 Solution

The canonical anticommutation relations arewith the mixed and equal-type anticommutators zero. Insert the mode expansion of a Dirac field. The phases cancel at equal time, and a givesSince $\bar{\psi }{\gamma }^0={\psi }^{†}$, the requested componentwise answer isThis is the Equal-time canonical anticommutator of a Dirac field: the canonical momentum conjugate to ${\psi }_{\alpha }$ is ${\pi }_{\alpha }=i{\psi }_{\alpha }^{†}$. It is the fermionic counterpart of the canonical field canonical commutation relation, ensuring the correct local canonical quantization. For this first-order Lagrangian density, the conjugate variables are constrained; the displayed relation is the standard result after handling that constraint. It does not assert a nonzero anticommutator between the fields at separated spatial points.