I am currently trying to understand the importance of the Green's function in QM and QFT. But there are instances where I don't understand what is taking place.
In the lecture, the green's function was defined but in the context of QM. For example on way of expressing the free green's function (free particle propagator) is by considering the eigenvalues and eigenfuctions of the Hamilton operator, meaning:
$$G^{+}(x, t_x, y, t_y) = \theta(t_x - t_y) \langle x(t_x) | y(t_y) \rangle = \theta(t_x - t_y) \sum_n \phi_n(x) \phi_n^*(y) e^{-i E_n (t_x - t_y)}. $$
where $\phi_n$ is the eigenvector and $E_n$ the corresponding eigenvalue of the Hamilton operator. This is one possible way of expressing the free propagator/greens function. Another one would be as a function of space but this time in the frequency/energy domain:
$$G^{+}(x, y, E) = \sum_n \frac{i \phi_n(x) \phi_n^*(y)}{E - E_n}. $$
Then, it is said that:
Actually, to ensure causality it’s necessary to replace $E_n$ $E_n - i\epsilon$ so we have:
$$G^{+}(x, y, E) = \lim_{\epsilon \to 0^{+}} \sum_n \frac{i \phi_n(x) \phi_n^*(y)}{E - E_n + i\epsilon}. $$
I have the following questions:
All the different expression and definitions for the green's function/propagator are made for non-rel. QM. When jumping in QFT, would and how the green's function expression change?
The fact that in most cases we consider the retarded greens function, is already a way to ensure microcausality. So it comes as a surprise to me that we still need to ensure causality and even more confusing is the fact that this is done by introducing the prescription $i\epsilon$. How does the introduction of this term, which simply shifts the poles, enforces proper time ordering of events?
