Editing Dirac equation (section)

=== 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.