NONLINEAR QUANTUM-MECHANICS AS WEYL GEOMETRY OF A CLASSICAL STATISTICAL ENSEMBLE

We derive nonlinear relativistic and non-relativistic wave equations for spin-0 and 1/2 particles. For a suitable choice of coupling constants, the equations become linear and Weyl gauge invariant in the spin-0 case. The Dirac particle is much more subtle. When a suitable gauge is chosen and, when the Compton wavelength of the particle is much larger than Planck's length, we recover the standard Dirac equation. Non-linear corrections to the Schrodinger equation are obtained and these appear as the first-order relativistic corrections to the non-relativistic Hamilton-Jacobi equation. Consequently, we construct nonbilinear homogeneous realizations of an approximate Galilean symmetry. We put forth the idea that not only a modification of quantum mechanics might be necessary in order to accommodate gravity, but quantum mechanics itself might have a geometrical origin with Planck's constant as the coupling between matter and curvature.