  It has long been conjectured that the entropy of quantum fields across
boundaries scales as the boundary area. This conjecture has not been easy to
test in spacetime dimensions greater than four because of divergences in the
von Neumann entropy. Here we show that the Renyi entropy provides a convergent
alternative, yielding a quantitative measure of entanglement between quantum
field theoretic degrees of freedom inside and outside hypersurfaces. For the
first time, we show that the entanglement entropy in higher dimensions is
proportional to the higher dimensional area. We also show that the Renyi
entropy diverges at specific values of the Renyi parameter (q) in each
dimension, but this divergence can be tamed by introducing a mass of the
quantum field.
