  A lattice Maxwell system is developed with gauge-symmetry, symplectic
structure and discrete space-time symmetry. Noether's theorem for Lie group
symmetries is generalized to discrete symmetries for the lattice Maxwell
system. As a result, the lattice Maxwell system is shown to admit a discrete
local energy-momentum conservation law corresponding to the discrete space-time
symmetry. These conservative properties make the discrete system an effective
algorithm for numerically solving the governing differential equations on
continuous space-time. Moreover, the lattice model, respecting all conservation
laws and geometric structures, is as good as and probably more preferable than
the continuous Maxwell model. Under the simulation hypothesis by Bostrom and in
consistent with the discussion on lattice QCD by Beane et al., the two
interpretations of physics laws on space-time lattice could be essentially the
same.
