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Lattice electromagnetic theory from a topological viewpoint. (English) Zbl 0968.78005

From the introduction: The main objective of this paper is to tackle the general problem of the consistency of lattice electromagnetic theory within the framework of algebraic topology. By general, we mean lattices with arbitrary metric and topological structures. We distinguish three basic classes of consistency requirements. The first class (based on topological considerations only) is common to all field theories cast on a discrete form, and it is associated with the correct implementation of the boundary operator on the lattice. Discrete schemes that satisfy this first class can be classified as divergence-preserving schemes. The second class (also based on topological considerations only) is related to the topological structure of EM theory and the dual nature of ordinary and twisted cell complexes. The third class is the metric-dependent one, associated with the Hodge operators. We point out that each requirement is a separate, necessary condition for an overall consistent lattice EM theory. In more detail, in Sec. II, the authors write Maxwell’s equations using the language of differential forms and discuss their factorization into topological and metric equations. In Sec. III, they review the discretization of differential forms on a lattice using algebraic topological tools. In Sec. IV, they put Maxwell’s equations on the lattice using the concepts of the previous sections, stressing that it provides an exact counterpart to the continuum theory that is invariant under homeomorphisms. They also discuss the topological consistency requirements associated with the correct implementation of the boundary operator, and their connection with the usual theorems of vector calculus.

In Sec. V, they discuss the concept of dual lattices and how it arises from the necessity of a proper discretization of the different geometrical objects representing the EM fields. In Sec. VI, they treat some additional algebraic properties of the resulting discrete Maxwell’s equations by discussing additional topological consistency requirements associated with the dual structure of the ordinary and twisted cell complexes (important to guarantee reciprocity of the discrete Maxwell’s equations). In Sec. VII, they discuss the problem of the discretization of the constitutive relations, where metric concepts are present and approximations are involved through the discretization of the Hodge operators. They do not present explicit constructions for the Hodge operators (these are highly problem specific); instead, they discuss general rationales for this, and describe basic requirements that any consistent version of the discrete Hodge should satisfy.

Finally, in Sec. VIII, they summarize the conclusions. They use a $\left(3+1\right)$ representation with the ${e}^{-i\omega t}$ time convention assumed.

##### MSC:
 78A25 General electromagnetic theory 58A10 Differential forms (global analysis)