  We construct an effective low-energy Hamiltonian from the classical action
via Monte Carlo with importance sampling. We use Monte Carlo (i) to compute
matrix elements of the transition amplitude and (ii) to construct
stochastically a basis. The MC Hamiltonian allows to obtain energies and wave
functions of low-lying states. It allows also to compute thermodynamical
observables in some temperature window (starting from temperature zero). We
present examples from lattice field theory (Klein-Gordon model).
