Relativistic two-, three- and four-body wave equations in scalar QFT

Abstract

We use the variational method within the Hamiltonian formalism of QFT to derive relativistic two-, three- and four-body wave equations for scalar particles interacting via a massive or massless mediating scalar field (the scalar Yukawa model). The Lagrangian of the theory is reformulated by using Green's functions to express the mediating field in terms of the particle fields. The QFT is then constructed from the resulting reformulated Hamiltonian. Simple Fock-space variational trial states are used to derive relativistic two-, three- and four-body equations. The equations are shown to have the Schrödinger non-relativistic limit, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. Ground-state solutions of the relativistic equations are obtained approximately for various strengths of coupling, for both massive and massless mediating fields, and a comparison of the two-, three- and four-particle binding energies is presented.

Study of virtual annihilation and retardation effects in relativistic two-, four-, and six-body wave equations in scalar QFT

“…The variational methods in Hamiltonian QFT is similar to Schrödinger type of representation of few-body systems, and can easily be applied to systems of more than two bodies [14].…”

“…The virtual annihilation interactions are mostly not included in other works. One should note that the solutions corresponding to four-body relativistic wave equations when the retardation effects are included and virtual annihilation interactions are eliminated have been done previously in [14] and they are added here for the comparison purposes with other cases in Tables 1 and 2. The numerical solutions for the variationally obtained trial wave functions are given in Table 1 for the massive-exchange case with l ¼ 0:15 m and for the massless-exchange case l ¼ 0 in Table 2.…”

Confinement interaction in nonlinear generalizations of the Wick–Cutkosky model

“…The use of many-particle Fock-space components in the variational trial states leads to wave equations with systematically improvable bound state energy levels, as has been shown, for example, on the simple scalar Yukawa model [12,13].…”

“…In order to have some understanding of the properties of the cluster interaction we consider the non-relativistic limit of the equation (5.5), in which case the kernels simplify considerably, and then perform the Fourier transformation into coordinate space. In this representation the equation is simply a Schrödinger equation for the three-particle eigenfunction Ψ(x 1 , x 2 , x 3 ) (see [12]) and eigenenergy ǫ = E − 3m:…”

Bound states in the tachyon exchange potential

“…On the second way one can proceed from the classical action-at-a-distance theory of Wheeler-Feynman type [18][19][20][21][22][23] in which the electromagnetic interaction is replaced by the tachyon one. The third possibility is the partially reduced quantum field theory [24][25][26][27][28][29] which takes advantages of both above approaches. Within this framework quantized matter fields interact via mediating field (the tachyon one in present case), variables of which are eliminated from the description at the classical level.…”