The aim of this paper is to find out the operator for generating fractional Fourier transform in Hermitian polynomial theory with the operator as the argument， and to incorporate fractional Fourier transform into quantum theory. The role of coordinate-momentum exchanging operator is explored in playing FFrT's addition rule. In the whole derivation the generalized generating function formula of operator Hermitian polynomials and the integration method within ordered product of operators are used. The core of operator Hermite polynomial theory is the operator identity $H_n\left(Q\right)=:{\left(\mathrm{2}Q\right)}^n:$， which turns the operator of complex special function into power series in normal ordering， as a result，this greatly simplifies calculations.