How to perform a integral in momentum space

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The discussion focuses on the process of performing integrals in momentum space, specifically addressing a transition between two lines in an integral expression. A participant questions the validity of the integral's transformation, suggesting an alternative form involving a factor of $$\frac{4 \pi}{(2 \pi)^3}$$ and an integral over momentum. The presence of a ##\cos(\theta)## term in the dot product is highlighted as a key factor that influences the integration process. This term leads to differences in the exponential functions when integrating over ##d(\cos(\theta))##. Understanding these nuances is crucial for accurately performing integrals in momentum space.
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I am not sure how does the integral was did here. More preciselly, How to go from the first line to the second line? Shouldn't it be $$\frac{4 \pi}{(2 \pi)^3} \int _{0} ^{\infty} p^2 e^{ip*r}/(2 E_p)$$ ? (x-y is purelly spatial)
 
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There is a ##\cos(\theta)## in the dot product which brings out the ipr and causes the difference in the exponentials when you integrate over ##d(\cos(\theta))##
 
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Thread 'Product of two boosts and then two inverse boosts is rotation matrix'

So is there some elegant way to do this or am I just supposed to follow my nose and sub the Taylor expansions for terms in the two boost matrices under the assumption ##v,w\ll 1##, then do three ugly matrix multiplications and get some horrifying kludge for ##R## and show that the product of ##R## and its transpose is the identity matrix with det(R)=1? Without loss of generality I made ##\mathbf{v}## point along the x-axis and since ##\mathbf{v}\cdot\mathbf{w} = 0## I set ##w_1 = 0## to...
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