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Helicity vs. Chirality?

  1. Aug 16, 2007 #1
    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?
  2. jcsd
  3. Aug 21, 2007 #2
  4. Aug 21, 2007 #3


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    It would be useful if you could give a link/reference/extract to what you read.

    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.
  5. Aug 22, 2007 #4
    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.
  6. Aug 23, 2007 #5
    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?
  7. Aug 23, 2007 #6
    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.
  8. Aug 23, 2007 #7


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    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

    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
    the direction of motion relative to your head reverses. This roughly
    illustrates what is meant when people say that a Lorentz boost (of your head) that
    overtakes the particle (your finger) changes the helicity that your head "measures"
    when observing your finger.

    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.

    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.)
  9. Aug 23, 2007 #8
    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?
  10. Aug 24, 2007 #9


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    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.
  11. Aug 24, 2007 #10
    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.)
  12. Aug 25, 2007 #11


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    Sorry, I don't know. Someone else will have to help with this question.
    Maybe Uncle Al over on sci.physics.research.

    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.
  13. Aug 25, 2007 #12

    Hans de Vries

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    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:

    [tex]\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)[/tex]

    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
    Last edited: Aug 25, 2007
  14. Aug 29, 2007 #13

    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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