Two mass relations for mesons from string-quark duality

We generalize a self-consistency equation derived previously by us for the Nambu string to strings carrying SU(6) quantum numbers.The self-consistency equations are derived by considering the propagation of a string with the quantum numbers of the physical vacuum. Since we can map the world sheet that the string sweeps out onto rectangles, and the wave function of the physical vacuum is a constant, the boundary conditions are the same on all four sides of the rectangles. We can then calculate the propagator in two ways and this leads to the self-consistency equations. For the strings carrying SU(6) quantum numbers we consider the propagator o strings with the quantum numbers of the physical vacuum all along the string except for having quark quantum numbers either attached or removed from the ends. We can solve these self-consistency equations in a certain approximation and for this case they lead to mass formulae for mesons that are well satisfied in nature.