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SU(2) as the three sphere $S^3$

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

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

group_theory/groups/su2.txt · Last modified: 2017/12/06 09:33 (external edit)