Vacuum Energy for a Scalar Field With Self-Interaction in (1 1) Dimensions⁺

We calculate the vacuum (Casimir) energy for a scalar field with ϕ4 self-interaction in (1 + 1) dimensions non perturbatively, i.e., in all orders of the self-interaction. We consider massive and massless fields in a finite box with Dirichlet boundary conditions and on the whole axis as well. For strong coupling, the vacuum energy is negative indicating some instability.