- #36

JD_PM

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Gaussian97 said:Well, this actually makes no sense. You are adding a spinnor with its adjoint, this operation is not defined at all.

My bad, I tend to overcomplicate things.

Gaussian97 said:I recommend you to take a look at the definition of the gamma matrices, that's all you need.

OK, recalling that

$$\{ \gamma_{\mu}, \gamma_{\nu} \} = 2 \eta_{\mu \nu}$$

We get

$$p\!\!\!/ k\!\!\!/ = p^{\mu} \gamma_{\mu} \gamma_{\nu} k^{\nu} = p^{\mu} ( 2\eta_{\mu \nu} - \gamma_{\nu} \gamma_{\mu}) k^{\nu} = -p^{\mu} \gamma_{\nu} \gamma_{\mu} k^{\nu} + 2 \eta_{\mu \nu}p^{\mu}k^{\nu} = -k\!\!\!/ p\!\!\!/ + 2 p_{\nu} k^{\nu}$$

Where ##p^{\mu}## and ##k^{\nu}## commute with ##\gamma_{\nu}## and ##\gamma_{\mu}## respectively as the former are not matrices.

But this does not lead to the desired Dirac solution; recalling that we are working with the relation

$$\left(p\!\!\!/_i + m \right)k\!\!\!/_i \frac{u(\vec p_i)}{2m}$$

And plugging what we have just obtained into it yields

$$\Big[ 2 p_{\nu} k^{\nu} + k\!\!\!/_i \left(-p\!\!\!/_i + m\right) \Big] \frac{u(\vec p_i)}{2 m}$$

Which does not give zero...

I know I must be missing something trivial here...