Soft Modes Contribution into Path Integral

Abstract

A method for nonperturbative path integral calculation is proposed. Quantum mechanics as a simplest example of a quantum field theory is considered. All modes are decomposed into hard (with frequencies $ω^2 >ω^2_0$) and soft (with frequencies $ω^2 <ω^2_0$) ones, $ω_0$ is a some parameter. Hard modes contribution is considered by weak coupling expansion. A low energy effective Lagrangian for soft modes is used. In the case of soft modes we apply a strong coupling expansion. To realize this expansion a special basis in functional space of trajectories is considered. A good convergency of proposed procedure in the case of potential $V(x)=λx^4$ is demonstrated. Ground state energy of the unharmonic oscillator is calculated.