Discrete-time classical electrodynamics
In this section we show that the discretization of time applied to Maxwell’s equations of electrodynamics gives difference equations that encode a discrete-time version of gauge invariance. To see how this works out, we review briefly the continuous-time Maxwell’s equations. We use the convention c = x0 = e0 = 1.
The continuous-time theory
There are four physical fields and an auxiliary electromagnetic potential field. The physical fields are the electric field E, the magnetic field B, the electric charge density p, and the charge current j. These satisfy Maxwell’s equations:
These equations give the continuity equation dtp + V- j = 0, which is consistent with charge conservation.
The electric and magnetic fields E, B can be written as E = - VA0 - 3tA, B = Vx A, where A0, A are the components of the electromagnetic four-vector potential A^ = (A0, A). The potentials are not physical fields, in that they can be replaced by the gauge transformed fields A'^ = A?“ + 3^x, where x is an arbitrary gauge field. The four-vector potential A?“ satisfies the Lorentz covariant equation
where j^ = (p, j) is the four-vector charge current. The physical fields are gauge invariant.