Entanglement entropy of a Maxwell field on the sphere

Abstract

We compute the logarithmic coefficient of the entanglement entropy on asphere for a Maxwell field in <span class="MathJax_Preview">d=4</span> dimensions. In spherical coordinates theproblem decomposes into one dimensional ones along the radial coordinate foreach angular momentum. We show the entanglement entropy of a Maxwell field isequivalent to the one of two identical massless scalars from which the mode of<span class="MathJax_Preview">l=0</span> has been removed. This shows the relation <span class="MathJax_Preview">c^M_{log}=2(c^S_{log}-c^{S_{l=0}}_{log})</span> between the logarithmic coefficient in theentropy for a Maxwell field <span class="MathJax_Preview">c^M_{log}</span>, the one for a <span class="MathJax_Preview">d=4</span> massless scalar<span class="MathJax_Preview">c_{log}^S</span>, and the logarithmic coefficient <span class="MathJax_Preview">c^{S_{l=0}}_{log}</span> for a <span class="MathJax_Preview">d=2</span>scalar with Dirichlet boundary condition at the origin. Using the acceptedvalues for these coefficients <span class="MathJax_Preview">c_{log}^S=-1/90</span> and <span class="MathJax_Preview">c^{S_{l=0}}_{log}=1/6</span>we get <span class="MathJax_Preview">c^M_{log}=-16/45</span>, which coincides with Dowker´s calculation, but doesnot match the coefficient <span class="MathJax_Preview">-rac{31}{45}</span> in the trace anomaly for a Maxwellfield. We have numerically evaluated these three numbers <span class="MathJax_Preview">c^M_{log}</span>,<span class="MathJax_Preview">c^S_{log}</span> and <span class="MathJax_Preview">c^{S_{l=0}}_{log}</span>, verifying the relation, as well aschecked they coincide with the corresponding logarithmic term in mutualinformation of two concentric spheres.