Entanglement entropy of a Maxwell field on the sphere

Abstract

We compute the logarithmic coefficient of the entanglement entropy on a sphere for a Maxwell field in $d=4$ dimensions. In spherical coordinates the problem decomposes into one dimensional ones along the radial coordinate for each angular momentum. We show the entanglement entropy of a Maxwell field is equivalent to the one of two identical massless scalars from which the mode of $l=0$ has been removed. This shows the relation $c_{\log }^M=2(c_{\log }^S-c_{\log }^{S_{l=0}})$ between the logarithmic coefficient in the entropy for a Maxwell field $c_{\log }^M$, the one for a $d=4$ massless scalar $c_{\log }^S$, and the logarithmic coefficient $c_{\log }^{S_{l=0}}$ for a $d=2$ scalar with Dirichlet boundary condition at the origin. Using the accepted values for these coefficients $c_{\log }^S=-1/90$ and $c_{\log }^{S_{l=0}}=1/6$ we get $c_{\log }^M=-16/45$, which coincides with Dowker's calculation, but does not match the coefficient $-\frac{31}{45}$ in the trace anomaly for a Maxwell field. We have numerically evaluated these three numbers $c_{\log }^M$, $c_{\log }^S$ and $c_{\log }^{S_{l=0}}$, verifying the relation, as well as checked they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.

18 pages, 5 figures

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