Spinor with Schr\" odinger symmetry and non-relativistic supersymmetry

We construct the $d$ dimensional “half” Schr"odinger equation, which is a kind of the root of the Schr"odinger equation, from the $d+1$ dimensional free Dirac equation. The solution of the “half” Schr"odinger equation also satisfies the usual free Schr"odinger equation. We also find that the explicit transformation laws of the Schr"odinger and the half Schr"odinger fields under the Schr"odinger 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-Schr"odinger 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.