a possibly very naive question about the standard $\mathbf{SU}(2)$ vs $\mathbf{SO}(3)$ thing: we know there is a two-to-one homomorphism $\mathbf{SU}(2)\to \mathbf{SO}(3)$, which is a universal covering etc. Mostly thinking in terms of qubits and Bloch sphere, we also know an "equivariant map" between these two representations:
given $\psi\in\mathbb C^2$, defining $f(\psi)=(2\bar\psi_0\psi_1,|\psi_0|^2-|\psi_1|^2)\in\mathbb R^3$ (there is some abuse of notation here, hopefully you can tell what I mean), and denoting with $\Phi:\mathbf{SU}(2)\to\mathbf{SO}(3)$ the homomorphism, we have $\Phi…
given $\psi\in\mathbb C^2$, defining $f(\psi)=(2\bar\psi_0\psi_1,|\psi_0|^2-|\psi_1|^2)\in\mathbb R^3$ (there is some abuse of notation here, hopefully you can tell what I mean), and denoting with $\Phi:\mathbf{SU}(2)\to\mathbf{SO}(3)$ the homomorphism, we have $\Phi…