The electron - described as a four-spinor in the Dirac equation - transforms according to the $(1/2,0)\oplus(0,1/2)$ representation of the Lorentz group, so it is actually a direct sum of a left- and right-handed Weyl spinor. So is the electron technically a set of two other massless fundamental particles?

  • $\begingroup$ I would just repeat my answer for the photon here, except adding that the Weyl degrees of freedom in a Dirac spinor do not decouple if the spinor is massive. $\endgroup$
    – ACuriousMind
    Feb 18, 2015 at 15:57
  • $\begingroup$ Haha.. Are you going to ask that for every particle? $\endgroup$ Feb 18, 2015 at 16:02
  • 3
    $\begingroup$ @both: Well actually I honestly thought these two questions are rather unrelated. The Dirac spinor really is the direct sum of two Weyl spinors, but can the same be said of the photon? Is it the direct sum of two helicity representations? And furthermore: does this type of "doubling" (fundamental particle arises as grouping together of two technically/mathematically even more fundamental entities) occur for any particle so that I could indeed ask the same question for every SM particle? $\endgroup$ Feb 18, 2015 at 16:14

1 Answer 1


You are basically asking a circular question of nomenclature. The Dirac quantum field is is a bispinor compactly packaging several degrees of freedom, such as the left- and right-handed Weyl spinors you wrote down the Lorentz transformation properties of. We call both left- and right-handed electrons "the electron", collectively, but of course they are distinct degrees of freedom with different couplings (e.g. electroweak ones) and, naturally, experimental signatures. They are also coupled, so you may think of them as mutating into each other, and so different facets of the same object.

Electrons may have different chiralities, charges (electrons vs positrons), momenta, positions in space, etc... The Dirac quantum field compactly describes all of these states, packaged in its components. So, when you say "the electron", you refer to all of these states, unless it makes sense to be more specific. Don't you do this for all particles? "Technically" does not have the same whiff of quibbling in physics that it has out there...


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