First of all, I'm not a physicist, I'm mathematics phd student, but I have one elementary physical question and was not able to find answer in standard textbooks.
Motivation is quite simple: let me fix some finite dimensional vector space $V$. Then we can think about Clifford algebra $Cl(V \oplus V^*)$ as algebra of odd differential operators on $\bigwedge V$ i.e. as canonical quantization of the algebra of classical fermionic observables. ($\bigwedge V$ is an analog of functions on a space and $Cl(V \oplus V^*)$ is an analog of a Weil algebra.)
From the other hand, action of a Clifford algebra is used in the construction of the Dirac operator. Thus my question: can one write Lagrangian for classical fermion on a space-time $\mathbb{R}^{3,1}$ such that canonical quantization of such system gives Dirac operator as quantum Hamiltonian?