# Can we make a Bloch sphere for Weyl spinors?

If spinors are the "square root" of 3-vectors [$$\mathrm{SU}(2)$$ double cover of $$\mathrm{SO}(3)$$], Weyl spinors can be thought of as the "square root" of 4-vectors [$$\mathrm{SL}(2,\mathbb{C})$$ double cover of $$\mathrm{SO}^+(1,3)$$]. Is is possible to make a Bloch-like sphere (probably not even a sphere) for a Weyl spinor? I am not completely sure in what dimension this object lies but hopefully some spatial dimensions (and global phase) can be removed for visualization.

Can a Weyl-Bloch sphere be constructed? What does it look like? Is it a hyperboloid?

1. Recall that the 2-component spinor of the subgroup $$SU(2)\subseteq SL(2,\mathbb{C})$$ is the restriction of the 2-component (left) Weyl spinor.
2. Recall that the density operators $$\rho~=~| \psi\rangle \langle\psi | ~=~\frac{1}{2}\left({\bf 1}_{2\times 2}+ \vec{r}\cdot \vec{\sigma}\right),$$ for the pure states of of the qubit is parametrized by the Bloch sphere $$\mathbb{S}^2$$, cf. e.g. my Phys.SE answer here.
3. The analogue for the (left) Weyl spinor $$\psi\in\mathbb{C}^2$$ is $$\psi\psi^{\dagger}~\in~u(2),$$ where $$(u(2),\det(\cdot))$$ is the set of Hermitian $$2\times 2$$ matrices equipped with a Lorentz invariant determinant.
4. $$(u(2),\det(\cdot))$$ is isometric to the Minkowski space $$(\mathbb{R}^{1,3},||\cdot||^2)$$, cf. e.g. my Phys.SE answer here.