Gauss Linking Number and Electro-magnetic Uncertainty Principle

TL;DR

This paper establishes a mathematical relationship between the Heisenberg uncertainty principle for electric and magnetic fluxes and the Gauss linking number, linking quantum uncertainty with topological invariants.

Contribution

It introduces a topological perspective to electromagnetic uncertainty by connecting flux uncertainties to linking numbers and framing of loops.

Findings

Uncertainty between electric and magnetic fluxes relates to Gauss linking number.

Regularization involves assigning a framing to loops.

Self-linking number determines flux uncertainty for a single surface.

Abstract

It is shown that there is a precise sense in which the Heisenberg uncertainty between fluxes of electric and magnetic fields through finite surfaces is given by (one-half $\hbar$ times) the Gauss linking number of the loops that bound these surfaces. To regularize the relevant operators, one is naturally led to assign a framing to each loop. The uncertainty between the fluxes of electric and magnetic fields through a single surface is then given by the self-linking number of the framed loop which bounds the surface.