On a Treatment of Many-paritcle Systems in Quantum Field Theory

Using the distribution of the virtual particles in the true vacuum and around a real particle, a complete set of states are defined, each of which may be interpreted as representing a prescribed number of real particles. The representation spanned by this set of states is non-orthogonal and the Schrödinger equation transformed into this representation includes the non-Hermitian interaction Hamiltonian. However, the usual definition of the transition probability can be used also in this representation. The equation has the advantage that the energy of the system appears as the difference from the vacuum energy and that the vacuum or the single particle self-energy process no longer occurs. An alternative Hermitian equation is also derived, which, applied to proton-neutron system, gives the non-adiabatic Dancoff potential and the finite single nucleon self-energy term to the second order in the coupling constant. To this order, the divergent vacuum self-energy term reappears, but it is shown to be cancelled by including the contributions of the higher order terms. The simple relations between the true vacuum state, the real one-particle states and the other unbound stationary states are also deduced. All these formulations may be used to treat the many-particle systems on the knowledge of the solutions of the one-particle systems.