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Given a linear operator $\hat{L}$ and $x \in \mathbb{R}^n$, a Green's function of $\hat{L}$ is a function that satisfies $$\hat{L}_xG(x,x^\prime) = \delta^{(n)}(x-x^\prime)$$ A "propagator" $G(x,x^\prime,t,t^\prime)$ of a linear PDE is essentially the integral kernel of the evolution operator. That is, we assign one variable to be the "time" variable $t$ and then if $\psi(x,t)$ solves the PDE then for $t>t^\prime$ $$\psi(x,t) = \int_{\mathbb{R}^{n-1}} G(x,x^\prime,t,t^\prime) \psi(x^\prime,t^\prime) d^{n-1}x^\prime$$ (where the dependence on $t^\prime$ on the RHS drops out). The Green's function and the propagator seem like very different objects but it turns out that if our PDE is $$\hat{L}\psi(x,t) = 0$$ where $$\hat{L} = i \frac{\partial}{\partial t} - \hat{H}_x$$ the two objects are the same. This is a standard result in quantum mechanics. This seems like a big "coincidence" and the proof relies heavily on the above form that $\hat{L}$ takes. Is there is a general relationship between these two objects? Could there be situations in which the two are different?

UtilityMaximiser
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    You are confusing yourself with the sloppy notation. This is one of the most frequent questions/misunderstandings in QM, and has been answered in PSE again and again. Indeed, the retarded Green's function of the inhomogeneous equation is a step function of times multiplying the kernel of the homogeneous one, which was defined for that very purpose. – Cosmas Zachos Dec 23 '18 at 20:35
  • I haven't made the mistake you think I've made because I separated out the time variable and required $t > t^\prime$. – UtilityMaximiser Dec 24 '18 at 23:14
  • My point is your first equation might have been $\hat L (\theta (t-t')G(x,t;x',t'))=\delta(t-t')\delta(x-x') $ in terms of the propagator, usually called K, and the answer would have been self-explanatory. Because the Green function and the propagator coincide for forward propagation, authors conflate the two, placing their readers in peril. – Cosmas Zachos Dec 25 '18 at 14:57

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The answer to my question is Duhamel's principle.

UtilityMaximiser
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  • A hint for the relation between Green functions and propagator is given here, eq. (10). One could also formally write a PDE as an ODE with value in a space of functions of time only: then the method of variation of constants gives exactly the Duhamel principle... – Noix07 Jan 31 '22 at 16:49