In natural units and with the slash notation, the action is then For this action, the conserved current above arises as the conserved current corresponding to the global symmetry through Noether's theorem for field theory. Gauging this field theory by changing the symmetry to a local, spacetime point dependent one gives gauge symmetry (really, gauge redundancy). The resultant theory is quantum electrodynamics or QED. See below for a more detailed discussion. Lorentz invariance The Dirac equation is invariant under Lorentz transformations, that is, under the action of the Lorentz group or strictly , the component connected to the identity. For a Dirac spinor viewed concretely as taking values in , the transformation under a Lorentz transformation is given by a complex matrix . There are some subtleties in defining the corresponding , as well as a standard abuse of notation. Most treatments occur at the Lie algebra level. For a more detailed treatment see here. The Lorentz group of real matrices acting on is generated by a set of six matrices with components When both the indices are raised or lowered, these are simply the 'standard basis' of antisymmetric matrices. These satisfy the Lorentz algebra commutation relations In the article on the Dirac algebra, it is also found that the spin generators satisfy the Lorentz algebra commutation relations. A Lorentz transformation can be written as where the components are antisymmetric in . The corresponding transformation on spin space is This is an abuse of notation, but a standard one. The reason is is not a well-defined function of , since there are two different sets of components (up to equivalence) which give the same but different . In practice we implicitly pick one of these and then is well defined in terms of Under a Lorentz transformation, the Dirac equation becomes Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents . Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents , which can be identified as the stress-energy tensor of the theory. The Lorentz current can be written in terms of the stress-energy tensor in addition to a tensor representing internal angular momentum. Historical developments and further mathematical details The Dirac equation was also used (historically) to define a quantum-mechanical theory where is instead interpreted as a wave-function. The Dirac equation in the form originally