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 Answer 1

  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.

  5. The Lorentz group orbits are hyperboloids.

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    $\begingroup$ we get the bloch sphere (as well as its higher dimensional analogs) as the locus of a certain set of algebraic equations. what would these equations be in this case? and could you point me in the direction of some further resources on the topic? $\endgroup$
    – paulina
    Apr 14 at 17:53

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