# Helicity vs. Chirality?

by JDługosz
Tags: chirality, helicity
 P: 346 When it was in the news about neutrinos having mass, I wondered what that meant about left-handed only-ness. You could catch up to one and pass it, or slow one down. More recently, I read that the helicity of the neutrino is invariant but the chirality can still change. What is the difference? What exactly are they?
 P: 346 *bump*
P: 1,468
 Quote by JDługosz When it was in the news about neutrinos having mass, I wondered what that meant about left-handed only-ness. You could catch up to one and pass it, or slow one down. More recently, I read that the helicity of the neutrino is invariant but the chirality can still change.
It would be useful if you could give a link/reference/extract to what you read.

 What is the difference? What exactly are they?
If you google for "helicity chirality" you'll gets heaps of explanations - far
more than I could write here. Have a read of some of them. If it's still not clear,
bump the thread again and say what remains mysterious.

P: 346

## Helicity vs. Chirality?

Actually, I didn't understand any of it. I saw explanations that dealt with the math, but no lucid explanation of what they were in semi-layman's terms. Are they two different states, with one permanent and the other mutable? What does that have to do with the single spin?

If you could point me to a link you think would help, I'll start there.
 P: 2,050 Wasn't it accepted that neutrinos always have the same helicity? But now if neutrinos have mass, then you can outrun them, from which perspective they will have opposite helicity. What gives?
 P: 77 My take= Helicity represents direction of motion and Chirality represents orientation. My Opinion= Both are ridiculous words that serve to confuse when put together and serve nothing better when separated. But then, I may be being a bit cynical. It's just that I've never heard either of them used in the context of a discussion. What's the origin of these strange words and why do you care? Why do I? These are good questions.
P: 1,468
 Quote by JDługosz Actually, I didn't understand any of it. I saw explanations that dealt with the math, but no lucid explanation of what they were in semi-layman's terms. Are they two different states, with one permanent and the other mutable?
I'm not very good at phrasing things in layman's terms, but here goes...

"Helicity" and "chirality" are not states, they are properties that particles/fields
possess.

To understand helicity heuristically, do the following experiment. Extend your index
finger out in front of you. As you extend it, rotate it clockwise (about the axis of the finger).
I.e: there's a clockwise twisting helical motion as you extend your arm forward.
Now do it again, this time rotating anti-clockwise. These two things correspond
roughly to opposite helicities: different rotations about the direction of motion.
Now imagine moving your body and head forward faster than you're extending
your arm. The rotation-sense (clockwise or anti-clockwise) doesn't change, but
illustrates what is meant when people say that a Lorentz boost (of your head) that

In contrast, to understand chirality heuristically, hold your left hand in
front of you and arrange the thumb, 1st finger, and 2nd finger so that they all
point in mutually perpendicular directions. Do the same thing simultaneously
with your right hand. The finger arrangements on your two hands form mirror
images of each other, and this corresponds to opposite "chirality".
Now, there is no way you can move you head around that will make your left
hand's finger arrangement look like your right hand's. That's what is
meant when people say that chirality is Lorentz-invariant (although one must
understand that this doesn't include the so-called discrete Lorentz transformation
of parity-reversal, i.e: exchange of "left" and "right" senses).

I think the word "chiral" has its origins in crystallography - when people
noticed that certain substances could occur in different crystalline forms,
mirror-images of each other which could not be superimposed upon
one another.

 What does that have to do with the single spin?
I don't know what you mean by that.

BTW, Idjot: you should definitely care about understanding the difference between
helicity and chirality, at least if you wish to understand particle physics and
its associated relativistic quantum theory. Many people get seriously confused
otherwise when trying to learn about the weak nuclear interaction for the first time.
(I speak from personal experience.)
 P: 346 Thank you strangerep, that starts to help. I understand from what you wrote that chirality is Lorentz-invariant and helicity depends on the observer. What about other particles, such as electrons? The spin along the motion vector in the frame of whatever interaction is about to take place is the helicity. But do electrons have two different types of chirality that are different from each other?
P: 1,468
 Quote by JDługosz Thank you strangerep, that starts to help. I understand from what you wrote that chirality is Lorentz-invariant and helicity depends on the observer. What about other particles, such as electrons? The spin along the motion vector in the frame of whatever interaction is about to take place is the helicity. But do electrons have two different types of chirality that are different from each other?
Alas, for this you'll need to understand more of the math in those google
references I alluded to earlier.

An electron (indeed, any massive fermion), does not have a deterministic
chirality in general. When an electron is a rest, it is a quantum superposition
of left- and right-handed chirality states in equal amounts. If (hypothetically) we
measured the chiralities of a large number of electrons at rest (separated far enough
from each other so we can neglect their electromagnetic interactions), we'd get the
result "left-handed" for half of them, and "right-handed" for the other half.
I.e: "chirality" is not a deterministic property of electrons. To make things
even more complicated, as an electron propagates forward in time along
its worldline, the left-handed and right-handed components of the
superposition will mix together. But you'll need to learn about the "Dirac eqn"
to understand more of that.

For a massless neutrino however, (at least, before we began to suspect there is
no such thing), the chirality is definite (i.e: deterministic). Massless neutrinos have
left-handed chirality (always), anti-neutrinos are right-handed (always).

Hope that helps.
 P: 346 How would you measure the chirality of an electron (as opposed to its helicity)? That is, what property is manifested. (As an aside, how do you, in principle make a measurement on a stationary particle? The observation is due to an interaction and the particle will be moving in the frame of the center of momentum of the interaction.)
P: 1,468
 Quote by JDługosz How would you measure the chirality of an electron (as opposed to its helicity)? That is, what property is manifested.
Sorry, I don't know. Someone else will have to help with this question.
Maybe Uncle Al over on sci.physics.research.

 (As an aside, how do you, in principle make a measurement on a stationary particle? The observation is due to an interaction and the particle will be moving in the frame of the center of momentum of the interaction.)
That's why I said "hypothetical". But theoretically, a moving electron is still in a
superposition of LH and RH chirality states, though it's no longer an even 50-50.
P: 1,136
 Quote by JDługosz Thank you strangerep, that starts to help. I understand from what you wrote that chirality is Lorentz-invariant and helicity depends on the observer. What about other particles, such as electrons? The spin along the motion vector in the frame of whatever interaction is about to take place is the helicity. But do electrons have two different types of chirality that are different from each other?
Fermions in the Weyl (chiral) representation have two bi-spinor parts which are each
others spatial inversions which can be seen from the definition of the gamma matrices:

$$\gamma^0 = \left(\begin{array}{cc} 0 & \ \ I\ \ \\ \ \ I\ \ & 0 \end{array} \right), \qquad \gamma^i = \left(\begin{array}{cc} 0 & +\sigma_i\\ -\sigma_i & 0 \end{array} \right)$$

Where the first represents the time component which isn't reversed and the other three
represent the spatial components (i=x,y,z) which are reversed. Now, an electron has
both chiralities but the amount of them depends on the relative speed. In the electron's
rest frame both are equal but at v = +c or v = -c only the first or the latter survives.
The chirality which survives depends on the sign of the speed but also on the sign of
the spin along the direction of motion.

Since both chiral components are each other spatial inversions, they couple differently
to normal or axial vectors, for instance: The spin coupling to the E field is opposite
for both chiralities because the electric field E is a normal vector which changes
sign under spatial inversion. However, the spin coupling to the B field is the same
for both chiral components since the magnetic field B is an axial vector which does
not change sign under spatial inversion. (Bz can be seen as the result of a circular
current in the xy-plane, reversing the x and y axis leaves the clockwise direction of
the current unchanged)

Thus an electron at rest has no spin coupling to the E field since both chiralities cancel
each other but it does couple to the B field: The electron has an intrinsic magnetic
moment. An electron at higher speed does couple with the E field. In classical electro
dynamics this is because the moving electron sees the E field partly transformed into
a B field in its own rest frame.

For the charge coupling (as opposed to the spin coupling), both chiral components
couple the same to the E and B fields because F = q( E + v x B), and both E as well
as v x B are normal vectors.

The Weyl (chiral) representation became a very important representation of the
electron after it was established that only one of the two chiralities couples to the
Weak force. From the above we can see that this is possible if the Weak force field
is a combination of both a V(ector) and A(xial) current with equal strength. The
couplings add for one chirality while they cancel each other for the other chirality.
This became the successful V-A theory of the Weak force.

Regards, Hans
P: 79
 Quote by JDługosz More recently, I read that the helicity of the neutrino is invariant but the chirality can still change. What is the difference? What exactly are they?
Hello,

I do not understand why the helicity of the neutrino should be invariant now it has a mass ?

Indeed if neutrinos have mass, it just implies that LH and RH chirality states are no more solutions of Weyl equations and so are no more eigenvalues of helicity operator.
Which means we can no more relate helicity to chirality exactly for neutrinos. Does it mean we could observe right helicity neutrinos even in the lab frame ?

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