traceless anti-hermitean part of SU(2) matrix

[simone-romiti]

Why here
https://github.com/urbach/su2/blob/7d4a9bfd9b7d6ef20e3f91e5bcbca2f42d038d11/include/su2.hh#L84

we don't have

const std::complex<double> a1 = 0.5 * (x.geta() - std::conj(x.geta()));
const std::complex<double> b1 = 0.5 * (x.getb() - std::conj(x.getb()));
return (_su2(a1, b1));

?

[urbach]

sorry, I don't understand the question!

…

Why here https://github.com/urbach/su2/blob/7d4a9bfd9b7d6ef20e3f91e5bcbca2f42d038d11/include/su2.hh#L84 we don't have ``` const std::complex<double> a1 = 0.5 * (x.geta() - std::conj(x.geta())) const std::complex<double> b1 = 0.5 * (x.getb() - std::conj(x.getb())) return (_su2(a1, b1)); ``` ?

[simone-romiti]

In the code we parametrize an $SU(2)$ matrix as:

$$ U = \begin{pmatrix} a & b \\ -b^* & a^* \end{pmatrix} $$

in the constructor of _su2 we pass 2 complex numbers $a$ and $b$.

The traceless, anti-hermitian part of the matrix is:

$$ A = (U - U^\dagger)/2 = \frac{1}{2} \begin{pmatrix} a - a^* & b - b^* \\ -b^* + b & a^* - a \end{pmatrix} $$

In we want to build an _su2 object representing it, we can use the same constructor but with $a_1 = (a - a^{*})/2$ and $b_1 = (b - b^{*})/2$ as arguments. Namely,:

return (_su2(a1, b1));

But in the line of code I quoted, we don't divide by 2 the second argument. It's like writing:

return (_su2(a1, 2.0*b1));

which is not what we want, is it?