C${}^{*}$-Algebras and the Description of Quantum Mechanics

Abstract

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.

Keywords:
Quantum Mechanics; C*-Algebras; Gelfand-Neumark Theorem