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Electromagnetic Theory with Quantum Principal Bundles

Gustavo Amilcar Saldaña Moncada

TL;DR

This work builds a rigorous NC geometric formulation of vacuum electromagnetism by placing Maxwell theory on the Moyal–Weyl spacetime $B$ via quantum principal bundles and quantum principal connections. The authors derive NC Maxwell equations from a NC Bianchi identity that includes the operator $S^ abla$, yielding vacuum electric and magnetic charges/currents generated by self-interactions encoded in the NC curvature $F^ abla$. They show that for a trivial NC bundle with a one‑dimensional invariant calculus, NC Yang–Mills dynamics reduce to a covariant Maxwell system, and they illustrate two explicit NC solutions that produce vacuum charges. Extending the framework, they introduce a two‑gauge-field model with electric/magnetic sectors, obtaining a symmetric Maxwell theory with a duality and demonstrating instantons that are not YM solutions. The results clarify how NC gauge theory should be formulated in this geometric setting and suggest avenues for generalization to other groups and NC spacetime geometries.

Abstract

The aim of this paper is to formulate a {\it non--commutative geometrical} version of the classical electromagnetic field theory in the vacuum with the Moyal--Weyl algebra as the space--time by using the theory of quantum principal bundles and quantum principal connections. As a result we will present the correct Maxwell equations in the vacuum of the model, in which we can appreciate the existence of electric and magnetic charges and currents. Finally, in the fourth section we are going to present a {\it mathematical model} for which there are instantons that are not solutions of the corresponding Yang--Mills equation.

Electromagnetic Theory with Quantum Principal Bundles

TL;DR

This work builds a rigorous NC geometric formulation of vacuum electromagnetism by placing Maxwell theory on the Moyal–Weyl spacetime via quantum principal bundles and quantum principal connections. The authors derive NC Maxwell equations from a NC Bianchi identity that includes the operator , yielding vacuum electric and magnetic charges/currents generated by self-interactions encoded in the NC curvature . They show that for a trivial NC bundle with a one‑dimensional invariant calculus, NC Yang–Mills dynamics reduce to a covariant Maxwell system, and they illustrate two explicit NC solutions that produce vacuum charges. Extending the framework, they introduce a two‑gauge-field model with electric/magnetic sectors, obtaining a symmetric Maxwell theory with a duality and demonstrating instantons that are not YM solutions. The results clarify how NC gauge theory should be formulated in this geometric setting and suggest avenues for generalization to other groups and NC spacetime geometries.

Abstract

The aim of this paper is to formulate a {\it non--commutative geometrical} version of the classical electromagnetic field theory in the vacuum with the Moyal--Weyl algebra as the space--time by using the theory of quantum principal bundles and quantum principal connections. As a result we will present the correct Maxwell equations in the vacuum of the model, in which we can appreciate the existence of electric and magnetic charges and currents. Finally, in the fourth section we are going to present a {\it mathematical model} for which there are instantons that are not solutions of the corresponding Yang--Mills equation.
Paper Structure (14 sections, 7 theorems, 244 equations)

This paper contains 14 sections, 7 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. The Trivial Quantum Principal $U(1)$--Bundle
  3. A Non--Commutative Minkowski Space--Time
  4. The Quantum Group of $U(1)$ and its Differential calculus
  5. The Quantum Bundle
  6. Non--Commutative Electromagnetic Field Theory
  7. Non--Commutative Geometrical (Topological) equation
  8. Non--Commutative Dynamical equation
  9. Some Solutions, Remarks and the Covariant Formulation
  10. Two Electromagnetic Gauge Fields
  11. The Quantum Bundle
  12. Non--Commutative Maxwell equations and Instantons
  13. Concluding Comments
  14. Classical Maxwell equations

Key Result

Proposition 2.1

The covariant derivative $D^\omega$ of an element of $\textsc{Mor}(\mathrm{ad}, \Delta_\mathrm{Hor})$ is just the differential $d$ of the element, for every qpc $\omega$.

Theorems & Definitions (25)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • ...and 15 more