Projection Operator Approach to Langevin Equations in $\phi^4$ Theory

We apply the projection operator method (POM) to $\phi^4$ theory and derive both quantum and semiclassical equations of motion for the soft modes. These equations have no time-convolution integral term in sharp contrast with other popular results obtained from the influence functional method (IFM) and the closed time path method (CTP). However, except for the fluctuation force terms, these equations are similar to the corresponding equation in IFM with the linear harmonic approximation which was introduced to remove the time-convolution integral. The quantum equation of motion in POM can be regarded as a kind of quantum Langevin equation where the fluctuation force is given in terms of the operators of the hard modes. The operators are then replaced with c-numbers by some procedure to obtain a semiclassical Langevin equation. It is pointed out that there are significant differences between the fluctuation forces introduced in this paper and that in IFM. Arbitrariness in the definition of the fluctuation force in IFM is also discussed