In the theory of relativistic wave equations, we derive the Dirac equation and the Klein-Gordon equation by using representation theory of the Poincare algebra.

For example, in this paper

http://arxiv.org/abs/0809.4942

the Dirac equation in momentum space (equation [52], [57] and [58]) can be derived from the 1-particle state of irreducible unitary representation of the Poincare algebra (equation [18] and [19]). The ordinary wave function in position space is its Fourier transform (equation [53], [62] and [65]).

Note that at this stage, this Dirac equation is simply a classical wave equation. i.e. its solutions are classical Dirac 4-spinors, which take values in $\Bbb{C}^{2}\oplus\Bbb{C}^{2}$.

If we regard the Dirac waves $\psi(x)$ and $\bar{\psi}(x)$ as 'classical fields', then the quantized Dirac fields are obtained by promoting them into fermionic harmonic oscillators.

What I do not understand is that when we are doing the path-integral quantization of the Dirac fields, we are, in fact, treating $\psi$ and $\bar{\psi}$ as Grassmann numbers, which are counter-intuitive for me. As far as I understand, we do path-integral by summing over all 'classical fields'. While the 'classical Dirac wave $\psi(x)$' we derived in the beginning are simply 4-spinors living in $\Bbb{C}^{2}\oplus\Bbb{C}^{2}$. How can they be treated as Grassmann numbers instead?

As I see it, physicists are trying to construct a 'classical analogue' of Fermions that are purely quantum objects. For instance, if we start from a quantum anti-commutators

$[\psi,\psi^{\dagger}]_{+}=i\hbar1$ and $[\psi,\psi]_{+}=[\psi^{\dagger},\psi^{\dagger}]_{+}=0$,

then we can obtain the Grassmann numbers in the classical limit $\hbar\rightarrow0$. This is how I used to understand the Grassmann numbers. The problem is that if the Grassmann numbers are indeed a sort of classical limit of anticommuting operators in Hilbert space, then the limit $\hbar\rightarrow0$ itself does not make any sense from a physical point of view since in this limit $\hbar\rightarrow0$, the spin observables vanish totally and what we obtain then would be a $0$, which is a trivial theory.

Please tell me how exactly the quantum Fermions are related to Grassmann numbers.