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

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