As is well known from classical electricity and magnetism,
the electric and magnetic field components are actually elements of a rank 2 Lorentz tensor.

This field tensor can simply be written in terms of the **vector potential**, (which is a Lorentz vector).

Note that is automatically antisymmetric under the interchange of the indices.

With the fields so derived from the vector potential, two of Maxwell's equations are automatically satisfied.
The remaining two equations can be written as one 4-vector equation.

We now wish to pick a scalar Lagrangian.
Since E&M is a well understood theory, the Lagrangian that is known to give the right equations is also known.

Note that (apart from the speed of light not being set to 1) the Lagrangian does not contain needless constants in this set of units. The last term is a source term which provides the interaction between the EM field and charged particles. In working with this Lagrangian, we will treat each component of as an independent field. In this case, the Euler-Lagrange equation is Maxwell's equation as written above.

The free field Hamiltonian density can be computed according to the standard prescription yielding

if there are no source terms in the region.

Gauge symmetry may be used to put a condition on the vector potential.

This is called the

Jim Branson 2013-04-22