On the Angular Momentum and Spin of Generalized Electromagnetic Field for $r$-Vectors in $(k,n)$ Space-Time Dimensions

TL;DR

This paper generalizes the concept of angular momentum and spin in electromagnetic fields to $r$-vector fields in arbitrary $(k,n)$ space-time dimensions, deriving conservation laws and flux expressions using exterior algebra.

Contribution

It introduces a novel derivation of the angular momentum tensor for generalized electromagnetic fields in arbitrary dimensions without relying on canonical tensors.

Findings

Derived the angular momentum tensor from Lorentz invariance for $r$-vector fields.

Provided an integral expression for flux across higher-dimensional surfaces.

Discussed the orbital angular momentum and spin in complex polarization states.

Abstract

This paper studies the relativistic angular momentum for the generalized electromagnetic field, described by $r$-vectors in $(k,n)$ space-time dimensions, with exterior-algebraic methods. First, the angular-momentum tensor is derived from the invariance of the Lagrangian to space-time rotations (Lorentz transformations), avoiding the explicit need of the canonical tensor in Noether's theorem. The derivation proves the conservation law of angular momentum for generic values of $r$, $k$, and $n$. Second, an integral expression for the flux of the tensor across a $(k+n-1)$-dimensional surface of constant $\ell$-th space-time coordinate is provided in terms of the normal modes of the field; this analysis is a natural generalization of the standard analysis of electromagnetism, i. e. a three-dimensional space integral at constant time. Third, a brief discussion on the orbital angular momentum and the spin of the generalized electromagnetic field, including their expression in complex-valued circular polarizations, is provided for generic values of $r$, $k$, and $n$.