Editing Dirac equation (section)

== U(1) symmetry ==
Natural units are used in this section. The coupling constant is labelled by convention with <math>e</math>: this parameter can also be viewed as modelling the electron charge.

=== Vector symmetry ===
The Dirac equation and action admits a <math>\text{U}(1)</math> symmetry where the fields <math>\psi, \bar\psi</math> transform as
<math display="block">\begin{align}
\psi(x) &\mapsto e^{i\alpha}\psi(x), \\
\bar\psi(x) &\mapsto e^{-i\alpha}\bar\psi(x).
\end{align}</math>
This is a global symmetry, known as the <math>\text{U}(1)</math> '''vector''' symmetry (as opposed to the <math>\text{U}(1)</math> '''axial''' symmetry: see below). By [[Noether's theorem]] there is a corresponding conserved current: this has been mentioned previously as
<math display="block">J^\mu(x) = \bar\psi(x)\gamma^\mu\psi(x).</math>

=== Gauging the symmetry ===
{{See also|Quantum electrodynamics}}
If we 'promote' the global symmetry, parametrised by the constant <math>\alpha</math>, to a local symmetry, parametrised by a function <math>\alpha:\mathbb{R}^{1,3} \to \mathbb{R}</math>, or equivalently <math>e^{i\alpha}: \mathbb{R}^{1,3} \to \text{U}(1),</math> the Dirac equation is no longer invariant: there is a residual derivative of <math>\alpha(x)</math>.

The fix proceeds as in [[scalar electrodynamics]]: the partial derivative is promoted to a covariant derivative <math>D_\mu</math>
<math display="block">D_\mu \psi = \partial_\mu \psi + i e A_\mu\psi,</math>
<math display="block">D_\mu \bar\psi = \partial_\mu \bar\psi - i e A_\mu\bar\psi.</math>
The covariant derivative depends on the field being acted on. The newly introduced <math>A_\mu</math> is the 4-vector potential from electrodynamics, but also can be viewed as a <math>\text{U}(1)</math> [[gauge field]] (which, mathematically, is defined as a <math>\text{U}(1)</math> [[Principal connection|connection]]).

The transformation law under gauge transformations for <math>A_\mu</math> is then the usual
<math display="block">A_\mu(x) \mapsto A_\mu(x) + \frac{1}{e}\partial_\mu\alpha(x)</math>
but can also be derived by asking that covariant derivatives transform under a gauge transformation as
<math display="block">D_\mu\psi(x) \mapsto e^{i\alpha(x)}D_\mu\psi(x),</math>
<math display="block">D_\mu\bar\psi(x) \mapsto e^{-i\alpha(x)}D_\mu\bar\psi(x).</math>
We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one:
<math display="block">S = \int d^4x\,\bar\psi\,(iD\!\!\!\!\big / - m)\,\psi = \int d^4x\,\bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi.</math>
The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term,
<math display="block">S_{\text{Maxwell}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\right].</math>
Putting these together gives
{{Equation box 1
|title='''QED Action'''
|indent=:
|equation = <math>S_{\text{QED}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(iD\!\!\!\!\big / - m)\,\psi\right]</math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}
Expanding out the covariant derivative allows the action to be written in a second useful form:
<math display="block">S_{\text{QED}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(i\partial\!\!\!\big / - m)\,\psi - eJ^\mu A_\mu\right]</math>

=== Axial symmetry ===
'''Massless''' Dirac fermions, that is, fields <math>\psi(x)</math> satisfying the Dirac equation with <math>m = 0</math>, admit a second, inequivalent <math>\text{U}(1)</math> symmetry.

This is seen most easily by writing the four-component Dirac fermion <math>\psi(x)</math> as a pair of two-component vector fields,
<math display="block">\psi(x) = \begin{pmatrix}
\psi_1(x)\\
\psi_2(x)
\end{pmatrix},
</math>
and adopting the [[gamma matrices|chiral representation]] for the gamma matrices, so that <math>i\gamma^\mu\partial_\mu</math> may be written
<math display="block">i\gamma^\mu\partial_\mu = \begin{pmatrix}
0 & i\sigma^\mu \partial_\mu\\
i\bar\sigma^\mu \partial_\mu\ & 0
\end{pmatrix}
</math>
where <math>\sigma^\mu</math> has components <math>(I_2, \sigma^i)</math> and <math>\bar\sigma^\mu</math> has components <math>(I_2, -\sigma^i)</math>.

The Dirac action then takes the form
<math display="block">S = \int d^4x\, \psi_1^\dagger(i\sigma^\mu\partial_\mu)\psi_1 + \psi_2^\dagger(i\bar\sigma^\mu\partial_\mu) \psi_2.</math>
That is, it decouples into a theory of two [[Weyl equation|Weyl spinors]] or Weyl fermions.

The earlier vector symmetry is still present, where <math>\psi_1</math> and <math>\psi_2</math> rotate identically. This form of the action makes the second inequivalent <math>\text{U}(1)</math> symmetry manifest:
<math display="block">\begin{align}
\psi_1(x) &\mapsto e^{i\beta} \psi_1(x), \\
\psi_2(x) &\mapsto e^{-i\beta}\psi_2(x).
\end{align}</math>
This can also be expressed at the level of the Dirac fermion as 
<math display="block">\psi(x) \mapsto \exp(i\beta\gamma^5) \psi(x)</math>
where <math>\exp</math> is the exponential map for matrices.

This isn't the only <math>\text{U}(1)</math> symmetry possible, but it is conventional. Any 'linear combination' of the vector and axial symmetries is also a <math>\text{U}(1)</math> symmetry.

Classically, the axial symmetry admits a well-formulated gauge theory. But at the quantum level, there is an [[anomaly (physics)|anomaly]], that is, an obstruction to gauging.

=== 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} }}.