the Pauli-Schr\"{o}dinger equation in its covariant form, that is, written in the light-cone of a five-dimensional De Sitter space-time. Following standard procedures, the analogue of the Dirac equation is derived, standing for a galilean spin 1/2 particle in the presence of a external field. Some results are important to be mention, such as the expected g-factor, but in a galilean (not Lorentz) context. In addition, considering interaction, the Pauli-Hartree-Fock equation is obtained following in parallel to the ideas used to construct the Dirac-Hartree-Fock equation.

Galilean covariance, Casimir effect and Stefan–Boltzmann law at finite temperature

The Galilean covariance, formulated in 5-dimensions space, describes the nonrelativistic physics in a way similar to a Lorentz covariant quantum field theory being considered for relativistic physics. Using a nonrelativistic approach the Stefan–Boltzmann law and the Casimir effect at finite temperature for a particle with spin zero and 1/2 are calculated. The thermo field dynamics is used to include the finite temperature effects.