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I struggle with integral notation in Quantum Field Theory , I see integrals like d4k/(2pi)^4 , but I dont know how to evaluate it! I think it will be good if I can understand the sign d4k and how to evaluate simple integrals in 4-dimension like in the picture enter image description here

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  • $\begingroup$ See in which reference? Which page? $\endgroup$
    – Qmechanic
    Sep 21, 2017 at 17:32
  • $\begingroup$ Peskin (Introduction to Quantum Field Theory) $\endgroup$
    – Moayd S.
    Sep 21, 2017 at 17:45

1 Answer 1

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It's because you're integrating a multivariate function: The function is defined not on a line (like, for example, the function $f(x) = x^2$), but on a four-dimensional space, like $g(x,y,z,t) = x^2 + y^2 + z^2 - ct^2$.

That little $d^4 k$ is the volume element. Without getting too much into the weeds, this means that you break the entire space you're integrating over down into little four-dimensional cubes of dimensions $dk \times dk \times dk \times dk$, evaluate the function in the center of each of these cubes, and sum the whole thing up. Then you take the limit as $dk \to 0$ and hope the answer converges.

Practically, this almost certainly means you evaluate over each coordinate separately and in whichever order you want. You can do this as long as Fubini's theorem holds, though most physicists I know just assume it holds and hope for the best.

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