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advanced_tools:group_theory:su2 [2018/03/17 16:01] jakobadmin [Student] |
advanced_tools:group_theory:su2 [2018/03/17 16:03] jakobadmin [Student] |
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Elements of $SU(2)$ can be written as | Elements of $SU(2)$ can be written as | ||
- | $$ U(x) = e^{i f(x) \vec{r} \vec{\sigma} },$$ | + | $$ U(x) = e^{i a \vec{r} \vec{\sigma} }= e^{i a \vec{r} \vec{\sigma}} = \cos(a) + i \vec{r} \vec{\sigma} \sin( a )$$ |
+ | |||
+ | where $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ are the usual Pauli matrices and $ \vec{r} $ is a unit vector. | ||
- | where $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ are the usual Pauli matrices and $ \vec{r} $ is a unit vector. The condition $U(x) \to 1$ for $|x| \to \infty $ therefore means $f(x) \to 2\pi n$ for $|x| \to \infty $, where $n$ is an arbitrary integer, because we can write the matrix exponential as | ||
- | $$e^{i f(x) \vec{r} \vec{\sigma}} = \cos(f(x)) + i \vec{r} \vec{\sigma} \sin( f(x) ) .$$ | ||
<tabbox Researcher> | <tabbox Researcher> | ||