# From relativistic equation to find Dirac matrices

Is this possible and then how? $$((\gamma \otimes \mathbf\sigma)\bullet\mathbf p)(\gamma^\prime\otimes\mathbf 1_2) = \gamma\gamma^\prime\otimes\sigma \bullet \mathbf p$$ where $$\gamma$$ and $$\gamma^\prime$$ are using to factorize later we see $$\gamma^0=\gamma^\prime\otimes1_2$$,$$\gamma^1=\gamma\otimes\sigma_1$$,$$\gamma^2=\gamma\otimes\sigma_2$$ $$\gamma^3=\gamma\otimes\sigma_3$$ where $$\gamma^1,\gamma^2,\gamma3$$ are Dirac Matrices, $$\mathbf\sigma$$ is Pauli Spin Matrix, p is momentum four vector $$p=(E/c,-p_1,-p_2,-p_3)$$ and $$1_2$$ is 2x2 unit matrix i.e$$1_2 = \left(\begin{matrix} 1 & 0 \\ 0 & 1\end{matrix}\right)$$ $$\sigma_1 = \left(\begin{matrix} 0 & 1 \\ 1 & 0\end{matrix}\right)$$ $$\sigma_2 = \left(\begin{matrix} 0 & -i \\ i & 0\end{matrix}\right)$$$$\sigma_3 = \left(\begin{matrix} 1 & 0 \\ 0 & -1\end{matrix}\right)$$ I know about matrix mixed product( ($$A\otimes B)(C \otimes D)=AC\otimes BD$$ ) but I can't understand how this possible I shall be happy to get any answer. Here I put this Image from Where I have got this query. Thanks.

I think this may be an answer.I am not quite sure about that. $$(\gamma^\prime\otimes 1_2)\{(\gamma\otimes\pmb\sigma)\}\bullet \pmb p\} =\{(\gamma^\prime\otimes 1_2)(\gamma\otimes\pmb\sigma)\}\bullet \pmb p$$ [Here I assume $$(\gamma^\prime\otimes 1_2)$$ just like scaler] $$=\{\gamma^\prime\gamma\otimes1_2\pmb\sigma\}\bullet \pmb p$$ [By using mixed product theorem $$(A\otimes B).(C\otimes D)=AC\otimes BD$$]$$=(\gamma^\prime\gamma\otimes\pmb \sigma)\bullet p$$ Similarly $$\{(\gamma\otimes\pmb \sigma)\bullet p\}.\{\gamma^\prime\otimes 1_2\}=\{(\gamma\otimes\pmb \sigma).(\gamma^\prime\otimes 1_2)\}\bullet \pmb p=(\gamma\gamma^\prime\otimes\pmb\sigma 1_2)\bullet p= (\gamma\gamma^\prime\otimes\pmb\sigma)\bullet p$$ Note:I am not sure that there exist any such type rule to exchange inner product with direct product.
• Well I'm glad you found what you were looking. You might want to see if you can start with Klein-Gordon relativistic wave equation and generate the Dirac equation - which is how Dirac found the $\gamma$ matrices - or the Dirac Clifford algebra – Cinaed Simson May 18 at 9:15