Helicity vs Chirality

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So I heard on different occasions that chirality it's a very confusing concept and it is often mixed with helicity. I read some definitions and examples from a book and as far as I can tell (at least for QED), helicity it's an operator that gives the component of the spin along the direction of motion and chirality is an eigenstate of the ##\gamma^5## matrix (which doesn't really have a reasonable physical explanation, but it is built in the theory). It seems to me that the distinction is pretty clear and the definition of chirality is quite simple and straightforward, at least mathematically. Am I missing something (I feel I am oversimplifying something and it shouldn't be that easy)? Is it more difficult for QCD, for example? What is the reason chirality is viewed as something complicated? Thank you!
 

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Orodruin
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If you do not have any problems, good for you. The typical problem people have is that they want something more tangible than a mathematical definition and for massless particles they are the same.
 
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If you do not have any problems, good for you. The typical problem people have is that they want something more tangible than a mathematical definition and for massless particles they are the same.
Thank you for your reply. However, what do you mean by tangible? I mean most of the quantities encountered in particle physics are not tangible. Like isospin, or color of quarks, even spin itself, which is not an actual arrow pointing in space, but an inner property of the particle, just like chirality. What is so special about chirality? (again, I just want to make sure I don't gloss over something deeper, that I actually don't understand)
 
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vanhees71
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You are right. Chirality is defined the way you learnt it, and it's hard to explain it beyond this mathematical definition in a proper analogy with something related to our "macroscopic" experience.

The most simple intuitive way is indeed to literally use your right and left hand (chirality=handedness) as examples. They cannot be transformed to each other by any geometrical transformation that is continuously connected with the identity of the symmetry group (i.e., the proper orthochronous Lorentz transformation), but you can map your right hand to your left hand by a spatial reflection, and indeed chirality eigenstates are flipped from one to the other by spatial reflection. To have a non-trivial realization of spatial reflections you have to use both helicity states. For spin-1/2 particles that implies that you need the Dirac field rather than a Weyl field in local relativistic QFT.

The intuitive example of our hands shows that chirality has nothing to do a priori with rotations and helicity, but of course, describing rotations in terms of axial vectors leads to such a connection, and that's why we use the "right-hand rule" to define the direction of a vector product or the orientation of coordinates in the three dimensional space (given the mirror symmetry of macroscopic physics, it's a pure convention; a particle physicist can define it uniquely by saying that the weak interaction couples to left-handed neutrinos and right-handed antineutrinos, i.e., he can uniquely tell an alien what he calls "left" and what "right" in terms of chirality without using his hands, which may be ununderstandable to the alien who may not have hands as we humans have ;-)).

Now it turns out that indeed for massless particles chirality and helicity eigenstates coincide. The reason for massless Dirac particles is that there's no mass term in the Hamiltonian that mixes left- and right-handed states. The intuitive consequence is that you always can overtake a massive particle, i.e., a Lorentz boost can be used to flip the helicity of a massive particle from one to the other frame, while this is not possible to overtake a massless particle which always moves with ##c##, and helicity is thus Lorentz invariant for massless particles.
 

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