proceeds exactly as with the case, replacing the Maxwell Lagrangian with the Yang–Mills Lagrangian where the Yang–Mills field strength or curvature is defined here as and is the matrix commutator. The action is then Physical applications For physical applications, the case describes the quark sector of the Standard model which models strong interactions. Quarks are modelled as Dirac spinors; the gauge field is the gluon field. The case describes part of the electroweak sector of the Standard model. Leptons such as electrons and neutrinos are the Dirac spinors; the gauge field is the gauge boson. Generalisations This expression can be generalised to arbitrary Lie group with connection and a representation , where the colour part of is valued in . Formally, the Dirac field is a function Then transforms under a gauge transformation as and the covariant derivative is defined where here we view as a Lie algebra representation of the Lie algebra associated to . This theory can be generalised to curved spacetime, but there are subtleties which arise in gauge theory on a general spacetime (or more generally still, a manifold) which, on flat spacetime, can be ignored. This is ultimately due to the contractibility of flat spacetime which allows us to view a gauge field and gauge transformations as defined globally on . See also Articles on the Dirac equation Dirac field Dirac spinor Gordon decomposition Klein paradox Nonlinear Dirac equation Other equations Breit equation Dirac–Kähler equation Klein–Gordon equation Rarita–Schwinger equation Two-body Dirac equations Weyl equation Majorana equation Other topics Fermionic field Feynman checkerboard Foldy–Wouthuysen transformation Quantum electrodynamics Quantum chromodynamics References Citations Selected papers Textbooks External links The Dirac Equation at MathPages The Nature of the Dirac Equation, its solutions, and Spin Dirac equation for a spin particle Pedagogic Aids to Quantum Field Theory click on Chap. 4 for a step-by-small-step introduction to the Dirac equation, spinors, and relativistic spin/helicity operators. BBC Documentary Atom 3 The Illusion of Reality 1928 introductions Fermions Partial differential equations Equation Quantum field theory Spinors