C*-Algebraic Formulation of Quantum Mechanics

Abstract

This manuscript is based on a series of lectures given in the XX Jacques-Louis Lions Spanish-French School on Numerical Simulations in Physics & Engineering. While quantum mechanics is widely presented within the Hilbert space formalism in undergraduate courses, in these notes we give an introduction to its algebraic formulation. It is based on the theory of C^{∗}-algebras, which is shortly outlined here. This viewpoint is more general than the usual formulation of quantum mechanics, because of the the non-uniqueness of irreducible representations of C^{∗}-algebras of infinite dimension. This is related to the Rosenberg theorem and Naimark's problem.W. de Siqueira Pedra has been supported by CNPq (309723/2020-5). J.-B. Bru is supported by the Basque Government through the grant IT1615-22 and the BERC 2022-2025 program, by the COST Action CA18232 financed by the European Cooperation in Science and Technology (COST), and by the grant PID2020-112948GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe"