Wave-particle duality and superconductivity in Weyl-Dirac theories

In the Gregorash-Papini-Wood approach to the Weyl-Dirac theory, some superconducting properties of the wave equation are analysed. In a special topology, for a constant modulus of the wavefunction, the phase gradient verifies an effect analogous to the gravitational Meissner effect (there is no momentum transfer between the wave and the particle, and the wave-particle duality is achieved by self-tunnelling). Through the gravitational Meissner effect the oscillation modes of a particle are cnoidal, and in a superconducting cylinder a quantum fluid moving along its axis is self-focusing. The matching between quantum hydrodynamics and wave mechanics is similar to the quantization of the gravitational fluxoid, and their unmatching is analogous to the gravitational analogue of the Aharonov-Bohm effect. The absence of the gravitational Meissner effect implies a permanent transfer of momentum between the wave and the particle such as the interior of the particle structured like a lattice. By defining a free energy we show that the wave-particle duality is equivalent to a normal-superconducting phase transition. In this context by generalizing the one-dimensional solution of the WD equation to the two-dimensional case, one can induce a superconducting state by in-phase oscillations of a vortex lattice.