Complex Lies, Real Physics: The Role of Algebra Complexification

Abstract

In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie algebras is necessary in order to derive irreducible representations of the Lie algebra and subsequently of the symmetry group. In this paper, we give a precise definition of the concept and prove step by step an important result $\left(\mathfrak{g}^\mathbb{R}\right)_\mathbb{C} \simeq \mathfrak{g} \times \bar{\mathfrak{g}}$. This result is used to determine the irreducible representations of the proper Lorentz group and thus the physical objects admissible when this symmetry is present. It is shown that finite representations of the proper Lorentz group are characterized by pairs of half-integers $(j_1,j_2)$, which determine unambiguously the physical object associated to the given representation. For example, the representation $(0,0)$ of dimension $1$ is called the scalar representation, it corresponds to the Higgs field, and $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ of dimension $4$ is called the Dirac spinor representation, it corresponds to matter particle called fermions. This means that the mathematical group structure determines the material content of the universe following this algebraic structure.