This is a common question that arises in the teaching of quantum-field-theory (QFT), i.e. whether the above is merely a prescription that works or are there deeper reasons behind it ? There are several related questions which may arise, e.g. is the prescription a purely relativistic effect ? What are the implication for "causality" in QFT ? What is the relation of this prescription to the symmetry of "particle-antiparticle-conjugation (C)"-symmetry and the "time-reversal symmetry(T)" ? How does one recover the purely causal-("retarded") propagator in the non-relativistic-limit ?- etc.
In short, the answer is that without regularization, the expression for propagator is meaningless. In Akhiezer-Berestetsky book there are examples of different types of regularization.
Of course there's a deeper reason, that doesn't have anything to do with regularization, however-it's a mathematical requirement that is needed in order for the solution for the Green function be unique, that's all.
This, then, can be given a physical interpretation, which was done by Feynman.
Cf. for instance: https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf
Dear Bimal,
The iε downstairs has a mathematical origin and it is in fact a prescription for how to handle the poles in the complex p0-plane. It automatically yields the Feynman contour of integration by shifting the poles appropriately. It is sometimes written. Besides it helps to get the time-ordered product of operators, which is the correct interpretation of such an expectation value from the perspective of the path integral or the correlated functions in QFT. Therefore it is related with the causality due to choose a given direction of the time in the timelike region of the Minkowki's spacetime.
Obviously this mathematical trick cannot change the finite symmetries of Lorentz associated to a physical system. C, P and T cannot depend at the end of the mathematical calculation of the different propagators.
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