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Another way to find an equivalence is to express the Green's function via the equation solutions - write down its spectral representation: $$G(x_1,x_2,t_1,t_2)=\sum_n \psi^*_n(x_1)\psi_n(x_2)e^{iE_n(t_1-t_2)}$$ or something like that.

9

First, the term "propagator" is usually defined as the Green's function of the first type, not the second type, i.e. as a solution to the diffential equation $\hat L G = \delta$. At any rate, those definitions are ultimately equivalent – when the details are correctly written down – because the Green's function defined as the correlator in the second ...

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If you plug the propagator into the equation of motion you'll get a $\delta$ function. The second "definition" is just a prescription to calculate the Green's function in a free field theory.

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Consider a harmonic oscillator wherein $$a(t) = a e^{-i \omega t} \quad \text{and} \quad a^\dagger (t) = a^\dagger e^{i \omega t} \, .$$ The derivatives are $$\dot{a}(t) = -i \omega a(t) \quad \text{and} \quad \dot{a}^\dagger (t) = i \omega a ^\dagger (t) \, .$$ Consider the observable $X(t)$ and $Y(t)$ defined by$^{[1]}$ \begin{align} X(t) &\equiv ...

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