Editing Dirac equation (section)

=== Extension to color symmetry ===
{{See also | quantum chromodynamics}}
We can extend this discussion from an abelian <math>\text{U}(1)</math> symmetry to a general non-abelian symmetry under a [[gauge group]] <math>G</math>, the group of [[color charge|color symmetries]] for a theory.

For concreteness, we fix <math>G = \text{SU}(N)</math>, the [[special unitary group]] of matrices acting on <math>\mathbb{C}^N</math>.

Before this section, <math>\psi(x)</math> could be viewed as a spinor field on Minkowski space, in other words a function <math>\psi: \mathbb{R}^{1,3}\mapsto \mathbb{C}^4</math>, and its components in <math>\mathbb{C}^4</math> are labelled by spin indices, conventionally Greek indices taken from the start of the alphabet <math>\alpha,\beta,\gamma,\cdots</math>.

Promoting the theory to a gauge theory, informally <math>\psi</math> acquires a part transforming like <math>\mathbb{C}^N</math>, and these are labelled by color indices, conventionally Latin indices <math>i,j,k,\cdots</math>. In total, <math>\psi(x)</math> has <math>4N</math> components, given in indices by <math>\psi^{i,\alpha}(x)</math>. The 'spinor' labels only how the field transforms under spacetime transformations.

Formally, <math>\psi(x)</math> is valued in a tensor product, that is, it is a function <math>\psi:\mathbb{R}^{1,3} \to \mathbb{C}^4 \otimes \mathbb{C}^N.</math>

Gauging proceeds similarly to the abelian <math>\text{U}(1)</math> case, with a few differences. Under a gauge transformation <math>U:\mathbb{R}^{1,3} \rightarrow \text{SU}(N),</math> the spinor fields transform as
<math display="block">\psi(x) \mapsto U(x)\psi(x)</math>
<math display="block">\bar\psi(x)\mapsto \bar\psi(x)U^\dagger(x).</math>
The matrix-valued gauge field <math>A_\mu</math> or <math>\text{SU}(N)</math> connection transforms as
<math display="block">A_\mu(x) \mapsto U(x)A_\mu(x)U(x)^{-1} + \frac{1}{g}(\partial_\mu U(x))U(x)^{-1},</math>
and the covariant derivatives defined 
<math display="block">D_\mu\psi = \partial_\mu \psi + igA_\mu\psi,</math>
<math display="block">D_\mu\bar\psi = \partial_\mu \bar\psi - ig\bar\psi A_\mu^\dagger</math>
transform as 
<math display="block">D_\mu\psi(x) \mapsto U(x)D_\mu\psi(x),</math>
<math display="block">D_\mu\bar\psi(x) \mapsto (D_\mu\bar\psi(x))U(x)^\dagger.</math>

Writing down a gauge-invariant action proceeds exactly as with the <math>\text{U}(1)</math> case, replacing the Maxwell Lagrangian with the [[Yang–Mills]] Lagrangian
<math display="block">S_{\text{Y-M}} = \int d^4x \,-\frac{1}{4}\text{Tr}(F^{\mu\nu}F_{\mu\nu})</math>
where the Yang–Mills field strength or curvature is defined here as
<math display="block">F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ig\left[A_\mu,A_\nu\right] </math>
and <math>[\cdot,\cdot]</math> is the matrix commutator.

The action is then
{{Equation box 1
|title='''QCD Action'''
|indent=:
|equation = <math>S_{\text{QCD}} = \int d^4x\,\left[-\frac{1}{4}\text{Tr}(F^{\mu\nu}F_{\mu\nu}) + \bar\psi\,(iD\!\!\!\!\big / - m)\,\psi\right]</math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}

==== Physical applications ====
For physical applications, the case <math>N=3</math> 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 <math>N=2</math> 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 <math>W</math> gauge boson.

==== Generalisations ====
This expression can be generalised to arbitrary Lie group <math>G</math> with connection <math>A_\mu</math> and a [[group representation|representation]] <math>(\rho, G, V)</math>, where the colour part of <math>\psi</math> is valued in <math>V</math>. Formally, the Dirac field is a function <math>\psi:\mathbb{R}^{1,3} \to \mathbb{C}^4\otimes V.</math>

Then <math>\psi</math> transforms under a gauge transformation <math>g:\mathbb{R}^{1,3} \to G</math> as
<math display="block">\psi(x) \mapsto \rho(g(x))\psi(x)</math>
and the covariant derivative is defined
<math display="block">D_\mu\psi = \partial_\mu\psi + \rho(A_\mu)\psi</math>
where here we view <math>\rho</math> as a [[Lie algebra]] representation of the Lie algebra <math>\mathfrak{g} = \text{L}(G)</math> associated to <math>G</math>.

This theory can be generalised to curved spacetime, but there are subtleties that arise in gauge theory on a general spacetime (or more generally still, a manifold), which can be ignored on flat spacetime. This is ultimately due to the [[contractible|contractibility]] of flat spacetime that allows us to view a gauge field and gauge transformations as defined globally on {{tmath|1= \mathbb{R}^{1,3} }}.