Abstract
The Golden-Thompson trace inequality which states that Tr e^H+K ≤ Tr e^H e^K has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here we make this G-T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H=Δ or H= -√-Δ +m and K= potential, Tr e^H+(1-u)Ke^uK is a monotone increasing function of the parameter u for 0≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai and Okubo (AHO): Tr X^sY^tX^1-sY^1-t ≤ Tr XY for positive operators X,Y and for 1/2 ≤ s,t ≤ 1 and s+t ≤ 3/2. The obvious conjecture that this inequality should hold up to s+t≤ 1, was proved false by Plevnik. We give a different proof of AHO and also give more counterexamples in the 3/2, 1 range. More importantly we show that the inequality conjectured in AHO does indeed hold in this range if X,Y have a certain positivity property -- one which does hold for quantum mechanical operators, thus enabling us to prove our G-T monotonicity theorem.