Proof of the quantum bound on specific entropy for free fields

The quantum bound on specific entropy for free fields states that the ratio of entropy S to total energy E of a system with linear dimension R cannot be larger than 2πR/ħc. Here we prove this bound for a generic system consisting of a noninteracting quantum field in three space dimensions confined to a cavity of arbitrary shape and topology. S(E) is defined as the logarithm of the number of quantum states (including the vacuum) accessible up to energy E. An integral equation is derived which relates an upper bound on S(E) to the one-particle energy spectrum in the given cavity. The spectrum may always be bounded from above by a power law in energy whose proportionality constant is the ζ function for the spectrum of the cavity. This last is not calculable in the generic case, but it is here proven to be bounded by that for a sphere which circumscribes the actual cavity. Thus the one-particle spectrum for all cavities that fit inside a given sphere is bounded by a generic formula which can be computed given the field. With the help of this result the integral equation is solved for a fictitious system whose entropy must bound that of the actual system. The resulting bound on S(E)/E proves to be smaller than 2πR/ħc with R interpreted as the radius of the enveloping sphere.