Classical field theory and relativistic quantum mechanics

Nowadays, the best theoretical framework we have to describe elementary particles' physics is the Standard Model, whose main language consists of quantum field theory. Historically, before its formulation, it has been attempted to make Schrödinger's quantum theory relativistic, in the framework of the so-called first quantization. The aim of this thesis is to make the reader aware of the problems of this procedure, necessary condition to understand the need to change paradigm and develop a new theory, known as second quantization. Since, in this context, the new fundamental physical entity is the quantum field, we shall introduce field theory, starting from the classical description of electromagnetism, making use of a formalism to make Maxwell's equations manifestly covariant. To obtain the latter directly from an action and to use the Lagrangian mechanics' tools, it's necessary to generalize the latter to a system with an infinite number of degrees of freedom. This will be achieved initially by discretizing the space and applying the known formalism into any elementary cell and, later, through a variational principle. Furthermore, we'll try to apply quantum mechanics to a relativistic particle, obtaining the Klein-Gordon equation, which will be interpreted as representing a field whose quantum is a massive particle without spin. We'll notice how, forcing a particular global symmetry of this equation to be locally valid, it'll be necessary to add some terms on the Lagrangian which can be interpreted as an interaction with the electromagnetic field. This allows us to introduce Gauge's principle, which is a fundamental tool to describe interactions in the Standard Model. Finally, this principle will be critically analyzed, leading to the conclusion that it's not correct to distinguish between the object and the mediator of an interaction.