A non-relativistic theory of quantum mechanics and gravity with local
  modulus symmetry
  Inspired by the similarities between quantum field theory and general
relativity, in that each theory encompasses two universal constants and a local
symmetry, we set out to construct a non-relativistic theory of quantum
mechanics and gravity based upon two assumptions: quantum system remains
invariant to local modulus transformation, and physical laws reduce to those of
conventional quantum mechanics in small enough region, i.e. a modified version
of the equivalence principle. Imposing local modulus symmetry brings a number
of changes, including the replacement of the complex conjugate of the wave
function by a new scalar function with the same phase but different modulus,
and particle momentum operator built on covariant derivative, with a connection
of purely imaginary velocity field that is identified as the gravitational
escape velocity. Three quantum metric functions are defined to signify the
kinematic change of quantum state brought by gravity. The modified equivalence
principle enables us to relate the escape velocity field with the quantum
metric functions. Equation of motion and field equation that are covariant to
local modulus transformation are constructed. New features in these equations
offer potential mechanisms to account for the dark energy, the mass
discrepancies in the universe, and the quantum state reduction of macroscopic
objects.
