  Standard entropy calculations in quantum field theory, when applied to a
subsystem of definite volume, exhibit area-dependent UV divergences that make a
thermodynamic interpretation troublesome. In this paper we define a
renormalized entropy which is related with the Newton-Wigner position operator.
Accordingly, whenever we trace over a region of space, we trace away degrees of
freedom that are localized according to Newton-Wigner localization but not in
the usual sense. We consider a free scalar field in d+1 spacetime dimensions
prepared in a thermal state and we show that our entropy is free of divergences
and has a perfectly sound thermodynamic behavior. In the high temperature/big
volume limit our results agree with the standard QFT calculations once the
divergent contributions are subtracted from the latter. In the limit of low
temperature/small volume the entropy goes to zero but with a different
dependence on the temperature.
