The Dirac equation describes the first quantized, spin one half fermion with nonzero mass in a fully Lorentz covariant way. It is a linear differential equation defining the evolution of a vector of four complex quantities (a bispinor) that transforms in a specific way under a Lorentz transformation. The equation's co-efficients, the so-called gamma matrices, are elements of the Clifford algebra $C\ell_{1,3}(\mathbb{R})$ and indeed generate this algebra. The solutions are superpositions of a fermion and its antiparticle and their collocation in the Dirac bispinor gives rise to the Klein paradox. The Dirac equation explains the fine structure of the Hydrogen spectrum but must be coupled to the electromagnetic field through $\partial_\mu \to\partial_\mu + i q A_\mu$ to explain the Lamb shift and spontaneous emission.

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