Relativistic field-theoretical representation of the Schrödinger equation and field-theoretical generalization of the inverse scattering method

Abstract

The relativistic three-dimensional equation ( 2 E p ′ − 2 E p ) 〈 p ′ / Ψ p 〉 = − ∫ V ( t ) d 3 p ″ 〈 p ″ / Ψ p 〉 with E p = m 2 + p 2 and t = ( E p′ − E p″ ) 2−( p′− p″) 2 is transformed into Schrödinger equation (Δ r + k 2)〈 r|ϕ k 〉=− v( r)〈 r|ϕ k 〉 with E p = k 2/2 m+ m using the Fourier transformation. The potential v( r) and the wave function 〈 r|ϕ k 〉 differ from the corresponding Fourier images of the relativistic potential V( t) and the relativistic wave function 〈 p′| Ψ p 〉 with the kinematical factors only. In the framework of the standard field-theoretical approach the three-dimensional relativistic equations for the NN scattering amplitude with the total NN potential V tot = V( t)+ V nonl are derived. The nonlocal part of this NN potential V nonl is constructed from the πNN vertex functions which are determined from the πN phase shifts. The other part of the NN potential V( t) consists of the π-, σ-, ρ-, ω-, … meson exchange diagrams and for the NN contact (overlapping) terms. In the case of the renormalizable Lagrangians V( t) reproduces exactly the One Boson Exchange (OBE) Bonn model of the NN potential. The inverse scattering problem for this V( t) potential is reduced to the construction of the potential v( r) of the ordinary Schrödinger equation from the NN phase shifts.