[Physics] Lagrangian for a free Dirac field equal zero

Question

The Lagrangian (density) for a free Dirac field is given as $${\mathcal L}_\mathrm{Dirac} = \bar{\psi} \left( i \gamma^{\mu} \partial_{\mu} – m \right) \psi,$$ but given that $\psi$ obeys the Dirac equation, $$\left( i \gamma^{\mu} \partial_{\mu} – m \right) \psi = 0$$
Doesn't this mean the Lagrangian (density) is zero?

Answer 1

What you have found out is that the lagrangian evaluated at the solution of the equation of motion is constant (and equal to zero).

However the lagrangian density is defined for a generic field configuration, not just the solution of the equation of motion.

Since eoms are found by stabilising the action $S = \int \mathcal{L}$, considering only the solution of the eoms in place of a generic field configuration is like considering a function $f(x)$ only at its stationary point; in general it is not sufficient because you need to know how the funciton behaves in a whole neighbourhood of such points.