From "Symmetry and the Standard Model: Mathematics and Particle Physics by Matthew Robinson", it states that 'SU(2) matrix has one of the real parameters fixed,leaving three real parameters'.(adsbygoogle = window.adsbygoogle || []).push({});

I don't really get this part and hope someone can clear my doubt.

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To ensure that the matrix is unitary, c=-b* & d=a* while a=a and b=b. To ensure that it is special, ad-bc=1

Together, $$ ad-bc=a.a^*-b.(-b^*)=|a|^2 + |b|^2 = 1 $$

$$a=Re(a)+i Im(a)$$ & $$b=Re(b)+i Im(b)$$ ← From here I got 4 parameters.

$$ |a|^2 + |b|^2 = (Re(a))^2 +(Im(a))^2 + (Re(b))^2 +(Im(b))^2 = 1 $$← I still see 4 parameters thus I am still not sure how the unit determinant fix one of the parameter.

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Can I think of it another way as in just like SO(3) rotation, in SU(2) we also have a 3D space except instead of z axis, we have the i(imaginary) axis. So is no different from rotation in 3D space other than just changing the one of the axis to i-axis.

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# I Why does SU(2) have 3 parameters/generators like the SO(3)?

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