FIZIKA B 13 (2004) 2, 625-644
RELATIVISTIC FIELD-THEORETICAL FORMULATION OF THE THREE-DIMENSIONAL
EQUATIONS FOR THE THREE FERMION SYSTEM
A. I. MACHAVARIANI
Joint Institute for Nuclear Research, Moscow Region, 141980 Dubna,
High Energy Physics Institute of Tbilisi State University, University Str. 9, 380086
Received 30 October 2003; Accepted
30 October 2004
Online 22 November 2004
A new kind of the relativistic equations for the three-fermion systems are suggested.
These equations are derived in the framework of the standard field-theoretical S-matrix
approach in the time-ordered three-dimensional form. Therefore the corresponding
relativistic covariant equations are three-dimensional from the beginning and the
considered formulation is free of the ambiguities which appear due to a three-dimensional
reduction of the four-dimensional Bethe-Salpeter equations. The solutions of the
considered equations satisfy automatically the unitarity condition, and for the leptons,
these equations are exactly gauge invariant even after the truncation over the
multiparticle (n > 3) intermediate states. Moreover, the form of these three-body
equations does not depend on the choice of the model Lagrangian and it is the same for the
formulations with and without quark degrees of freedom. The effective potential of the
suggested equations is defined by the vertex functions with two on-mass shell particles.
It is emphasized that these INPUT vertex functions can be constructed from experimental
data. Special attention is given to the comparison with the three-body Faddeev equations.
Unlike these equations, the suggested three-body equations have the form of the
Lippmann-Schwinger-type equations with the connected potential. In addition, the
microscopical potential of the suggested equations contains the contributions from the
three-body forces and from the particle creation (annihilation) mechanism on the external
particles. The structure of the three-body forces, appearing in the considered
field-theoretical formulation, is analyzed.
PACS numbers: 11.80.-a, 11.80.-m, 11.10-z, 25.80.Hp
Keywords: three-fermion systems, relativistic equations, Faddeev equations,