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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 the 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 its poles, cuts, etc.?

Usually when discussing such objects authors will just write this as 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 subtleties 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!).

Edits:

  1. One suggestion is just to use this procedure, but I believe the starting point there involves a time-ordered correlation function (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.
Qmechanic
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Tim
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  • 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". – Phoenix87 Dec 19 '16 at 21:45
  • 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. – Tim Dec 19 '16 at 21:47
  • 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. – Tim Dec 19 '16 at 21:50
  • @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!" – Tim Dec 19 '16 at 21:57
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    It is done in Eqn(71) in https://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. – user168471 Sep 07 '17 at 09:32

1 Answers1

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These sorts of expressions are discussed in older quantum field theory books, such as Bjorken & Drell and Bogoliubov & Shirkov. In Quantum Fields, by Bogoliubov & Shirkov, the expression for the two-point function of a massive scalar field in Minkowski spacetime can be read from the expressions given on Appendix V.2. Specifically, one has (up to typos as I copy down the expression) $$\langle 0 | \phi(x) \phi(0) |0 \rangle = \frac{\mathrm{sign}(x^0) \delta(\lambda)}{4\pi i} - \frac{m \Theta(\lambda)}{8 \pi i \sqrt{\lambda}}\left[\mathrm{sign}(x^0) J_1(m \sqrt{\lambda}) - i N_1(m \sqrt{\lambda}) \right] + \frac{m \Theta(-\lambda)}{4 \pi^2 \sqrt{-\lambda}} K_1(m \sqrt{-\lambda}),$$ where $\lambda = (x^0)^2 - ||\vec{x}||^2$, $J_1$, $Y_1$, and $K_1$ are the Bessel functions, $\Theta$ is Heaviside's theta function, and $\mathrm{sign}$ is the sign function.

There are some discussions about this and other related propagators on n-Lab. n-Lab also lists this table on commutator functions and propagators that might be useful to understand the pole structures and so on.

Níckolas Alves
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