Gerenuk on Nostr: **How spinors square to 1** Spinors can be mapped to their bilinear covariants ...

**How spinors square to 1**

Spinors can be mapped to their bilinear covariants
\[\begin{aligned}
\Omega_1&=\psi^\dagger\gamma_0\psi\\J^\mu&=\psi^\dagger\gamma_0\gamma^\mu\psi\\
S^{\mu\nu}&=\psi^\dagger \gamma_0 i \gamma^{\mu\nu}\psi\\
K^\mu&=\psi^\dagger \gamma_0 u \gamma^{0123}\gamma_\mu\psi\\
\Omega_2&=\psi^\dagger \gamma_0 \gamma^{0123}\psi
\end{aligned}\]
These covariants can be collected into multivectors of the Clifford algebra Cl(1,3) (Lounesto Ch 10.5)

The spinor can be recovered from the covariants uniquely up to a complex phase (Lounesto Ch 11.2)

Since the spinor had only 8 parameters, but the covariants have 16 all together, the covariants satisfy some identities known as Fierz identities (in Clifford algebra)
\[\begin{aligned}
J^2&=\Omega_1^2+\Omega_2^2\\
&=-K^2\\
J\cdot K&=0\\
J\wedge K&=-(\Omega_2+\Omega_1\gamma_{0123})S
\end{aligned}\]

But what is surprising now and what I haven't seen in any other source is, that these identities are (almost) exactly what you need for a general element from the Clifford algebra to square to a real number:\[\begin{aligned}
A&=\Omega_1+J+S+I K+I\Omega_2\\
A^2&\in\mathbb{R}
\end{aligned}
\]

Therefore there is a 1-to-1 correspondence between spinors and multivectors (or equally real 4x4 matrices) squaring to a real number.
Has anyone seen that discussed somewhere?

There is a small difference in one sign and the Fierz identities with this choice covers only one type of element squaring to 1, but elements squaring to 1 thus describe the Fierz identities more generally.