Franz Gross (William & Mary, USA)

Paving the Relativistic Way: Past, Present and Future

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Vladimir Karmanov (Lebedev Phys. Ins., Russia)

The Bethe-Salpeter Equation and the Nakanishi representation

We present a method of nding the Bethe-Salpeter (BS) amplitude in Minkowski space.
The method is based on representation of the BS amplitude (as a function of two variables) via a double integral from other function g( ; z) (Nakanishi representation). This integral incorporates analytically the singularities of the BS amplitude, that simplies  nding g( ; z). We derive an equation for g( ; z) which is well solvable numerically (in contrast to initial BS equation). Then, knowing g( ; z), we restore the BS amplitude both in Minkowski and in Euclidean spaces (and in the complex plane, if needed).
The method, initially developed for spinless particles, after recent modications and improvements, was applied to two-fermion system. For a test, we compare binding energies, found by our method, with our solution in Euclidean space and with the Euclidean solutions given in literature. For any system (scalars or fermions) these three sets of binding energies coincide with each other with high accuracy.

Wayne Polyzou (Iowa Univ., USA)

From field theory to the Poincare' invariant three-body problem

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Tobias Frederico (ITA, Sao Jose` dos Campos, Brazil)

Projecting the Bethe-Salpeter equation onto the light-front

The projection of the four-dimensional two-body Bethe-Salpeter equation to the threedimensional light-front hypersurface using the quasi-potential formalism will be discussed with relation to the valence component of the wave function. Examples for mass operator equation for two-bosons and two-fermions systems will be given. The conserved electromagnetic current operator acting on the valence component of the light-front wave function
will be also shown. The generalization of the formalism to three-particles will be discussed.

Noriyoshi Ishii (Tokyo Univ., Japan)

Lambda-Nucleon and Nucleon-Nucleon interactions on the Lattice

Recent lattice study of nuclear forces is reviewed. Scattering phase shift is an important experimental observable for two particle system. In lattice QCD, phase shifts are calculated from long distance behavior of Bethe-Salpeter(BS) wave functions by Luscher's  nite volume method. For applications to nuclear physics of multi-nucleon system, it is more advantageous to keep the information of phase shifts in the form of potentials. We therefore extend the method to generate the potentials from BS wave functions. These potentials are faithful to scattering phase shift by construction, because they can reproduce the long distance behaviors of BS wave functions. The method was rst applied to the central force in NN system. It is now applied to many objects, such as tensor force,
hyperon forces, energy dependence of nuclear force, and investigations of the repulsive core at short distance.