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I have a particular derivation of a four-vector integration measure, basically changing the measure to some related more useful measure - but I'd like to do this in 3-vector notation. Here it is, from the integral:

[tex]-i\lambda\int_{-\infty}^{\infty}\frac{d^4k}{(2\pi)^4}\frac{i}{k^2-m^2}[/tex]

where k is the four vector: [tex]k=(E,\mathbf{p})[/tex]

We transform the integration measure like so:

[tex]d^3k=4\pi k^{2}dk\: ,\: \frac{dk^2}{dk}=2k\: ,\: d^3k=2\pi kdk^2[/tex]

Where the first step is done due to spherical symmetry (can anyone explain why this is spherically symmetric? I'm guessing the 'k' is isotropic in some sense?)

Anyway we attain that the integration measure is:

[tex]\frac{d^4k}{(2\pi)^4}=\frac{k^2dk^2}{16\pi^2}[/tex]

My main question is how to change this derivation to a 3-vector one? I'm guess I pretty much subsitute 'k' for 'p' but I'm not sure.

Thanks,

SK

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# Transforming a 4-vector integration measure

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