In real space and in equilibrium, the retarded Green function $G_{i,j}(t)$ gives the amplitude probability for propagation of a particle transfering from j to i during time t. The Laplace tranformation of this Green function gives $G_{i,j}(z)$ where $z=\omega+i\delta$. I would like to know the meaning of $G_{i,j}(z)$. Can we similarly interpret it as the probability amplitude for a particle to transfer from $j$ to $i$ with energy change as much as $z$?
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$\begingroup$ Related: Why are time-ordered Greens functions equal to retarded Greens functions at zero temperature? $\endgroup$– Roger V.Commented May 25, 2023 at 11:37
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$\begingroup$ Thanks, but this post is not related to my question. $\endgroup$– H. KhaniCommented May 26, 2023 at 9:31
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I am hesitating on putting any deeper meaning into the definitions of the various Green-functions. The phrase that you wrote above can also be found in the Mahan-book, and it is the only interpretation that Mahan gives to the zoo of Green-functions. Negele and Altland and, if I remember correctly, also Flensberg hesitate on giving any physical interpretation to the Green-functions. They just "appear" when one wants to calculate thermodynamic quantities, responses and conductivities and susceptebilities, and the zoo of Green-functions being related through the variety of Lehmann-identities is imho just a tool to calculate these observables... If you are keen on putting deeper interpretations into these definitions you can probably associate the Fourier-transform as probability of some event happening within a certain energy window... (And similarly for lesser and greater Green-function the probability of having some states within some energy-window (un)-occupied. However... I prefer Feynmans "Shut up and calculate"-attitude.
