On the Euclidean Version of the Photon Number Integral

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.