In this article we study the discretization of the Maxwell-Klein-Gordon equation from a variational point of view. We first solve the problem with an action corresponding to the Yee scheme for the Maxwell part, which is automatically gauge invariant, and a gauge invariant action for the Klein-Gordon part given by the Lattice Gauge Theory. The action is showed to be consistent with the continuous formulation, and the equations to be solved are derived from a discrete stationary action principle.
Due to the gauge invariance, the local electric charge can be shown to be conserved through Noether's theorem. As this is an essential feature of the continuous model, this conservation can be viewed as the great achievement of this scheme.
Thereafter we compare the above described scheme with a scheme that uses a standard finite difference approximation of the derivatives, and where the coupling between the scalar field and the gauge field is done in the simplest way. This scheme will posess a global gauge symmetry which ensures the conservation of global charge as in the hybrid case, but the scheme has no local symmetry and no locally conserved charge.
At last we present some numerical results in the temporal gauge, shedding light on the theoretical discussion.