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Using the three Pauli spin matrices

$$\boldsymbol{\sigma}_{1}=\left(\begin{array}{ll}{0} & {1} \\ {1} & {0}\end{array}\right), \quad \boldsymbol{\sigma}_{2}=\left(\begin{array}{cc}{0} & {-i} \\ {i} & {0}\end{array}\right), \quad \boldsymbol{\sigma}_{3}=\left(\begin{array}{cc}{1} & {0} \\ {0} & {-1}\end{array}\right)$$ and the matrix $\boldsymbol{R},$ which is a complex valued $2\times 2$ matrix, I'm trying to extract some identity from the following calculation:

$$(\boldsymbol{\sigma}_{3}-\boldsymbol{R}\boldsymbol{\sigma}_{3}\boldsymbol{R})(\boldsymbol{\sigma}_{3}+\boldsymbol{R}\boldsymbol{\sigma}_{3}\boldsymbol{R})-(\boldsymbol{\sigma}_{3}+\boldsymbol{R}\boldsymbol{\sigma}_{3}\boldsymbol{R})(\boldsymbol{\sigma}_{3}-\boldsymbol{R}\boldsymbol{\sigma}_{3}\boldsymbol{R})\\ =-2(\boldsymbol{R}\boldsymbol{\sigma}_{3}\boldsymbol{R}\boldsymbol{\sigma}_{3}+\boldsymbol{\sigma}_3\boldsymbol{R}\boldsymbol{\sigma}_3\boldsymbol{R})$$

Is there any trick that can be used to simplify the last line?

rami_salazar
  • 457
  • hint: try writing $R=a_0 I+\sum a_i \sigma_i $ and then use commutations to push all $\sigma_3$ to one side – user619894 Aug 21 '19 at 13:19

0 Answers0