Quantum bound on the specific entropy in strongly coupled scalar field theory

We discuss the (g0φp)d self-interacting scalar field theory, in the strong-coupling regime. We assume the presence of macroscopic boundaries confining the field in a hypercube of side L. We also consider that the system is in thermal equilibrium at temperature β-1. For spatially bounded free fields, the Bekenstein bound states that the specific entropy satisfies the inequality (S/E)<2πR, where R stands for the radius of the smallest sphere that circumscribes the system. Employing the strong-coupling perturbative expansion, we obtain the renormalized mean energy E and entropy S for the system up to the order (g0)-(2/p), presenting an analytical proof that the specific entropy also satisfies in some situations a quantum bound. Defining εd(r) as the renormalized zero-point energy for the free theory per unit length, the dimensionless quantity ξ=(β/L) and h1(d) and h2(d) as positive analytic functions of d, for the case of high temperature, we get that the specific entropy satisfies (S/E)<2πR(h1(d)/h2(d))ξ. When considering the low-temperature behavior of the specific entropy, we have (S/E)<2πR(h1(d)/εd(r))ξ1-d. Therefore the sign of the renormalized zero-point energy can invalidate this quantum bound. If the renormalized zero-point energy is a positive quantity, at intermediate temperatures and in the low-temperature limit, there is a quantum bound