  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.
