un 2 00 8 Quantum Bound on the Specific Entropy in Strong - Coupled Scalar Field Theory

We discuss the (g0 φ )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 ε (r) d 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) ε (r) d ξ 1−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. PACS numbers: 03.70+k, 04.62.+v e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]