  Fourier and fractional-Fourier transformations are widely used in theoretical
physics. In this paper we make quantum perspectives and generalization for the
fractional Fourier transformation (FrFT). By virtue of quantum mechanical
representation transformation and the method of integration within normal
ordered product (IWOP) of operators, we find the key point for composing FrFT,
and reveal the structure of FrFT. Following this procedure, a full family of
generalized fractional transformations are discovered with the usual FrFT as
one special case. The eigen-functions of arbitrary GFrT are derived explicitly.
