# 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.

For references, see the article on the Dirac equation.

How to Cite This Entry:
Dirac spinor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirac_spinor&oldid=40009
This article was adapted from an original article by V.D. Kukin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article