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Hi, I'm studying QFT (quantum field theory) again, with the help of Peskin&Schroeder. Altough the book is much better to read for a second time, there are some things that keep me wondering.
It's about shifting physical quantities by a small imaginary constant in order to be sure that expressions converge. For instance, in the Feynman propagator on page 31:
<br /> D_{F}(x-y) = int \frac{d^{4}p}{(2\pi)^{4}} \frac{i}{p^{2}-m^{2} + i \epsilon} e^{-ip\cdot(x-y)}<br />
What is the exact reason that we are allowed to shift the energy by a small imaginary constant? Isn't contradicting this the idea that physical measurable quantities should always be real? I understand that we hit a pole if we don't, but to me this whole business looks ill-defined.
A second time this comes by is that the time T in the integral of the vertices is taken to be
<br /> T \rightarrow \infty (1-i\epsilon) <br />
for instance on page 95. P&S call this an integration along a contour which is slightly rotated away from the real p^{0} just like in calculating the Feynman propagator; again, we are talking imaginary energies here!
Can anyone give a satisfactory answer to this? Why are we allowed to do so? Maybe this has come along some times earlier, but it's really bothering me; I find it hard to take this sort of juggling seriously without seeing a serieus justification for it other than "or else it will diverge!".
It's about shifting physical quantities by a small imaginary constant in order to be sure that expressions converge. For instance, in the Feynman propagator on page 31:
<br /> D_{F}(x-y) = int \frac{d^{4}p}{(2\pi)^{4}} \frac{i}{p^{2}-m^{2} + i \epsilon} e^{-ip\cdot(x-y)}<br />
What is the exact reason that we are allowed to shift the energy by a small imaginary constant? Isn't contradicting this the idea that physical measurable quantities should always be real? I understand that we hit a pole if we don't, but to me this whole business looks ill-defined.
A second time this comes by is that the time T in the integral of the vertices is taken to be
<br /> T \rightarrow \infty (1-i\epsilon) <br />
for instance on page 95. P&S call this an integration along a contour which is slightly rotated away from the real p^{0} just like in calculating the Feynman propagator; again, we are talking imaginary energies here!
Can anyone give a satisfactory answer to this? Why are we allowed to do so? Maybe this has come along some times earlier, but it's really bothering me; I find it hard to take this sort of juggling seriously without seeing a serieus justification for it other than "or else it will diverge!".