On Thermal Entropy in Quantum Field Theory

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. Such divergences cannot be reabsorbed by local counterterms although, with a rather unusual procedure, they can be subtracted from the final result. A key issue here is which set of quantum degrees of freedom, i.e. which quantum subsystem of the theory, is associated with the region of space under consideration. In the case of a free scalar field in d+1 spacetime dimensions we show that by using an alternative localization scheme – related to the Newton-Wigner position operator – the entropy of a thermal state is free of divergences and has a perfectly sound thermodynamic behavior. In the high temperature/big volume limit our results are consistent with the standard calculation, once the divergent contribution is 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.