Neuman-Dirichlet tadpoles as new string states and quantum mechanical particle-wave duality from world-sheet t-duality

We consider new objects in bosonic open string theory — ND tadpoles, which have N(euman) boundary conditions at one end of the world-sheet and D(irichlet) at the other, must exist due to s − t duality in a string theory with both NN strings and D-branes. We demonstrate how to interpolate between N and D boundary conditions. In the case of mixed boundary conditions the action for a quantum particle is induced on the boundary. Quantum-mechanical particle-wave duality, a dual description of a quantum particle in either the coordinate or the momentum representation, is induced by world-sheet T-duality. The famous R = αR relation is equivalent to the quantization of the phase space area of a Planck cell ∮ pdx = 2 πh̷. We also introduce a boundary operator Z− a “Zipper” which changes the boundary condition from N into D and vice versa.