Entropy Constraint on Effective Field Theory

In effective field theory, the positivity bounds of higher derivative operators are derived from analyticity, causality, and unitarity. We show that the positivity bounds on some effective field theories can be derived for a class of UV theories with a specific sign of interactions between heavy and light degrees of freedom by the non-negativity of relative entropy. For such a class of theories, we prove that the interactions increase thermodynamic entropy at fixed energy and charge, which is intimately connected with the extremality relations of black holes exhibiting Weak-Gravity-Conjecture. The entropy constraint is a consequence of the Hermiticity of Hamiltonian, and any theory violating the non-negativity of entropy would not respect the second law of thermodynamics.Comment: 17 pages, 2 figur