For the spin-1/2 fermion field $\psi$, we may choose it to be a spinor which needs to be

  • Grassmann variable

but can also be

  • complex $\mathbb{C}$ Grassmann. (Dirac or Weyl spinor/fermion)
  • We can ask: Can be in real $\mathbb{R}$ Grassmann. (Majorana or Majorana-Weyl spinor/fermion)

What are the legal values of spin-1 field can take? real $\mathbb{R}$, complex $\mathbb{C}$, quaternion $\mathbb{H}$, .. (Grassmann)? (p.s. I remember S Adler tries to construct quaternion QFT -- is this related to the quaternion field.)

  • $\begingroup$ In how many spacetime dimensions and in what signature? Related: physics.stackexchange.com/q/53318/2451 $\endgroup$ – Qmechanic May 28 at 6:33
  • $\begingroup$ when I say complex and real - I mean complex Grassmann or real Grassmann -- I say this in separate order -- let me fix it $\endgroup$ – annie heart May 29 at 18:18

Fermion field can NOT be complex $\mathbb{C}$ or real $\mathbb{R}$. That is a common mistake in some text books. Fermion field must be real/complex/quaternion Grassmann variable.

Case in Point: the Majorana mass term would not be possible, if not for the Grassmann nature of the Fermion field.

In addition to real $\mathbb{R}$/complex $\mathbb{C}$/quaternion $\mathbb{H}$ Grassmann variable, you may go further up the division algebra ladder and toy with the idea of non-associative octonions $\mathbb{O}$ here

If that wets you appetite, you might want to check out sedenions $\mathbb{S}$ here.

Mind that one should be talking about octonion/sedenion Grassmann variables, the authors of the above linked papers failed to do so.


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