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I am currently stuck on problem 5.3 (c) about spinor products in PS, where one needs to prove the Fierz identity:

$$ \bar{u}_{L}(p_{1}) \gamma^{\mu} {u}_{L}(p_{2}) [\gamma_{\mu}]_{ab} = 2 [u_{L}(p_{2})\bar{u}_{L}(p_1) +u_{R}(p_{1})\bar{u}_{R}(p_2) ]_{ab} $$

They say that a Dirac matric M satisfies:

$$ \gamma^{5} [M]=-[M]\gamma^{5}$$

hence should be of the form:

$$ [M]= \left( \frac{1-\gamma^{5}}{2} \right) \gamma_{\mu} V^{\mu} + \left( \frac{1+\gamma^{5}}{2} \right) \gamma_{\mu} W^{\mu} $$

But then, to get the answer, I suppose that:

$$ V^{\mu} = u_{L}(p_{2})\bar{u}_{L}(p_1) $$

$$ W^{\mu} = u_{R}(p_{1})\bar{u}_{R}(p_2) $$

Honestly, I don't see exactly why. Can someone point me out the way to make this identification?

Next, in part (d), we should get an amplitude of the form:

$$ i\mathcal{M} = (-ie)^{2} \bar{v}_{R} (k_{2}) \gamma^{\mu} u_{R} (k_{1}) \frac{-i}{s} \bar{u}_{R}(p_{1})\gamma_{\nu} v_{R}(p_2)$$

Thus, we have terms in u and v. However, all the spinor product formalism has been developed in terms of u. My question is: can one define a spinor product $$s(p_1,p_2)=\bar{v}_{R}(p_1) u_{L}(p_2) $$?

Thanks,

Emmanuel

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# A Spinor product in Peskin-Schroeder problem 5.3

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