However, as Feynman points out, it fails to explain why particles such as the electron have the masses they do. "There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them – what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem." Mathematical formulation QED action Mathematically, QED is an abelian gauge theory with the symmetry group U(1), defined on Minkowski space (flat spacetime). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action where are Dirac matrices. a bispinor field of spin-1/2 particles (e.g. electron–positron field). , called "psi-bar", is sometimes referred to as the Dirac adjoint. is the gauge covariant derivative. e is the coupling constant, equal to the electric charge of the bispinor field. is the covariant four-potential of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a connection. is the external field imposed by external source. m is the mass of the electron or positron. is the electromagnetic field tensor. This is also known as the curvature of the gauge field. Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field set to zero for simplicity) where is the conserved current arising from Noether's theorem. It is written Equations of motion Expanding the covariant derivative in the Lagrangian gives For simplicity, has been set to zero. Alternatively, we can absorb into a new gauge field and relabel the new field as From this Lagrangian, the equations of motion for the and fields can be obtained. Equation of motion for ψ These arise most straightforwardly by considering the Euler-Lagrange equation for . Since the Lagrangian contains no terms, we immediately get so the equation of motion can be written Equation of motion for Aμ Using the Euler–Lagrange equation for the field, the derivatives this time are Substituting back into () leads to which can be written in terms of the current as Now, if we impose the Lorenz gauge condition the equations reduce to which is a wave equation for the four-potential, the QED version of the classical Maxwell equations in the