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arXiv · 0710.4276

On the Euclidean Version of the Photon Number Integral

S. Ruijsenaars·L. Stodolsky
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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 $n_{ellipse}=(ξ^{-1}+ξ)π^2$, where $ξ$ is the ratio of the minor and major axes. This is in agreement with the previous result $n_{circle}=2π^2$ and also with the conjecture that the minimum value of $n$ for any plane curve occurs for the circle.

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S. Ruijsenaars, L. Stodolsky. 2007-10-23. On the Euclidean Version of the Photon Number Integral. https://doi.org/10.1063/1.2836411

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