Positivity bounds from thermal field theory entropy

We present an approach to deriving positivity bounds on effective field theories by analyzing the thermodynamic behavior of thermal quantum field systems. Focusing on scalar theories with higher-dimensional operators, we compute the finite-temperature entropy using thermal field theory techniques. We argue that consistency with fundamental thermodynamic principles--specifically, the expectation that entropy increases with the introduction of new degrees of freedom--imposes nontrivial constraints on Wilson coefficients. In particular, we show that the coefficient of the leading dimension-8 operator must be strictly positive. This thermodynamic perspective offers an alternative to traditional S-matrix-based derivations of positivity bounds and provides a complementary perspective into the interplay between entropy, unitarity, and causality in quantum field theory.