Time Dependent Effective Actions at Finite Temperature ∗

I study derivative expansions of effective actions at finite temperature, illustrating how the standard methods are badly defined at finite temperature. I then show that by setting up the initial conditions at a finite time, these problems are solved. 1.1 Zero temperature I will work with a simple model of two real fields φ and η 1 L[φ, η] = 1 2 η∆ −1 η − 1 2 gφη 2 + L 0 [φ], (1) and integrating out the η field gives