  We prove that any positive solution for the critical nonlinear Hartree
equation $$-\Laplacian\fct{u}{x} -\int_{\R^6} \frac{\abs{\fct{u}{y}}^2 }{
  \abs{x-y}^4 }\odif{y} \,\fct{u}{x}=0,\qtq{} x\in\R^6.$$ is nondegenerate.
Firstly, in terms of spherical harmonics, we show that the corresponding linear
operator can be decomposed into a series of one dimensional linear operators.
Secondly, by making use of the Perron-Frobenius property, we show that the
kernel of each one dimensional linear operator is finite. Finally, we show that
the kernel of the corresponding linear operator is the direct sum of the kernel
of all one dimensional linear operators.
