I am wanting to get the expression up to ##O(\epsilon^{2}) ## :(adsbygoogle = window.adsbygoogle || []).push({});

To show that ##\frac{1}{2w_{k}} (\frac{1}{w_{k}-k_{0}-i\epsilon} + \frac{1}{w_{k}+k_{0}-i\epsilon})##

##=##

## \frac{1}{k_{v}k^{v} + m^{2} - i\epsilon}##, [2]

where ##w_{k}^{2}=k^{2}+m^{2}##, ##k## the variable, and (this seemed weird for me but it is in my lecture notes) that ##k_{0}=w-w_{k}## in the second term in the sum above and ##k_{0}=w_{k}-w## in the first term in the sum above.

where ##k^{v}k_{v} ## is a summation over 4 dimensions of space time where the relevant metric is the minkowski metric.

My attempts:

So so far I have tried a binomial expansion of ##({w_{k}-k_{0}-i\epsilon}) ^{-1}## and the other term, but of course this brings all terms in the 'numerator' , I don't see how I'll get any ##\epsilon## in the denominator to get something that will look like [2].

and, since this failed, cross multiplying to get a common denominator, but this didn't look promising either... [2]

What is the best way to procceed?

Many thanks

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# I QFT Feynman Propagator Algebra

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