What does the two point position space function of massive scalars look like in Minkowski space?

$$\langle 0| \phi(x) \phi (y)|0\rangle =\ ?$$

I've been trying to better understand analytic behavior of some simple correlators and realized I don't quite know how to write down such a function. Is there a closed form which I can learn about it's poles, cuts, etc.?

Usually when discussing such objects authors will with just write this is a Fourier transform of momentum space, $D(x,y)=\langle 0| \phi(x) \phi (y)|0\rangle =\int e^{-i p(x-y)} D(p)$, without actually evaluating the Fourier transform. I'm worried I am missing some subtitles when I try to do it myself. Other authors will argue that if we think about the physics, we should really be concerned with causal objects and so we should consider a related object like $\langle0| [\phi(x), \phi (y)]|0\rangle $. (For example Peskin & Schroeder pp 27-29)

This seems like a basic object so I am assuming I have some very basic misunderstanding. The only time I ever see such an object discussed in physics is when dealing with a CFT. Here, however, one is always interested in a massless theory (we don't need no scales!).


  1. One suggestion is just to use this procedure, but I believe the starting point there involves a time-ordered correlation functions (i.e. Feynman propagator). I'm specifically interested in the non-time-ordered case.
  2. Another suggestion is that I should be careful about my language. This seems the providence of Wightman functions and more general distributions in field theory, but alas this is not my forte.
  • $\begingroup$ Although that is known as the 2 point function, it doesn't mean that it is a "function" in the proper sense. This is unfortunate terminology, as Wightman functions are tempered distributions in each of the "variables". $\endgroup$ – Phoenix87 Dec 19 '16 at 21:45
  • $\begingroup$ I had an inkling I should have been using the language of more generalized distributions, Wightman functions, but wasn't sure I was on the right track. $\endgroup$ – Tim Dec 19 '16 at 21:47
  • $\begingroup$ AccidentalFourierTransform: I had already been through that post and unfortunately I don't think it answers my question. There they show that one can take the position space Feynman propagator, take the massless limit, and they recover the result for a massless propagator. The starting point, Feynman Propagator, is defined for a time-ordered product. I'm specifically interested in the correlator without time ordering. $\endgroup$ – Tim Dec 19 '16 at 21:50
  • $\begingroup$ @AccidentalFourierTransform No problem! Indeed, I ended up reading through a whole boat load of similar responses like that thinking "gah my question must have been asked before!" $\endgroup$ – Tim Dec 19 '16 at 21:57
  • $\begingroup$ It is done in Eqn(71) in arxiv.org/abs/hep-th/9908140 Given in terms of BesselK. May be I should post this as a comment. I haven't check it. Hope this hope helps. $\endgroup$ – user168471 Sep 7 '17 at 9:32

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