Relativistic wave equations of combined three particles and three antiparticles in scalar quantum field theory

Abstract

The recent observation of a positronium molecule (e−e+e−e+) by Cassidy and Mills (D B Cassidy and A P Mills 2007 Nature (London) 449 195) raises interest in other (fermionic) exotic systems. Similarly, the question arises whether larger bosonic systems can be formed. The variational method in a reformulated Hamiltonian formalism of quantum field theory is used to derive relativistic wave equations for a system consisting of three scalar particles and three scalar antiparticles (six-body problem) interacting via a massive or massless mediating scalar field (the scalar Yukawa model). Simple Fock-space variational trial states are used to derive relativistic six-body wave equations. The equations are shown to have the Schrödinger non-relativistic limits, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. Approximate ground-state solutions of the six-body relativistic equations are obtained for various strengths of coupling, for both massive and massless mediating fields.

Relativistic wave equations ofn-body systems of particles and antiparticles of various masses in scalar quantum field theory

“…One of the alternatives to the BS approach is the variational method within the Hamiltonian formalism of quantum field theory (see [5][6][7][8]). This approach has a number of appealing features, among which is that it can be cast in a form similar to the Schrödinger-type description of few-body systems and that it is straightforwardly generalizable to systems of more than two particles [9][10][11].…”

Variational method using basis functions

“…a Coulomb-like potential, is often used for the short-range quark interaction and is motivated by high-momentum studies of quantum chromodynamics [7]. These potentials can be inserted into the non-relativistic Schrödinger equation or, with appropriate modifications, can be used in three-dimensional psuedo-relativistic wave equations [8][9][10][11][12][13][14][15][16] or the fully relativistic bound-state Bethe-Salpeter equation [17][18][19]. Relativistic effects are important because they are needed to describe various phenomenological properties, such as Regge trajectories [20].…”