Chirality and duality

The terms $chirality$, $helicity$ and $handedness$ are used in a somewhat sloppy way, often as synonyms.
Just like left and right are defined relative to the front of our elongated bodies, the handedness of an elementary particle is defined in relation to some direction, or axis, related to the particle. To a particle in motion is associated the axis defined by its momentum, and its $helicity$ is defined by the projection of the particle's spin on this axis, as $\vec{s}\hat{p}$ which gives the component of angular momentum along the momentum $\vec{p}$. The helicity operator thus projects out two physical states, with the spin along or opposite the direction of motion, whether the particle is massive or not. The helicity is measurable, however not Lorentz invariant. Since helicity is the scalar product of two vectors, it is however invariant under space rotations. If the particle has vanishing mass, its helicity is also invariant under Lorentz transformations.
For a massless fermion, the Dirac equation reads

$$\gamma^{\mu}\partial_{\mu}\psi=0,$$ (50)

which is also satisfied by $\gamma_{5}\psi$,

$$\gamma^{\mu}\partial_{\mu}(\gamma_{5}\psi)=0$$ (51)

where the combination of the $\gamma$ matrices, $\gamma_{5}=i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$ has the properties $\gamma_{5}^{2}=1$ and $\{\gamma_{5},\gamma_{\mu}\}=0$. This allows us to define the $chirality$ operators which project out left-handed and right-handed states,

$$\psi_{L}=\frac{1}{2}(1-\gamma_{5})\psi{\hspace{3mm}}{\rm {and}} {\hspace{3mm}}\psi_{R}=\frac{1}{2}(1+\gamma_{5})\psi$$ (52)

where $\psi_{L}$ and $\psi_{R}$ satisfy the equations $\gamma_{5}\psi_{L}$ = -$\psi_{L}$ and $\gamma_{5}\psi_{R}$ = $\psi_{R}$, so the chiral fields are eigenfields of $\gamma_{5}$, regardless of their mass. Massive chiral states are however not physical since $\gamma_{5}$ does not commute with the free Hamiltonian for a massive fermion. So in the general case, contrary to the helicity, the chirality is not directly measurable. It is however Lorentz invariant: chiral fields transform among themselves under the Lorentz group.

In the massless case, the Dirac equation splits into two independent reformulated as the Weyl equations

$$\vec{\sigma}\hat{p}[\frac{1}{2}(1 \pm \gamma_{5})\psi] = \pm \frac{1}{2}(1 \pm \gamma_{5})\psi$$ (53)

The massless chiral states $(1/2)(1 \pm \gamma_{5})\psi$ are thus physical, since they correspond to eigenstates of the helicity operator, with eigenvalues $\pm1$.
We can always express a fermion as $\psi=\psi_{L}+\psi_{R}$, and a massive particle has got L-handed as well as R-handed components. In the massless case $\psi$ however "disintegrates" into separate helicity states. A massless particle, which is in perpetual motion, thus has an unchangeable handedness, that is, its momentum can be altered by a Lorentz transformation, while its helicity remains unchanged. For a massive particle, on the other hand, we can perform a Lorentz transformation along the particle's momentum direction with a velocity larger than the particle's, so the particle's momentum direction changes. The direction of its spin however remains the same, meaning that its helicity changes.