The matrix Hamiltonian for hadrons and the role of negative - energy components

The world-line (Fock-Feynman-Schwinger) representation is used for quarks in arbitrary (vacuum and valence gluon) field to construct the relativistic Hamiltonian. After averaging the Green's function of the white q ¯ q system over gluon fields one obtains the relativistic Hamiltonian, which is matrix in spin indices and contains both positive and negative quark energies. The role of the latter is studied in the example of the heavy-light meson and the standard einbein technic is extended to the case of the matrix Hamiltonian. Comparison with the Dirac equation shows a good agreement of the results. For arbitrary q ¯ q system the nondiago-nal matrix Hamiltonian components are calculated through hyperfine interaction terms. A general discussion of the role of negative energy components is given in conclusion. 1 Introduction The quest for the Hamiltonian which contains main features of QCD-confinement and Chiral Symmetry Breaking (CSB) exists from the very beginning , when fundamental field-theoretical QCD Hamiltonians, have been constructed in different gauges [1]. Unfortunately (nonlocal) confinement cannot be seen in these local FTh Hamiltonians and for practical purposes another sort of Hamiltonians – Effective Hamiltonians (EH) have been modelled containing minimal relativity and string-type potentials [2]. A lot of 1