C*-Algebra in Quantum Mechanics: Proving the Limitations of Our Typical Representations and the Need for C*-Algebra

In thesis we will build up our operator theory for finite and infinite dimensional systems. We then prove that Heisenberg and Schrodinger representations are equivalent for systems with finite degrees of freedom. We will then make a case to switch to a C*-algebra formulation of quantum mechanics as we will prove that the Schrodinger and Heisenberg pictures become inadequate to full describe systems with infinitely many degrees of freedom because of inequivalent representations. This becomes important as we shift from single particle systems to quantum field theory, statistical mechanics, and many other areas of study. The goal of this thesis is to introduce these mathematical topics rigorously and prove that they are necessary for further study in particle physics.