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Complex Lies, Real Physics: The Role of Algebra Complexification

Tanguy Marsault, Laurent Schoeffel

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.

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 . 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 , which determine unambiguously the physical object associated to the given representation. For example, the representation of dimension is called the scalar representation, it corresponds to the Higgs field, and of dimension 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.

Paper Structure

This paper contains 6 sections, 6 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Complexification of a real Lie algebra
  3. Application to $\mathfrak{sl}(2,\mathbb{C})$ and consequences
  4. Conclusion
  5. Appendix, proof of Lemma (\ref{['theorem:so13sl2C']})
  6. Appendix : Poincaré group and fields

Key Result

Theorem 2.3

Let $\mathfrak{g}$ be a complex Lie algebra, $\mathfrak{g}^\mathbb{R}$ its scalar restriction and $\left(\mathfrak{g}^\mathbb{R}\right)_\mathbb{C}$ the complexification of this scalar restriction, we have where the conjugate Lie algebra $\overline{\mathfrak{g}}$ of $\mathfrak{g}$ is the Lie algebra with the same vector space structure as $\mathfrak{g}$, the same Lie bracket but with the scalar mu

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 5 more