In the presence of an idealized potential on two parallel planes represented by two one-dimensional delta -functions at x3=-d/2 and x3=+d/2 the authors discuss the Feynmann propagators for relativistic scalar and spinor fields. These propagators take into account bound states, scattering states and resonances. The Casimir energy for this configuration is calculated. For massive fields the Casimir force decreases exponentially with rising distances. In the scalar case they find an attractive force and in the spinor case a repulsive force. An attempt to treat the same problem for a massive scalar field using nonrelativistic quantum field theory leads to a vanishing Casimir force.