Discretization Errors and Rotational Symmetry: The Laplacian Operator on Non-Hypercubical Lattices

Discretizations of the Laplacian operator on non-hypercubical lattices are discussed in a systematic approach. It is shown that order a2 errors always exist for discretizations involving only nearest neighbors. Among all lattices with the same density of lattice sites, the hypercubical lattices always have errors smaller than other lattices with the same number of spacetime dimensions. On the other hand, the four dimensional checkerboard lattice (also known as the Celmaster lattice) is the only lattice which is isotropic at order a2. Explicit forms of the discretized Laplacian operators on root lattices are presented, and different ways of eliminating order a2 errors are discussed. Typeset using REVTEX 1