On the Euclidean Version of the Photon Number Integral

TL;DR

This paper studies a Euclidean photon number integral for smooth curves, demonstrating its conformal invariance and explicitly calculating it for ellipses, supporting the idea that circles minimize this integral among plane curves.

Contribution

It provides a detailed analysis of the Euclidean photon number integral, including explicit evaluation for ellipses and confirmation of its invariance and minimality properties.

Findings

The integral is conformally invariant.

Explicit formula for ellipses: $n_{ellipse}=(\xi^{-1}+\xi)\pi^2$.

The circle minimizes the integral among plane curves.

Abstract

We reconsider the Euclidean version of the photon number integral introduced in ref 1. This integral is well defined for any smooth non-self-intersecting curve in $\R^N$. Besides studying general features of this integral (including it s conformal invariance), we evaluate it explicitly for the ellipse. The result is $n_{ellipse}=(\xi^{-1}+\xi)\pi^2$, where $\xi$ is the ratio of the minor and major axes. This is in agreement with the previous result $n_{circle}=2\pi^2$ and also with the conjecture that the minimum value of $n$ for any plane curve occurs for the circle.