Nonlinear quantum mechanics as weyl geometry of a classical statistical ensemble

Abstract

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. Nonlinear corrections to the Schrödinger 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 anapproximate 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.