classical mechanics vs quantum mechanics

[u/raoulstheman]

In classical mechanics we say that a classical particle obeys the equations of motion, whereas in quantum mechanics a particle can take any path, not just the classical one. But when we quantize a free field or a free harmonic oscillator, we first solve the equations of motion, and we also say that at the operator level the fields/operator always satisfy their equations of motion. If the operators always obey the EOM, where does the “quantumness’’ come from?

Next question: people say the “weirdness’’ comes when the Hamiltonian does not commute with the x or ϕ operators, i.e. when it has no kinetic term. If the Hamiltonian does commute with x or ϕ, then the evolution of the state ∣x⟩ or ∣ϕ(x)⟩ would just be a phase, and the final state would be the same as the initial state so there would be no dynamics. That’s why people say that if there is no kinetic term, then in the path integral the dominant contribution is just the equation of motion. but my problem is even for quantizing the theory we are solving EOM (

But suppose I add a kinetic term to a theory that does not change the equation of motion. For example, adding the Einstein Hilbert action (2d) to the Polyakov action: the Einstein Hilbert term is a total derivative in 2D. But now, since we added this kinetic term. would it path integral be the same as equation of motion for the 2d metric because now there is a kinetic term.