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Tetrads in General Relativity

VI. Physical Geometry

Dirac Equation

   In a tetrad Minkowski space $\{ {x^a}\} $, the Dirac equation for a free spin-1/2 particle having mass $m$ may be written in the covariant form

6.10

\[(i \hbar {\gamma ^a}{\partial _a} - m)\psi (x) = 0{\qquad}a = 0,1,2,3\]

Plank’s constant $\hbar$ is henceforth set equal to one. The Dirac spinor $\psi (x)$ is a 1x4 complex matrix field and the four gamma matrices $\{ {\gamma ^a}\} $ are a set of complex 4x4 matrices which satisfy the defining anti-commutation relations:

6.11

\[{\gamma ^a} \cdot {\gamma ^b} : = \frac{1}{2}({\gamma ^a}{\gamma ^b} + {\gamma ^b}{\gamma ^a}) = {\eta ^{ab}}I\]

The Dirac algebra is a Clifford algebra ${\text{Cl(}}1,3)(\mathbb{C})$ with the gamma matrices as generators. It is conventional to suppress component indices of the gamma matrices and the spinor. Usually, the identity matrix at the right-hand side is also omitted.

   Under Lorentz transformations ${x'^a} = {\Lambda ^a}_b\,{x^b}$, the Dirac spinor transforms as

6.12

\[{\psi }(x) \to {\psi '}(x) := S{(\Lambda )}{\psi}({\Lambda ^{ - 1}}x)\]

The 4x4 matrices $S(\Lambda )$ form the spin representation of the restricted Lorentz group ${\text{S}}{{\text{O}}^ + }(1,3)$. The transformed Dirac equation then becomes

6.13

\[S(\Lambda )\left[ {i{S^{ - 1}}(\Lambda ) {\gamma ^a}S(\Lambda ){{({\Lambda ^{ - 1}})}^b}_a{\partial _b} - m} \right]\psi ({\Lambda ^{ - 1}}x) = 0\]

The Lorentz transformation acting on the Lorentz index, commutes with internal Lorentz transformation $S(\Lambda )$ that acts on spinor objects: $[{\Lambda ^a}_b,S(\Lambda )] = 0$.

   Between the brackets, the original Dirac operator is regained if

6.14

\[{S^{ - 1}}(\Lambda ){\gamma ^a}S(\Lambda ) = {\Lambda ^a}_b{\gamma ^b}\]

Hence, in flat space, the Dirac equation is Lorentz invariant, provided equality (6.14) is true. This may be shown to be the case by considering Lorentz transformations near the identity. [Wikipedia: Dirac Equation]