Chirality

Chirality¶

Just as we say that a particle can have left- or right-handed Helicity, we also say that a particle can have left- or right-handed chirality. As we said above, for massless particles the chirality and helicity are the same. A massless left-chiral particle also has left-helicity.

However, a massive particle has a specific chirality. A massive left-chiral particle may have either left- or right-helicity, depending on your reference frame relative to the particle(imagine overtaking a vehicle). In all reference frames, the particle will still be left-chiral, no matter what helicity it is.

It is an inherently quantum mechanical sense in which a particle is left- or right-handed. For now, let us focus on fermions, which are “spin one-half.” Recall that this means that if you rotate an electron by 360 degrees, you don’t get the same quantum mechanical state: you get the same state up to a minus sign! This minus sign is related to quantum interference. A fermion’s chirality tells you how it gets to this minus sign in terms of a complex number.

What happens when you rotate a left- vs right-chiral fermion 360 degree about its direction of motion. Both particles pick up a -1, but the left-chiral fermion goes one way around the complex plane, while the right-chiral fermion goes the other way. The circle on the right represents the complex phase of the particle’s quantum state; as we rotate a particle, the value of the phase moves along the circle. Rotating the particle 360 degrees only brings you halfway around the circle in a direction that depends on the chirality of the fermion.

Physically, it means that the phase of the wavefunction changes. Rotating a fermion shifts its quantum wavefunction in a way that depends on the fermion chirality.

Particles with different chiralities are really different particles(see below image) If we have a particle with left-handed helicity, then we know that there should also be a version of the particle with right-handed helicity. On the other hand, a particle with left-handed chirality needn’t have a right-chiral partner.

(electron and positron have same charge but opposite chirality.)