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  1. Is there a simple intuitive way to understand the difference between left-handed and right-handed fermions (electrons say)?
  2. How to experimentally distinguish between them?
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For the first question, no there can be no such general rule. The reason is the same why there is no intuitive way to understand the difference between your right and left shoe -- they are just reflections of each other but neither one is more fundamental (assuming you are not a pirate).

In other words, all the difference (that really matters) comes from the given theory. If the theory is $P$-symmetric (such as electromagnetism) you have no way to distinguish between these (that's why you haven't heard anyone talking about right- or left-handed electrons in the classical physics courses; there were just electrons). On the other hand, if the theory violates $P$-symmetry you have a chance since one type of particles will interact differently than the other. E.g. weak interactions violate the $P$-symmetry so that only left-handed electrons and left-handed neutrinos can form a weak isospin doublet (and so can right-handed antiparticles).

I will leave the experimental part of the question to someone more educated on these matters. But e.g. for neutrinos you might be able to measure helicity (which is as good as chirality since neutrinos are almost massless) which is a projection of spin on to the momentum. You know the momentum direction (since you know where the collision happened and where you registered the particle) and I suppose there are standard methods for measuring spin projection to arbitrary direction (but I admit complete ignorance on these experimental matters).

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Can we say that left-handed and right-handed electrons are different particles? –  Revo Aug 18 '11 at 0:22
    
@Revo: hm, I wouldn't probably say different particles. They are just different polarizations of the same particle (similarly like for light you get two circular polarizations). On the other hand, thanks to weak interactions these two polarizations behave in certain situations completely differently and so it may be more useful to view them as different particles. In any case, whether they are different or same is just a label depending on your definition of "different" and "particle". There is not much physics in this discussion. –  Marek Aug 18 '11 at 6:30
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@Revo--- I like to think of right handed and left handed electrons as different particles, because the left handed electron forms a weak SU(2) doublet with the neutrino, so weak SU(2) rotates these into each other, while the right-handed electron is all alone. Absorbing Higgs condensate particles keeps flipping right handed electrons into left handed electrons all the time, so the physical mass eigenstate, the physical electron, has a mass and can't be separated right and left. –  Ron Maimon Aug 19 '11 at 9:05
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Examples of devices used to measure electron helicity (the current polarization state, not the handedness) are the Møller Polarimeter (which uses e.g. an Iron target where the spins of the iron atoms are aligned using an external magnetic field) and the Compton polarimeter (which uses a polarized laser beam, i.e. photons as scattering partners).

The helicity measurement works by scattering the electrons off another polarized particle (i.e. with well known polarization) and exploting the fact that the interaction rate (cross section) depends on the helicity of the electron.

The (electron) neutrino helicity was experimentally measured with the Goldhaber Experiment (there seems to be only a German version of this wikipedia article).

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