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Noetherian Conservation Laws for Photons

Michael K. -H. Kiessling, A. Shadi Tahvildar-Zadeh

TL;DR

This work develops Noetherian, gauge-invariant conservation laws for a Lorentz-covariant photon wave equation expressed as a rank-two bispinor on Minkowski space, embedding the theory in a Clifford-algebra framework. It establishes a 10-dimensional family of conserved currents, of which eight yield global, ADM-like charges that converge to gauge-invariant, asymptotic quantities defining a covariantly constant self-dual bispinor. The analysis extends to curved spacetimes, where boundary currents are constructed to obtain ADM-like invariants at infinity, and an explicit Minkowski-space example demonstrates the existence of a nontrivial bispinor at infinity while preserving gauge invariance. The results illuminate how Noether's theorems continue to yield physically meaningful, global conserved quantities for gauge theories of massless spin-one particles, even in settings with limited spacetime symmetries, and suggest a robust link between asymptotic charges and self-duality in the photon sector.$

Abstract

We review the formulation of a Lorentz-covariant bispinorial wave function and wave equation for a single photon on a flat background. We show the existence of a 10-dimensional set of conservation laws for this equation, and prove that 8 of these can be used to obtain global, gauge-invariant, ADM-like quantities that together define a covariantly constant self-dual bispinor.

Noetherian Conservation Laws for Photons

TL;DR

This work develops Noetherian, gauge-invariant conservation laws for a Lorentz-covariant photon wave equation expressed as a rank-two bispinor on Minkowski space, embedding the theory in a Clifford-algebra framework. It establishes a 10-dimensional family of conserved currents, of which eight yield global, ADM-like charges that converge to gauge-invariant, asymptotic quantities defining a covariantly constant self-dual bispinor. The analysis extends to curved spacetimes, where boundary currents are constructed to obtain ADM-like invariants at infinity, and an explicit Minkowski-space example demonstrates the existence of a nontrivial bispinor at infinity while preserving gauge invariance. The results illuminate how Noether's theorems continue to yield physically meaningful, global conserved quantities for gauge theories of massless spin-one particles, even in settings with limited spacetime symmetries, and suggest a robust link between asymptotic charges and self-duality in the photon sector.$

Abstract

We review the formulation of a Lorentz-covariant bispinorial wave function and wave equation for a single photon on a flat background. We show the existence of a 10-dimensional set of conservation laws for this equation, and prove that 8 of these can be used to obtain global, gauge-invariant, ADM-like quantities that together define a covariantly constant self-dual bispinor.

Paper Structure

This paper contains 20 sections, 3 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Review of Photon Wave Function and Equation on Minkowski Space
  3. Algebraic Preliminaries
  4. The Clifford algebra associated with a spacetime
  5. Spinors and differential forms
  6. Lorentz group $O(1,3)$ and its spinorial representation
  7. Relativistic wave functions and equations
  8. Photon wave function and equation
  9. Consequences of the photon wave equation
  10. The diagonal blocks of the photon wave function
  11. The off-diagonal components of the photon wave function.
  12. Photon probability current
  13. Photons in Curved Spacetime
  14. Conservation Laws
  15. Gauge invariance
  16. ...and 5 more sections

Key Result

Proposition 1

Let ${\psi_{\hbox{\rm\tiny ph}}} = \left( \right)$ be a solution of (eq:DirPH). There exists a gauge transformation (eq:DIRACmZEROeqnGAUGE), such that after applying it, the $\chi_\pm$ are Hermitian matrices (equivalently, the $\boldsymbol{\mathfrak{a}}_\pm$ are real-valued.)

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 2.2
  • Proposition
  • Proposition 3.1
  • proof
  • Remark 3.2
  • THEOREM 3.3
  • proof