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Let's see what relation can we find between $\alpha, \beta, \alpha^i, \beta^i$ and $\gamma, \delta, \gamma^i, \delta^i$ First using Baker Campbell Hausdorff lemma we deduce two things: $$\alpha + \beta = \gamma \text{ and } \alpha - \beta = \delta$$ because $\mathbb{1}$ commutes with $\sigma$. And $$e^{i\vec{\alpha}\cdot \vec{\sigma}} = \mathbb{1}\text{cos ...



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