# Dirac spinor

A four-component complex function defined on four-dimensional space-time that satisfies the Dirac equation. In spinor analysis, the Dirac spinor is defined as a bi-spinor of the first rank, realizing the irreducible linear representation of the general Lorentz group on $\mathbb{R}^{4}$ equipped with the pseudo-Euclidean metric $(\cdot,\cdot): \mathbb{R}^{4} \times \mathbb{R}^{4} \to \mathbb{R}$ given by $$\forall \underbrace{(x^{0},x^{1},x^{2},x^{3})}_{x},\underbrace{(y^{0},y^{1},y^{2},y^{3})}_{y} \in \mathbb{R}^{4}: \qquad (x,y) \stackrel{\text{df}}{=} \sum_{\alpha,\beta = 0}^{3} \eta_{\alpha \beta} x^{\alpha} y^{\beta} = x^{0} y^{0} - \sum_{k = 1}^{3} x^{k} y^{k}, \qquad \text{where} \quad [\eta_{\alpha \beta}] \stackrel{\text{df}}{=} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}.$$ The Dirac matrices, which form part of the Dirac equation, are defined up to an arbitrary unitary transformation, so that the Dirac spinor is also defined up to such a unitary transformation. This property makes it possible to select the most physically convenient representation of the Dirac matrices and, consequently, of the Dirac spinor.