Energy Levels of Interacting Fields in a Box

Abstract

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 $\chi$ interacting with another scalar field $\varphi$ through the lagrangian ${\mathcal{L}}_{int}=3/2g{\varphi }^2{\chi }^2$ 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 $L\sim 2.932\sqrt{2}/M$ the ground state level splits, which we interpret as particle-antiparticle production under small perturbations of box size. Below another critical sizes $(L\sim 6/102\sqrt{2}/M)$ and $(L\sim 1.712\sqrt{2}/M)$ of the box, the ground state and first excited state merge in the continuum part of the spectrum.

11 pages, LaTex, 3 ps-figs. to appear in Int.J.Theor.Phys. (1999)