text: Vector gauge theories with massless Dirac fermion fields    exhibit chiral symmetry, i.e., rotating the left-handed and the right-handed components independently makes no difference to the theory. We can write this as the action of rotation on the fields:
  :   and  
  or
  :   and   
  With  flavors, we have unitary rotations instead: .
  More generally, we write the right-handed and left-handed states as a projection operator acting on a spinor. The right-handed and left-handed projection operators are
  and
  Massive fermions do not exhibit chiral symmetry, as the mass term in the Lagrangian, ,  breaks chiral symmetry explicitly.
  Spontaneous chiral symmetry breaking may also occur in some theories, as it most notably does in quantum chromodynamics.
  The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as vector symmetry, and a component that actually treats them differently, known as axial symmetry. (cf. Current algebra.) A scalar field model encoding chiral symmetry and its breaking is the chiral model.
  The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference.
  The general principle is often referred to by the name chiral symmetry. The rule is absolutely valid in the classical mechanics of Newton and Einstein, but results from quantum mechanical experiments show a difference in the behavior of left-chiral versus right-chiral subatomic particles.