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I know about the commutator relationship given in this question: $$\left[\gamma^{\mu},\gamma^{\nu}\right]=2\gamma^{\mu}\gamma^{\nu}-2\eta^{\mu\nu}$$ then, my question is if there exist a similar relationship for the following $$\left[\gamma \cdot p,\gamma\cdot q\right] $$ where $p$ and $q$ are 4-vectors, I have tried solve it using the following Dirac Salsh relation
$${a\!\!\!/}{b\!\!\!/}=\left[a\cdot b-ia_{\mu }\sigma ^{\mu \nu }b_{\nu }\right]$$ where $\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]$, thus $$\left[\gamma \cdot p,\gamma\cdot q\right]=i\left(q_{\mu}\sigma^{\mu\nu}p_{\nu}-p_{\mu}\sigma^{\mu\nu}q_{\nu}\right)$$ therefore, is $\left[\gamma \cdot p,\gamma\cdot q\right]=0$?

Qmechanic
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Lluis Gerardo
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1 Answers1

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If you write $[\gamma\cdot p,\gamma\cdot q] = [\gamma^\mu p_{\mu}, \gamma^\nu q_{\nu}]$ then you can treat $p_{\mu}$ and $q_{\nu}$ as just numbers, i.e. they are "scalars" with respect to the spinor indices of the gamma matrices and can be pulled out of the commutator.

Then we can use your result $[\gamma^\mu p_{\mu}, \gamma^\nu q_{\nu}] = p_\mu q_\nu [\gamma^\mu, \gamma^\nu] = p_\mu q_\nu (2\gamma^\mu \gamma^\nu - 2\eta^{\mu\nu}) = 2{\not}p{\not}q - 2 p\cdot q$

where I've used the slash notation to imply ${\not}p = \gamma^{\mu}p_{\mu}$

CStarAlgebra
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