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

The world-line (Fock-Feynman-Schwinger) representation is used for quarks in an arbitrary (vacuum and valence gluon) field to construct the relativistic Hamiltonian. After averaging the Green’s function of the white \(q\bar q\) system over gluon fields, one obtains the relativistic Hamiltonian, which is a matrix in spin indices and contains both positive and negative quark energies. The role of the latter is studied using the example of the heavy-light meson and the standard einbein technique is extended to the case of the matrix Hamiltonian. Comparison with the Dirac equation shows good agreement of the results. For an arbitrary \(q\bar q\) system, the nondiagonal matrix Hamiltonian components are calculated through hyperfine interaction terms. A general discussion of the role of negative-energy components is given in conclusion.