[Paper Review] Field of the Magnetic Monopole

Ali R. Hadjesfandiari|ArXiv.org|Jan 19, 2007

Geomagnetism and Paleomagnetism Studies3 references3 citations

TL;DR

This paper argues that the magnetic monopole field cannot be represented by a vector potential due to mathematical inconsistencies with the Helmholtz decomposition theorem, challenging the foundational assumption in Dirac's quantization condition. The key result is that the vector potential formalism, essential to predicting quantized magnetic charge, is mathematically invalid for monopoles, implying a fundamental flaw in the standard theoretical framework for magnetic monopoles.

ABSTRACT

This paper shows that based upon the Helmholtz decomposition theorem the field of a stationary magnetic monopole, assuming it exists, cannot be represented by a vector potential. Persisting to use vector potential in monopole representation violates fundamentals of mathematics. The importance of this finding is that the vector potential representation was crucial to the original prediction of the quantized value for a magnetic charge.

Motivation & Objective

• To re-express the magnetic monopole field using rigorous mathematical decomposition theorems.
• To identify the mathematical inconsistency in representing a monopole field via a vector potential.
• To challenge the foundational validity of the vector potential formalism in monopole theory.
• To re-evaluate the theoretical basis for Dirac's prediction of quantized magnetic charge.
• To establish that the vector potential representation violates fundamental mathematical principles for monopole fields.

Proposed method

• Application of the Helmholtz decomposition theorem to analyze the structure of the magnetic field of a stationary monopole.
• Demonstration that the magnetic field of a monopole is irrotational and has a non-zero divergence, violating conditions required for vector potential representation.
• Mathematical proof that a vector potential cannot represent a monopole field due to non-vanishing divergence of the field.
• Use of vector calculus to show that the standard representation fails at the fundamental level of field decomposition.
• Analysis of the implications for gauge theory and the Dirac quantization condition.
• Comparison of the monopole field structure with standard electromagnetic fields to highlight the mathematical incompatibility.

Experimental results

Research questions

• RQ1 Can the magnetic field of a stationary monopole be represented by a vector potential according to the Helmholtz decomposition theorem?
• RQ2 What mathematical conditions must be satisfied for a vector potential to represent a magnetic field, and do they hold for a monopole?
• RQ3 How does the non-zero divergence of the monopole field contradict the existence of a vector potential?
• RQ4 What are the consequences for Dirac's quantization condition if the vector potential representation is invalid?
• RQ5 Is the standard theoretical framework for magnetic monopoles mathematically sound?

Key findings

• The magnetic field of a stationary monopole cannot be represented by a vector potential because it has non-zero divergence, violating the solenoidal condition required for such a representation.
• The Helmholtz decomposition theorem proves that a vector potential cannot describe a field with non-zero divergence, which applies directly to the monopole field.
• Persisting with the vector potential representation for monopoles violates fundamental principles of vector calculus and mathematical consistency.
• The standard derivation of quantized magnetic charge via the vector potential is therefore mathematically invalid.
• The paper concludes that the vector potential formalism is fundamentally incompatible with the existence of magnetic monopoles as classically defined.
• The findings imply a need to re-express monopole theory using alternative mathematical frameworks beyond gauge potentials.