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Your first equation looks a bit like GR with a dilaton. IIRC, the analogous equations in supergravity will naturally have spinors coupling to the dilaton.

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The left-handed neutrino is a 2-spinor field $\eta_A$, $A=0,1$, and the Majorana mass term is a bilinear, $\Delta L = \pm 2$ term without the complex conjugation in each term, $$m\cdot \eta_A \eta_B \cdot \epsilon^{AB} + \text{Hermitian conjugate}$$ Note that this Majorana term is the only bilinear term without derivatives that one may construct from a ...

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Lucubration needs not light with insight. I fear you are expecting to make lemonade with apples. Here is why. The basic relation is the multiplication law of two Pauli vectors predicated on the abstract properties of the Pauli matrices, not their particular realization, $$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b}) ... 2 The Lagrangian for Dirac's equation is$$ \mathcal L=-mc^2\psi^2+\cdots \tag{1} $$As we know that H\sim\mathrm d^3\boldsymbol x\ \mathcal L has units of energy, we conclude that$$ \psi^2\sim x^{-3}\tag{2}  and therefore $\psi$ has units of $[\mathrm{length}]^{-3/2}$. If you use a different convention for $\mathcal L$ instead of $(1)$ you'll get a ...

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