Neste artigo apresentamos a Teoria das C *-Álgebras. Demonstramos dois importantes resultados que são: o Teorema de Gelfand, que associa toda C *-Álgebra Abeliana às funções contínuas sobre um espaço Hausdorff compacto, e o Teorema de Gelfand-Neumark, que relaciona toda C *-Álgebra não Abeliana aos operadores lineares sobre um espaço de Hilbert. Em seguida, mapeamos a Mecânica Clássica dentro da teoria da C *-Álgebra, obtendo uma prescrição algébrica para os estados e às observáveis clássicas. Ao estendermos essa construção para o caso Quântico, preservando a prescrição algébrica e utilizando o Princípio da Incerteza, obtemos que os estados quânticos devem ser descritos por vetores do Espaço de Hilbert enquanto que os observáveis quânticos são os operadores auto-adjuntos sobre este espaço; discutimos brevemente sobre a Álgebra de Weyl.

In this paper we present the theory of C *-Algebras. We show two major results that are Gelfand's Theorem, which associates every Abelian C *-Algebra as continuous functions on a compact Hausdorff space, and the Gelfand-Neumark Theorem, which relates all non-Abelian C *-Algebra to linear operators on a Hilbert space. Then we map the Classical Mechanics into the C *-Algebra's theory, obtaining an algebraic prescription for classical states and observables. When we extend this construction to the Quantum case, preserving the algebraic prescriptions and using the Uncertainty Principle, we obtain that quantum states must be described by vectors in a Hilbert space while quantum observables are the self-adjoint operators on this space; we briefly discussed the Weyl's Algebra.

It is presented a theoretical study on electron -CO collisions in the (5-20) eV energy range. Specifically, elastic differential cross sections (DCS) are calculated using the configuration interaction (CI) method to describe the target. Static and exchange contributions to the interaction potential are obtained exactly from a CI wave function with single and double excitations. The DCS are obtained using the Schwinger variational iterative method (SVIM). It is observed that the discrepancies between Hartree-Fock and CI results are more significant for low energies and small angles.

The Duffin-Kemmer-Petiau (DKP) relativistic equation has been recently uded to study the interactions of spinless mesons with nuclei. In view of this interest and also the interest to determine the phase space picture for hadronic quantum theory, we present a derivation of the DKP equation of the generalized phase space proposed by Bohm and Hiley. Our development is based on the algebraic calculus introduced by Schönberg and uses the idea of algebraic spinors due to Riesz and Cartan. The free DKP particle and the more general case of the DKP particle in a prescribed external electromagnetic field are considered, and we obtain the DKP Liouville type equations for these cases.

A nonequilibrium statistical operator (NSO) is built for classical sistems using a field theory in classical phase space, which is but a classical version of the thermofield dynamics formalism. The approach developed here starts with a second-quantized version of this phase-space field theory. Then elements of symmetry are analysed and invariants of the theory are introduced. The local conservation laws are derived and used to make explicit NSO. Such a method is applied to derive the Fokker-Planck-Kramers equation.