Wigner’s Classification Theorem and Construction of Fields in Quantum Theory (2025)
The purpose of this thesis is to give a self-contained introduction to the mathematical underpinnings of Lie groups, their representations, and apply the machinery to understand Wigner's classification of quantum fields. The presentation is well-suited for two kinds of readers. On the one hand, a mathematically curious reader who wishes to follow the full development can read the thesis linearly, starting from the appendices and then following through the main body of the text. In doing so, one will encounter the necessary background on manifolds, using the modern language of sheaves, fibre bundles, Lie algebras, and Hilbert spaces, and eventually see how all these ingredients fit together into the representation-theoretic formulation of quantum theory. Taken this way, the thesis can be read almost like a guided tour through the subject, with the appendices serving as a companion to fill in background knowledge when needed. The mathematically-inclined reader may be happy to know the approach to the theory of smooth manifolds taken here is mostly coordinate-free. On the other hand, it is perfectly acceptable for the reader to take a more pragmatic approach: many of the results in this text are stated together with proofs, but one may safely skip the technical details without losing track of the main ideas. For example, it is not necessary to master every step in the construction of fibre bundles or to check every computation in the theory of Lie algebras in order to appreciate the physical consequences discussed later on. The structure of the text is such that the statements of theorems, propositions, and examples are easy to extract, allowing the reader to "take them on faith", if so desired. We begin with an introduction to Lie groups and their actions on manifolds, which encode the language in which the theory of symmetries of physical spaces is encoded. Subsequently, we discuss fibre bundles, which give us the right framework to talk about vector fields and provide us with a necessary tool for the construction of induced representations later on. Lie algebras make their appearance in the third chapter, they are used to infinitesimally describe Lie groups and provide methods to find covering spaces of Lie groups. From there, we turn to projective Hilbert spaces and unitary representations, we introduce methods that build up the machinery that is eventually applied to Wigner's program of particle classification in the context of quantum fields. The appendices provide a compact reference on manifolds and differential calculus for those who want a firmer mathematical foundation, though these sections can be read independently of the main chapters. In short, this work can be read as either a careful mathematical journey or as a conceptual overview, depending on the reader’s preference. Both paths should, I hope, lead to a clearer appreciation of the mathematical formulation behind Wigner’s classification and its role in quantum theory. [Tomado del prefacio]