لخّصلي

The theory is therefore regarded as a great success.
This gave us the motivation to use this theory in the field of numerical analysis to find classical
solutions of the fields. We started out with the simplest gauge group, the U(1) group, which corresponds to pure electromagnetism, and compared the results with the classical Yee-scheme[5]. One
peculiar fact in this case is that LGT with gauge group U(1) gives you a set of non-linear equations to
solve, while Maxwell’s equations describing the electromagnetic field are linear. However, numerical
experiments indicate a good agreement between the LGT-scheme and the Yee-scheme.
General Relativity can be viewed as a gauge theory, with the Poincare group as gauge group,
and should have wave-like solutions. One would like a geometric discretization of these nonlinear
equations. This is exactly what LGT provides for the Yang-Mills equations, so this is an attempt
towards creating a good numerical scheme for GR. In view of that, some of the purpose of this article
is to bring LGT to the attention of numerical analysts, and as such the paper is partly expository.
In this article we expand the earlier work to also include a complex scalar field, implying that the
equations to be solved are the Maxwell-Klein-Gordon equations. Since it may seem a bit too much to
use the Lattice Gauge Theory formalism on the Maxwell part of the action, which in any case is gauge
2
invariant, we will instead study a hybrid scheme, consisting of the 2. order Yee action for the Maxwell
part and the LGT-action for the Klein-Gordon part. The scheme we then end up with is locally gauge
invariant. Because of this symmetry we can use a discrete version of Noether’s theorem to extract
the conserved quantity. As in the continuous case, we get a locally conserved charge consistent with
Maxwell’s equations. The conservation of the local charge implies of course the conservation of a
global charge as well.
We will also compare this hybrid scheme with a more standard finite difference scheme for solving
the Maxwell-Klein-Gordon-equations. This scheme uses the 2. order Yee action for the Maxwell part
of the action, as the hybrid scheme does, and a finite difference approximation of the derivatives in
the KG action. This scheme does not possess a local U(1) symmetry, hence Noether’s theorem can
not be used to deduce a locally conserved charge. However, the scheme is invariant with respect to
global U(1) transformations, implying a globally conserved charge.
The paper is organized as follows: In §2 we introduce the continuous Maxwell-Klein-Gordon
equation from a variational point of view and prove constraint preservation. In §3 we discretize the
MKG-action using an LGT inspired action for the Klein-Gordon part and a Yee-action for the Maxwell
part, and we show that the resulting discrete action is indeed gauge invariant. In §4 a numerical scheme
which uses standard finite difference discretization is discussed. This scheme does not possess the
local gauge symmetry, so it does not preserve the constraint. In §5 a discussion of the energy of the
schemes is included. Finally in §6 we present some numerical results.