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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.
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.