What kind of quantum number is Chirality? Helicity, being the projection of spin in the direction of the momentum, is like a component of spin, and therefore, additive in nature. For a process, $A\to B+C$, the helicity of the LHS should match with sum of the helicities of B and C on the RHS. Is it the same about the chirality?

  • $\begingroup$ Is it a qn? I'd say it was rep. of Lorentz group, that coincides with helicity in massless limit $\endgroup$ – innisfree Mar 7 '17 at 10:54
  • $\begingroup$ @innisfree I'm not sure. $\gamma_5$ is the chirality operator. Isn't it? When acting on chiral projections of fermionic field, it gives a number times the field. $\endgroup$ – SRS Mar 7 '17 at 11:20

Chirality is not a quantum number. In representations of the Lorentz group where it makes sense to talk about it, it is the eigenvalue of the $\gamma^5$ (as one would denote it i four dimensions) that lives in the top degree of the Clifford algebra, but not every representation of the Lorentz algebra is one of the Clifford algebra, so speaking of chirality does not make sense for arbitrary objects.

In particular, $\gamma^5$ is not an observable - not even a quantum operator at all - and cannot act on states - it only exists in the finite-dimensional representation a Weyl/Dirac spinor transforms in, but not in the infinite-dimensional unitary representations where the states of your quantum theory live. The unitary representations are labeled by the mass and spin (for massive states) and helicity (for massless states), but not by anything that would lend itself to be interpreted as chirality that would be distinct from there. The available operator is helicity (and other spin components, of course), not chirality. Note that for massless fermions, the notions of chirality and helicity coincide, and that for massive Dirac fermions, the two chiral parts do not decouple, making the notion not particularly useful in the first place.


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