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In addition to the well known momentum-space representation of the propagator, e.g. for the scalar particle $$ D(z)=\int \frac{d^4p}{(2\pi)^4} \frac{e^{-i(pz)}}{p^2-m^2+i \varepsilon} $$ there also exist a position-space formula, namely: $$ D(z) = \frac{1}{4\pi}\delta(z^2)-\frac{m}{8\pi} \frac{H_1^{(2)}(m\sqrt{z^2})}{\sqrt{z^2}} \theta(z^2) + \frac{im}{4\pi^2} \frac{K_1(m\sqrt{-z^2})}{\sqrt{-z^2}} \theta(-z^2) $$ While the first formula hides all the spacelike/timelike details of the propagation, the second one allows to study this effects separately.

The question is, whether the separate study of spacelike and timelike propagation is of any practical (or theoretical) use? Are there any papers that concern this issue?

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  • $\begingroup$ Not really sure what you're asking about, but the propagator is crucial to the definition of the QFT model. In fact, free QFTs are completely defined by their propagator. It encodes physics such as the Yukawa potential for the interaction mediated by the quantum field, the Compton wavelength, etc. $\endgroup$
    – Prof. Legolasov
    Commented Aug 22, 2021 at 15:14
  • $\begingroup$ The propagator as a whole, yes. But the propagator can be written in different forms. E.g., In momentum space it is used to simplify the computation of amplitudes. But what are its usages in position space and, especially, is there any relevance of considering its terms (with delta and thetas) separately? $\endgroup$
    – stanislav-iablokov
    Commented Aug 22, 2021 at 16:33
  • $\begingroup$ As of my knowledge, we can not distinguish whether some process occurs due to spacelike or timelike propagation. Therefore it seems that considering spacelike and timelike terms separately has no physical significance, only their sum is relevant. Or maybe I'm wrong? $\endgroup$
    – stanislav-iablokov
    Commented Aug 22, 2021 at 16:36

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