a gauge-invariant Dirac action by promoting the partial derivative to a covariant one: The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term, Putting these together gives Expanding out the covariant derivative allows the action to be written in a second useful form: Axial symmetry Massless Dirac fermions, that is, fields satisfying the Dirac equation with , admit a second, inequivalent symmetry. This is seen most easily by writing the four-component Dirac fermion as a pair of two-component vector fields, and adopting the chiral representation for the gamma matrices, so that may be written where has components and has components . The Dirac action then takes the form That is, it decouples into a theory of two Weyl spinors or Weyl fermions. The earlier vector symmetry is still present, where and rotate identically. This form of the action makes the second inequivalent symmetry manifest: This can also be expressed at the level of the Dirac fermion as where is the exponential map for matrices. This isn't the only symmetry possible, but it is conventional. Any 'linear combination' of the vector and axial symmetries is also a symmetry. Classically, the axial symmetry admits a well-formulated gauge theory. But at the quantum level, there is an anomaly, that is, an obstruction to gauging. Extension to color symmetry We can extend this discussion from an abelian symmetry to a general non-abelian symmetry under a gauge group , the group of color symmetries for a theory. For concreteness, we fix , the special unitary group of matrices acting on . Before this section, could be viewed as a spinor field on Minkowski space, in other words a function , and its components in are labelled by spin indices, conventionally Greek indices taken from the start of the alphabet . Promoting the theory to a gauge theory, informally acquires a part transforming like , and these are labelled by color indices, conventionally Latin indices . In total, has components, given in indices by . The 'spinor' labels only how the field transforms under spacetime transformations. Formally, is valued in a tensor product, that is, it is a function Gauging proceeds similarly to the abelian case, with a few differences. Under a gauge transformation the spinor fields transform as The matrix-valued gauge field or connection transforms as and the covariant derivatives defined transform as Writing down a gauge-invariant action