 
FIZIKA B 13 (2004) 2, 625644
RELATIVISTIC FIELDTHEORETICAL FORMULATION OF THE THREEDIMENSIONAL
EQUATIONS FOR THE THREE FERMION SYSTEM
A. I. MACHAVARIANI
Joint Institute for Nuclear Research, Moscow Region, 141980 Dubna,
Russia
High Energy Physics Institute of Tbilisi State University, University Str. 9, 380086
Tbilisi, Georgia
Received 30 October 2003; Accepted
30 October 2004
Online 22 November 2004
A new kind of the relativistic equations for the threefermion systems are suggested.
These equations are derived in the framework of the standard fieldtheoretical Smatrix
approach in the timeordered threedimensional form. Therefore the corresponding
relativistic covariant equations are threedimensional from the beginning and the
considered formulation is free of the ambiguities which appear due to a threedimensional
reduction of the fourdimensional BetheSalpeter 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 threebody
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 onmass 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 threebody Faddeev equations.
Unlike these equations, the suggested threebody equations have the form of the
LippmannSchwingertype equations with the connected potential. In addition, the
microscopical potential of the suggested equations contains the contributions from the
threebody forces and from the particle creation (annihilation) mechanism on the external
particles. The structure of the threebody forces, appearing in the considered
fieldtheoretical formulation, is analyzed.
PACS numbers: 11.80.a, 11.80.m, 11.10z, 25.80.Hp
UDC 539.128.3
Keywords: threefermion systems, relativistic equations, Faddeev equations,
LippmannSchwingertype equations
