On the Euclidean Version of the Photon Number Integral

Details On the Euclidean Version of the Photon Number Integral On the Euclidean Version of the Photon Number Integral

2007-10-23

arxiv.org/abs/0710.4276v1

Jnl. Math. Phys. {\bf 49} 023502 (2008)

Physics

Mathematical Physics

13 pages including one figure

Scientific paper

10.1063/1.2836411

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.

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