✎ SU(2) [The Physics Travel Guide]

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====== SU(2) ======

<tabbox Why is it interesting?> 
$SU(2)$ is one of the most important groups in modern physics. The group is a crucial ingredient of the [[advanced_tools:gauge_symmetry|gauge symmetry]] of the [[models:standard_model|standard model]] of particle physics and, in some sense, explains the structure of weak interactions.

In addition, the fundamental spacetime symmetry group called Lorentz group, is usually analyzed in terms of its Lie algebra. This Lie algebra can be understood as two copies of the $SU(2)$ Lie algebra. Hence, by understanding the Lie algebra of $SU(2)$, we understand almost everything about the Lorentz group. This analysis is crucial for the understanding what [[basic_notions:spin|spin]] is, which is one of the most important properties of [[advanced_notions:elementary_particles|elementary particles]]. 
<tabbox Layman>

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Student>

Every $SU(2)$ transformation can be written as $$ g(x) = a_0(x) 1 + i a_i(x) \sigma ,$$ where $\sigma$ are the Pauli matrices. The defining conditions of $SU(2)$ are $g(x)^\dagger g(x)=1$ and $det(g(x)=1$, and thus we have $$ (a_0)^2 +a_i^2=1 , $$ which is the defining condition of $S^3$.

Source: page 23 in http://www.iop.vast.ac.vn/theor/conferences/vsop/18/files/QFT-4.pdf

This is also shown nicely at page 164 in the book Magnetic Monopoles by Shnir.

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Elements of $SU(2)$ can be written as

$$ U(x) = e^{i f(x) \vec{r} \vec{\sigma} },$$

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>

* [[https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/|A Journey to The Manifold SU(2)]]

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