Uma introdução à Equação de Dirac: álgebra, matrizes e espinores

Abstract

The integrated understanding of Quantum Mechanics and Special Relativity is essential for describing elementary particles in high-energy regimes, yet this unification is rarely explored in depth during the initial training of physics teachers. Motivated by this educational gap, we developed a conceptually coherent and mathematically grounded introduction to relativistic quantum mechanics, with emphasis on covariant wave equations and, in particular, on the Dirac equation. Beginning with the incompatibility between the Schrödinger equation and the principles of Special Relativity, we reviewed the necessary foundations of both theories with the goal of constructing a consistent framework for discussing relativistic wave equations. Building on this basis, we presented the Klein-Gordon equation as the first attempt at a covariant formulation. Although mathematically consistent, we showed that it exhibits important physical limitations, such as the non-positivity of the probability density. We then developed the derivation of the Dirac equation, motivated by the need for a wave equation linear in both temporal and spatial derivatives. We introduced the matrices $\boldsymbol{\alpha}$ and $\beta$, discussing their anticommutation properties in the context of Clifford algebra, and analyzed how this matrix structure reconciles the relativistic energy-momentum relation with the probabilistic interpretation of Quantum Mechanics. We also examined the solutions of the Dirac equation, showing the natural emergence of spin $1/2$, the existence of negative-energy states, and their modern interpretation as antiparticles, a prediction later confirmed experimentally with the discovery of the positron. Finally, we highlight the formative relevance of this study by reconstructing, in an accessible and systematic manner, concepts such as spinors, matrix operators, conserved densities, and Lorentz transformations. This trajectory made it possible to emphasize not only the conceptual depth of the Dirac equation and its historical and epistemological role, but also the type of integrated view of modern theoretical physics that can be offered to undergraduate students in teacher-training programs. Such integration reinforces the importance of symmetries, mathematical representations, and the articulation between different areas of physics, thereby expanding the conceptual repertoire of future teachers and strengthening their ability to understand physics as a body of knowledge in continuous development, supported by fundamental ideas, mathematical rigor, and a permanent dialogue between theory and natural phenomena.