# What are the differences between chiral states and helical states in condensed matter physics?

As we know in particle physics, chirality corresponds to eigenvalues of the fifth gamma matrix, and helicity corresponds to the value of the projection of spin onto momentum. So in condensed matter physics, are they the same as the particle-physics definition? And how does one distinguish chiral states from helical states?

• Possible duplicate: physics.stackexchange.com/q/232591/50583. However, your question is somewhat unclear: You exhibit the correct (and obviously different!) definitions of chirality and helicity, so what do you mean when you ask "are they the same and original definition"? When you ask for a method to distinguish, are you asking for an experimental method or for how to distinguish the state in the abstract formalism? Oct 13 '17 at 12:08
• I mean how to disitinguish the state in the abstract formalism.Only when the particle is massless, helicity equals to chirality.I'm just confused about when I should choose chirality or helicity to discrible system ?Thanks:) Oct 13 '17 at 12:46

The condensed matter physics typically works for non-relativistic systems described by Schroedinger-like hamiltonian. For these systems the notion of chirality is simply not defined. Thus is because the chirality is the property of the representation defining the way in which it is transformed under the Lorentz group, and it doesn't survive in non-relativistic limit.

However, in some systems the spectrum of excitations in the solid body can be Dirac- or Weyl-like. This is the thing which happens for semimetals (3D case) and for graphene (2D case). In this case, you can effectively (i.e., near the corresponding Dirac and Weyl points) define the chirality. This is, of course, not the chirality defined by the Lorentz group representation.

As for helicity, its definition - the projection of the total angular momentum on the direction of motion - doesn't suffer from taking of the non-relativistic limit. However, there isn't clear relation of the helciity and effective chirality described above, unlike the case of massless representations of the Poincare group, for which helicity and chirality are the same up to the sign.