Energy Levels of Interacting Fields in a Box

We study the influence of boundary conditions on energy levels of interacting fields in a box and discuss some consequences when we change the size of the box. In order to do this we calculate the energy levels of bound states of a scalar massive field χ interacting with another scalar field φ through the lagrangian Lint = 3 2gφ2χ2 in an one-dimensional box, on which we impose Dirichlet boundary conditions. We have found that the gap between the bound states changes with the size of the box in a non-trivial way. For the case the masses of the two fields are equal and for large box the energy levels of Dashen-Hasslacher-Neveu (DHN model) (Dashen et al, 1974) are recovered and we have a kind of boson condensate for the ground state. Below to a critical box size L ∼ 2.93 2 √ 2 M the ground state level splits, which we interpret as particle-antiparticle production under small perturbations of box size. Below another critical sizes (L ∼ 6 10 2 √ 2 M ) and (L ∼ 1.71 √ 2 M ) of the box, the ground state and first excited state merge in the continuum part of the spectrum.