Can I really take the classical field equations at face value in QFT?

Question

To be concrete, let's say I have a relativistic $\phi^4$ theory [with Minkowski signature $(+,-,-,-)$]

$$ \tag{1} \mathcal{L} ~=~ \frac{1}{2} \left ( \partial_{\mu} \phi \partial^{\mu} \phi - m^2 \phi^2\right ) - \frac{\lambda}{4!} \phi^4. $$

The classical equation of motion for $\phi$ is:

$$ \tag{2} (\Box + m^2) \phi + \frac{\lambda}{3!} \phi^3 ~=~ 0. $$

I knew that canonical quantization is basically replacing all Poisson' brackets with (anti-)commutators. From that point of view, I would expect a classical field equation to remain valid as an operator equation even after quantization. Am I wrong?

If I am indeed correct, then specifically to the $\phi^4$ example, does that mean

$$ \tag{3} \left \langle \left [ (\Box + m^2) \phi + \frac{\lambda}{3!} \phi^3 \right ] \mathcal{O} \right \rangle ~=~ 0 $$

for any operator $\mathcal{O}$, in the full interacting theory?

And how do I reconcile this with the path integral picture?

Only the classical paths follow classical equations of motion to the letters. But to quantize a theory, every path is assigned a weight $e^{iS}$, and obviously none of these new inclusions will follow the classical equations. Then, how can the field equations still hold?

Answer 1

OP's equation is a special case of

$$\left< \Omega \left| T\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(x)}\right\}\right| \Omega \right>_J~=~0, \tag{A}$$ where the Euler-Lagrange expression is $\frac{\delta S[\phi;J]}{\delta \phi^{\alpha}(x)}.$

NB: Be aware that the time-ordering usually used in the literature is the covariant time-ordering $T_{\rm cov}$, i.e. time-differentiations inside its argument should be taken after/outside the usual time ordering $T$. This induces quantum corrections/contact terms, so that eq. (A) can be formally rewritten into the standard Schwinger-Dyson equations:

$$ \left< \Omega \left| T_{\rm cov}\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(x)}\right\}\right| \Omega \right>_J~=~i\hbar\left< \Omega \left| T_{\rm cov}\left\{\frac{\delta F[\phi]}{\delta \phi(x)} \right\}\right| \Omega \right>_J ~. \tag{B}$$ One should realize that this is just the beginning into a long discussion about operator ordering ambiguities, time ordering, and quantum correction, cf. e.g. this & this Phys.SE posts.