From Classical Mechanics to Quantum Field Theory, A Tutorial

The following sections are included:

• Preface
• Acknowledgments
• Contents

The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then at presenting the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader…

Within these lectures, I review the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced viewpoint essentially based on the orthomodular lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and theorems also related to the representation of physical symmetries. The final step consists of an elementary introduction of the so-called (C*-) algebraic formulation of quantum theories.

We review the basic principles of Quantum Field Theory in a brief but comprehensive introduction to the foundations of Quantum Field Theory. The principles of Quantum Field Theory are introduced in canonical and covariant formalisms. The problem of ultraviolet divergences and its renormalization is analyzed in the canonical formalism. As an application, we review the roots of Casimir effect. For simplicity, we focus on the scalar field theory but the generalization for fermion fields is straightforward. However, the quantization of gauge fields requires extra techniques which are beyond the scope of these lectures. The special cases of free field theories and conformal invariant theories in lower space-time dimensions illustrate the relevance of the foundations of the theory. Finally, a short introduction to functional integrals and perturbation theory in the Euclidean formalism is included in the last section.

The following section is included:

• Index