Euclidean Methods on Dirac Quantization and Gauge Representation

TL;DR

This paper presents a geometric approach to U(1) gauge theory with Dirac monopoles, deriving the Dirac quantization condition without assuming single-valued gauge representations, emphasizing the role of boundary conditions and phase gauge transformations.

Contribution

It introduces a Euclidean geometric formulation of U(1) gauge theory on Dirac monopoles, deriving the quantization condition from boundary conditions rather than canonical assumptions.

Findings

Dirac quantization condition derived geometrically from boundary conditions.

Gauge representation can be multi-valued without violating physical consistency.

Phase gauge transformation preserves the form of the gauge equation.

Abstract

Geometrical interpretation on U(1) gage theory of Dirac monopole, introduced here from the line integral\cite{Brandt} form of vector potentials, shows the gauge representation be multi-valued. In this paper, we construct Euclidean form of U(1) gauge theory on Dirac monopole based on the geometrical interpretation, derive the Dirac quantization condition geometrically. Even several theoretical methods which leads to Dirac quantization $eg=l/2$ has been developed, these are only possible under the postulate that gauge representation is single-valued. Using the line integration representation of vector potential, we derived the gauge equation which satisfies under the `phase gauge transformation' ($\mathbf{A}\to\mathbf{A}'=\mathbf{A}, \ \psi\to\psi'=e^{ie\Lambda}\psi$), also showed that the Dirac quantization condition can be derived by the geometrical boundary condition on this gauge equation, without any supposing of canonical quantization condition. We finally explain geometrical meaning of constant phase of gauge and how the Dirac quantization holds single valued gauge representation.