Lagrangian formulation, generalizations and quantization of null Maxwell's knots

TL;DR

This paper develops a Lagrangian and Hamiltonian framework for knotted electromagnetic solutions, generalizes null conditions to other fields, and explores quantization methods for these configurations.

Contribution

It introduces a comprehensive formulation for null electromagnetic knots and extends the null condition concept to scalar and two-form fields, setting the stage for quantization.

Findings

Formulated Lagrangian and Hamiltonian for knotted electromagnetic solutions

Generalized null conditions to scalar and two-form fields

Outlined quantization approaches for null configurations

Abstract

Knotted solutions to electromagnetism are investigated as an independent subsector of the theory. We write down a Lagrangian and a Hamiltonian formulation of Bateman's construction for the knotted electromagnetic solutions. We introduce a general definition of the null condition and generalize the construction of Maxwell's theory to massless free complex scalar, its dual two form field, and to a massless DBI scalar. We set up the framework for quantizing the theory both in a path integral approach, as well as the canonical Dirac method for a constrained system. We make several observations about the semi-classical quantization of systems of null configurations.