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I think we can be pretty sure that fermions exist. We have several ways to describe them (Dirac, Weyl, Majorana, maybe someone I'm missing?), with different equations and number of components. My question is: are physical fermions
(1) either Dirac or Majorana or Weyl?
(2) just fermions, which we can describe in whatever way we find suitable for the theory we are currently studying?

If (1) is right, then what kind of fermions do we have in our universe?

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    $\begingroup$ $\uparrow$ in 3+1D? $\endgroup$
    – Qmechanic ♦
    Commented May 30, 2021 at 15:56
  • $\begingroup$ @Qmechanic I'm studying String Theory so I would love an explanation in generic D, but I'd settle for a 3+1D case too. $\endgroup$
    – Mauro Giliberti
    Commented May 30, 2021 at 16:01
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    $\begingroup$ Since you explicitly restrict to "physical fermions", you are presumably aware that in $D\equiv2\text{ mod }8$ there exist spinors that are both Majorana and Weyl? $\endgroup$
    – Nihar Karve
    Commented May 30, 2021 at 16:19
  • $\begingroup$ @NiharKarve After checking my notes on Clifford algebras, yes I know that (thank you for pointing me in that direction), but I do not understand how this is related to "physical fermions". I just meant fermions that actually exist, not predictions/toy models. $\endgroup$
    – Mauro Giliberti
    Commented May 30, 2021 at 16:50
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    $\begingroup$ Does this answer your question? Why do people say that neutrinos are either Dirac or Majorana fermions? $\endgroup$
    – tparker
    Commented Nov 3, 2023 at 12:07

2 Answers 2

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All of the fermions in the Standard Model, with or without right-handed neutrinos, can be modeled with Weyl spinor fields (left-handed or right-handed or a mix of both, it doesn't matter).

There's an isomorphism between Weyl and Majorana spinors in 3+1 dimensions which means that they can also all be modeled with Majorana spinors, according to Howard E. Haber, "Massless Majorana and Weyl fermions cannot be distinguished". (That's an unpublished note, but I believe it's correct.)

A Dirac spinor is two Weyl/Majorana spinors. You can always pair up Weyl/Majorana spinors if you have an even number of them. The naturalness of it depends on how much you can then write operations on the pairs as a unit in the Lagrangian, versus just projecting out the original spinors all the time. In the Standard Model, you can pair up the spinors according to the Higgs "mass" couplings (either leaving the neutrinos unpaired or adding right-handed neutrinos), so that the mass terms look similar to Dirac mass terms, but you then have the $SU(2)\times U(1)$ electroweak symmetry acting asymmetrically on the two halves of the spinors. There's nothing to suggest that the universe is built out of Dirac spinors fundamentally.

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  • $\begingroup$ I think this explanation is oversimplified. All fermion fields can be expressed entirely in terms of Weyl spinor fields, or Dirac bispinor fields, or Majorana bispinor fields. Which one is the best to use is largely just a matter of convenience. There's even a textbook that chooses to omit Weyl and Dirac fields entirely and to express the field content of the SM entirely in terms of Majorana bispinors. $\endgroup$
    – tparker
    Commented Nov 3, 2023 at 13:29
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    $\begingroup$ @tparker I rewrote the answer. $\endgroup$
    – benrg
    Commented Nov 4, 2023 at 4:09
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In the Standard Model, fermions are in left-handed doublets and right-handed singlets, all represented with left-handed Weyl spinors (the singlets are actually antiparticles), the representation being with respect to the SU(2) part of the electroweak interaction. In the low energy theory, particles with left- and right-handed components in the SM pair off and look like they are either Dirac or Majorana. As far as we know, all of the charged leptons and quarks are Dirac particles since Majorana particles allow violation of particle number by two units and baryon number seems to be conserved and lepton number seems to be conserved in the cases with charged leptons that we can readily observe. Neutrinos could either be Dirac or Majorana but as far as we're aware not both and they can't be just Weyl as they are in the SM because that wouldn't allow them to have a mass.

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