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New Projection Operators for Planar Electrodynamics

Flávio P. Cruz, José A. Santos, Victor J. V. Otoya

TL;DR

The paper develops a projector-based algebraic framework to efficiently invert the wave operator in planar (2+1) electrodynamics, yielding explicit propagators for Maxwell–Lee–Wick–Chern–Simons and Maxwell–Deser–Jackiw models. By introducing covariant and projector bases, it derives pole structures, residues, and unitarity conditions, including a Lee–Wick treatment for certain parameter regimes. The analysis shows a real-pole spectrum with relativistic dispersion or, when appropriate, damped resonances, and demonstrates that the MDJ model supports a single unitary massive mode while the MLWCS model can feature ghost states and requires careful contour prescriptions to preserve macroscopic causality. The framework is presented as general and extensible, with potential applications to nonlocal, interacting, and gravitational planar theories.”

Abstract

In this article, we provide a new method for obtaining the propagator of two three-dimensional models of electrodynamics (Maxwell-Lee-Wick-Chern-Simons and Maxwell-Deser-Jackiw). This method introduce a new set of projection operators. Then we perform a causality and unitarity analysis.

New Projection Operators for Planar Electrodynamics

TL;DR

The paper develops a projector-based algebraic framework to efficiently invert the wave operator in planar (2+1) electrodynamics, yielding explicit propagators for Maxwell–Lee–Wick–Chern–Simons and Maxwell–Deser–Jackiw models. By introducing covariant and projector bases, it derives pole structures, residues, and unitarity conditions, including a Lee–Wick treatment for certain parameter regimes. The analysis shows a real-pole spectrum with relativistic dispersion or, when appropriate, damped resonances, and demonstrates that the MDJ model supports a single unitary massive mode while the MLWCS model can feature ghost states and requires careful contour prescriptions to preserve macroscopic causality. The framework is presented as general and extensible, with potential applications to nonlocal, interacting, and gravitational planar theories.”

Abstract

In this article, we provide a new method for obtaining the propagator of two three-dimensional models of electrodynamics (Maxwell-Lee-Wick-Chern-Simons and Maxwell-Deser-Jackiw). This method introduce a new set of projection operators. Then we perform a causality and unitarity analysis.

Paper Structure

This paper contains 24 sections, 4 theorems, 102 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Mathematical Formalism
  4. Solutions of the Problem
  5. Operators $\omega$, $\theta$ and $S$
  6. Calculation of the Propagators
  7. General Propagator
  8. The Maxwell-Lee-Wick-Chern-Simons Propagator
  9. The Maxwell-Deser-Jackiw Propagator
  10. General analysis of causality and unitarity
  11. Analysis on the base $\{P_1,P_2,P_3\}$
  12. Pole structure and general spectral analysis
  13. Real poles
  14. Complex poles
  15. Residues and tree–level unitarity
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $P : V \to V$ be a projector. Then one has: In other words, a projector $P$ decomposes $V$ as a direct sum of its image and kernel. Moreover, $\dim(V) = \dim(\ker P) + \dim(\mathrm{Im}\, P)$.

Theorems & Definitions (5)

  • Definition 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 4.1