Special relativity is formulated on Minkowski spacetime with interval $s^2 = \eta_{\mu\nu}x^\mu x^\nu$. Lorentz transformations preserve $s^2$, ensuring that the speed of light is invariant and that causal structure is maintained. Four-vectors and tensors organize relativistic quantities in a manifestly covariant way.

The proper orthochronous Lorentz group is locally isomorphic to $SL(2,\mathbb{C})$. This enables the construction of Weyl spinors and Dirac spinors, whose transformation properties are essential for describing half-integer spin particles. Chirality and helicity become especially important in the massless limit.

The Dirac equation $\left(i\gamma^\mu \partial_\mu - m\right)\psi = 0$ combines Lorentz covariance with first-order dynamics. Its structure implies antiparticles, conserved probability current $j^\mu = \bar\psi\gamma^\mu\psi$, and a natural coupling to gauge fields via $\partial_\mu \to D_\mu$.

The Klein-Gordon equation $(\Box + m^2)\phi = 0$ describes free scalar relativistic fields, while Maxwell theory can be expressed covariantly using $F_{\mu\nu}$. Causality and consistency strongly constrain relativistic field equations, especially once interactions are included.