Spinor with Schroedinger symmetry and non-relativistic supersymmetry

We construct the d-dimensional "half" Schroedinger equation, which is a kind of the root of the Schroedinger equation, from the (d+1)-dimensional free Dirac equation. The solution of the "half" Schroedinger equation also satisfies the usual free Schroedinger equation. We also find that the explicit transformation laws of the Schroedinger and the half Schroedinger fields under the Schroedinger symmetry transformation are derived by starting from the Klein-Gordon equation and the Dirac equation in d+1 dimensions. We derive the 3- and 4-dimensional super-Schroedinger algebra from the superconformal algebra in 4 and 5 dimensions. The algebra is realized by introducing two complex scalar and one (complex) spinor fields and the explicit transformation properties have been found.