I learn the chiral anomaly by Fujikawa method. The text book "Path Integrals and Quantum Anomalies, Kazuo Fujikawa", in the page 151, says that
…one can define a complete orthonormal set $\{\phi_n\}$and the expansion of fermionic variables $$D\phi_n(x)=\lambda_n\phi_n(x)$$ $$\int d^2x \phi_n^\dagger(x)\phi_m(x)=\delta_{nm}$$ $$\phi(x)=\sum_na_n\phi_n(x)$$ where $a_n$ is the Grassmann numbers.
Here $\phi$ is any fermionic matter field, and the $D$ is the Dirac operator twisted by gauge field. Since it is self-dual, the eigenvalues $\lambda_n$ are real numbers.
My question is, what is the reason why the expansion $\phi(x)=\sum_na_n\phi_n(x)$ with Grassmann coefficients holds? In mathematics, fermion (interacted by gauge field) $\phi$ is interpreted by a section of the Dirac bundle $S\otimes E$ that is the spinor bundle $S$ twisted by the vector bundle $E$. Then the orthonormal basis $\phi_n$ is complete in the $L^2$-space $L^2(S\otimes E)$. So I understand the expansion $\phi(x)=\sum_nc_n\phi_n(x)$ with complex coefficient $c_n$, in the sense of $L^2$-norm. But why Grassmann numbers? I know that the bundle $S\otimes E$ has a Grassmann algebra module structure, canonically. Possibly, do the $\phi_n$ form a basis with respect to the Grassmann module? Or, are there operators $a_n$ such that $a_n\phi_n=c_n\phi_n$ and that $\{a_n,a_m\}=0$?