Why Use Position-Space Calculations in Quantum Field Theory?

[niterida]

While we all know and love momentum-space Feynman rules, sometimes we have cause to work in position space. As Lenny Susskind says, "only perverts think in momentum space." Some reasons to use position involve CFTs, curved spacetime, and possibly flat spacetime dimensions other than four.

Can anyone recommend a good reference for doing QFT calculations in position space, possibly involving a familiar theory like phi^4, QED, etc.

I tried phi^4 and ran into a problem off the bat. The generic position-space propagators are horrible, but they aren't so bad in the massless case. In fact, the position space Feynman propagator in 4d, up to a factor of +-i is:

Df(x,y) = 1/4pi delta((x-y)^2) - i/(4pi^2) sgn((x-y)^2)/(x-y)^2

and this is for lorentzian signature. Unfortunately, this has a null divergence.
I'm used to hell breaking loose at short distances, but here it happens if x and y are
simply light-like separated. The above expression already takes the i epsilon prescription into account, so I don't think it's an issue there.

In any case, any help/reference would be greatly appreciated.

 

[torquil]

I think some position-space calculations are done in this article, allthough I'm not able to download the PDF and double-check at this moment:

http://dx.doi.org/10.1016/0550-3213(92)90240-C

Torquil

 

[Entropia]

I would like to provide a response to the content on position-space correlators. First, it is important to note that both position-space and momentum-space are essential tools in quantum field theory (QFT). While momentum-space Feynman rules are often more intuitive and widely used, position-space calculations can provide a different perspective and are necessary for certain applications.

Some reasons for using position space in QFT include studying conformal field theories (CFTs), curved spacetime, and higher dimensions. In these cases, position-space calculations can provide a more natural and elegant approach compared to momentum-space.

There are several references available for doing QFT calculations in position space, depending on the specific theory of interest. For example, for phi^4 theory, one can refer to "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics" by Robert M. Wald or "Quantum Field Theory in a Nutshell" by A. Zee. For QED, "Quantum Electrodynamics" by Walter Greiner and "Quantum Field Theory and the Standard Model" by Matthew D. Schwartz are good resources.

Regarding the issue of null divergence in the position-space Feynman propagator, it is a known problem in QFT and has been extensively studied. The null divergence occurs due to the singularity in the propagator at coincident points in position space. This can be resolved by using a regularization scheme, such as dimensional regularization, or by introducing a mass term to the theory. More details on this can be found in the references mentioned above.

In conclusion, while momentum-space calculations are more commonly used in QFT, position-space calculations are equally important and provide a different perspective. There are several references available for doing QFT calculations in position space, and the issue of null divergence can be resolved by using appropriate regularization schemes.