Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 69 / 1 / a / Solution

Statistical homogeneity makes the two-point correlation depend on the displacement $\mathbf{u}=\mathbf{r}-{\mathbf{r}}^{\prime }$. Substituting the Fourier transforms and changing variables from $(\mathbf{r},{\mathbf{r}}^{\prime })$ to $(\mathbf{u},{\mathbf{r}}^{\prime })$ givesThe last integral is a Dirac delta function, soThe plus sign occurs because neither Fourier transform has undergone complex conjugation. For a real magnetic field, the version with ${\hat{B}}_j({\mathbf{k}}^{\prime }{)}^{\ast }$ instead has $\delta (\mathbf{k}-{\mathbf{k}}^{\prime })$. Homogeneity determines dependence on the displacement vector; dependence only on its norm needs statistical isotropy as well.