# What's the role of field equation in QFT?

For free field theory, it seems the solutions of a field equation are used to give a representation of Poincare group, and the field equation is still satisfied after quantization. However for an interacting theory, such as QED, the field equation:

$$\partial_\nu F^{\nu \mu} ~=~ e \bar{\psi} \gamma^\mu \psi.$$

I don't see any possibility of satisfying this equation using operators since LHS will only act nontrivially on bosonic sector of the Hilbert space and RHS only on fermionic sector, and I don't think perturbation technique can fix this since it's qualitatively not satisfied. Yet nobody seems to be bothered by this, so I'm wondering what's the role of field equation in QFT, especially for interacting ones.

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 In QFT equations of motion are only satisfied in sense of expectation values i.e. expectation value of equation of motion in any given state should vanish. – user10001 Aug 13 '12 at 17:38