On the degrees of freedom of lattice electrodynamics

TL;DR

This paper demonstrates that in lattice electrodynamics, the dynamic degrees of freedom of electric and magnetic fields are equal, using geometric discretization and dual lattices, providing new physical and geometric insights.

Contribution

It establishes a fundamental equality of degrees of freedom in lattice electrodynamics using discrete differential forms and dual lattices, offering new physical and geometric interpretations.

Findings

Electric and magnetic degrees of freedom are equal in lattice electrodynamics.

Euler's formula underpins the equality of degrees of freedom.

Provides geometric and physical interpretations for Euler's formula and Hodge decomposition.

Abstract

Using Euler's formula for a network of polygons for 2D case (or polyhedra for 3D case), we show that the number of dynamic\textit{\}degrees of freedom of the electric field equals the number of dynamic degrees of freedom of the magnetic field for electrodynamics formulated on a lattice. Instrumental to this identity is the use (at least implicitly) of a dual lattice and of a (spatial) geometric discretization scheme based on discrete differential forms. As a by-product, this analysis also unveils a physical interpretation for Euler's formula and a geometric interpretation for the Hodge decomposition.