# Physics Notes by Jakob Schwichtenberg

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group_theory:groups:su2

# SU(2)

## 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 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.