# Does the Chirality match the helicity?

• I
Gold Member
In trying to understand the Neutrino where it has mass and its chirality is the same as its helicity, I have always had trouble visualizing a particle. I recently ran into this particle. I believe the "the chirality is the same as helicity" as in one direction it would feed things through the center, but if spinning the other way, it would act like a solid shield or disk and could not flow.

Does this left-handed particle have "chirality match the helicity"?

mfb
Mentor
I have always had trouble visualizing a particle
Don't visualize particles. It just leads to misconceptions because you try to apply everyday intuition to things that don't follow this intuition.

Staff Emeritus
I agree with mfb. And whatever that animation is supposed to clarify, it doesn't.

Gold Member
What is the best way to explain "the chirality matches the helicity"? Why not include a visual?

ChrisVer
Gold Member
I guess it would help you figuring out what chirality and what helicity is:
https://en.wikipedia.org/wiki/Chirality_(physics)#Chirality_and_helicity

Then
"why not include a visual?"
I think mfb explained the reason. Such visualizations will take you several light-years away from what you want to obtain. You can only visualize shapes (like those cables revolving in your picture), but there are no shapes.

vanhees71
Gold Member
Thou shalt not make images! In QT it's almost always wrong to make images too much related to classical thinking.

Then it's important to remember that chirality and helicity are only the same for massless spin-1/2 fermions, not for massive ones. In the latter case chirality is a lorentz-invariant concept, helicity not. You can flip the helicity by an appropriate Lorentz boost to a frame which moves faster than the particle. This is not possible for massless particles, and that's why for massless particles the helicity is a frame-independent concept, but not for massive ones. Here you rather define the spin eigenstates in the rest frame of the particle and then construct the non-zero momentum states by a rotation-free Lorentz boost (Wigner bases of the irreducible representations of the proper orthochronous Lorentz group). For details, see Appendix B in

http://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

Gold Member
.. it's important to remember that chirality and helicity are only the same for massless spin-1/2 fermions, not for massive ones. ...

I am not sure I agree, certainly the photons has that property, but also the neutrino, that does appear to have mass.

From wiki: https://en.wikipedia.org/wiki/Chirality_(physics)

With the discovery of neutrino oscillation, which implies that neutrinos have mass, the only observed massless particle is the photon. The gluon is also expected to be massless, although the assumption that it is has not been conclusively tested. Hence, these are the only two particles now known for which helicity could be identical to chirality, and only one of them has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames. It is still possible that as-yet unobserved particles, like the graviton, might be massless, and hence have invariant helicity like the photon.

Staff Emeritus
I am not sure I agree

Should we have a vote?

edguy99 and vanhees71
vanhees71
Gold Member
Then, Wikipedia should define chirality for a photon. I don't know, how that should make sense. The photon (and also gluons, but that's a bad example either, because it's pretty certain that there are no free gluons to be ever seen) is the massless realization of the Poincare group based on the representation (1/2,1/2) for the Lorentz subgroup. I don't see, how you could define a chirality transformation. Of course helicity is well defined, and since the photon is massless, it's Lorentz-invariant.

To define chirality you need a representation, where something "flips" under space reflections. To that end you have to extend the proper orthochronous Poincare group ##\mathrm{SO}(1,3)^{\uparrow}## (at least) to the orthochronous Lorentz group ##\mathrm{O}(1,3)^{\uparrow}##. For spin-1/2 particles this leads to the direct sum of the two Weyl-spinor reps. to the Dirac-spinor rep. (which is reducible wrt. to the former group but irreducible wrt. the latter), ##(1/2,0) \oplus (0,1/2)##. The Dirac bispinors can always be split into the two Weyl-spinor parts, defined by the eigenvalues of the Dirac matrix ##\gamma_5=\mathrm{i} \gamma^0 \gamma^1 \gamma^2 \gamma^3##. Then the parity operation is realized by switching (1/2,0) to (0,1/2), i.e., it flips the chirality, and that's why this quantity is called "chirality" in the first place. For massive particles, chirality is, however, not conserved, because the mass term in the Hamiltonian mixes left- and right-handed components. For massless particle you have chiral symmetry (modulo anomalies, but that's another story). It also turns out that for massless particles chirality and helicity are the same, indeed also helicity flips under space reflections, because momentum is a polar and angular momentum an axial vector, i.e., ##\vec{J} \cdot \vec{P}## is a pseudo scalar under space reflections.

edguy99 and dextercioby
Gold Member
... For massive particles, chirality is, however, not conserved, because the mass term in the Hamiltonian mixes left- and right-handed components. For massless particle you have chiral symmetry (modulo anomalies, but that's another story). It also turns out that for massless particles chirality and helicity are the same, indeed also helicity flips under space reflections, ...

People talk about chirality in 2 different ways and it is often unclear.

1/ From the wiki neutrino page: "An experiment done in 1956 by C. S. Wu at Columbia University showed that neutrinos always have left-handed chirality."

I think when they go on to say the neutrinos chirality matches its helicity, they mean when viewed from where it was created, ie. the center of mass of all the particles. Neutrinos always have left-handed chirality, Anti-Neutrinos always have right-handed chirality. Sometimes you also see it worded as "the spin matches the direction of movement".

2/ Clearly as you point out for a massive particle like the neutrino that travels at less then the speed of light. If you were travelling faster then the neutrino and slower then the speed of light and looked back at the neutrino it would be moving away from you and that same neutrino would have right-handed chirality. You would see it "spinning in the opposite direction to its movement" and clearly its chirality does not match its helicity.

As for a vote as to which is right ... please let me know, they both seem to have some valid points ...

vanhees71