RANDOM DYNAMICS TABLE OF CONTENTS SEQUENCE OF DERIVATIONS
Helicity and Chirality
HELICITY AND CHIRALITY

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.

Helicity refers to the relation between a particle's spin and direction of motion. 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 $\vec{s}$ on this axis: the helicity is $\vec{s}\hat{p}$, i.e. the component of angular momentum along the momentum.
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. If the spin is projected parallell on the direction of motion, the particle is of right helicity, if the projection is antiparallell to the direction of motion, the particle has left helicity.
The helicity is measurable, however not Lorentz invariant, unless the particle is massless. Since helicity is the scalar product of two vectors, it is however invariant under space rotations.

Something is $chiral$ when it cannot be superimposed on its mirror image, like for example our hands: you can only superimpose your left hand on your right hand if you put them back to back or inside to inside, you cannot superimpose the two hands turned to you (the word chirality comes from the Greek word for hand, cf. the word chiropractor or the French word for surgery, la chirurgie). Like our hands, chiral objects are classified into left-chiral and right-chiral objects. For a massless fermion, the Dirac equation reads

\begin{displaymath}
\gamma^{\mu}\partial_{\mu}\psi=0,
\end{displaymath}

which is also satisfied by $\gamma_{5}\psi$,
\begin{displaymath}
\gamma^{\mu}\partial_{\mu}(\gamma_{5}\psi)=0
\end{displaymath}

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,
\begin{displaymath}
\psi_{L}=\frac{1}{2}(1-\gamma_{5})\psi{\hspace{3mm}}{\rm {and}}
{\hspace{3mm}}\psi_{R}=\frac{1}{2}(1+\gamma_{5})\psi
\end{displaymath}

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, unlike helicity, chirality is not directly measurable. It is however Lorentz invariant: chiral fields transform among themselves under the Lorentz group. Chirality and Lorentz invariance are thus closely connected.
We can express any fermion as $\psi=\psi_{L}+\psi_{R}$, a massive particle always has a L-handed as well as a R-handed component. In the massless case $\psi$ however "disintegrates" into separate helicity states: the Dirac equation splits into two independent parts, reformulated as the Weyl equations
\begin{displaymath}
{\bf {\sigma}}\hat{p}[\frac{1}{2}(1 \pm \gamma_{5})\psi]
= \pm \frac{1}{2}(1 \pm \gamma_{5})\psi
\end{displaymath}

where ${\bf {\sigma}}\hat{p}$ is the helicity operator expressed in terms of the Pauli spin matrices ${\bf {\sigma}}$.
The Weyl fermions, i.e. the massless chiral states $(1/2)(1 \pm \gamma_{5})\psi$, are physical since they correspond to eigenstates of the helicity operator.
A massless particle, which is in perpetual motion, thus has an unchangeable helicity. The reason is that its momentum cannot be altered, and its spin of course remains unchanged.
For a massive particle however, we can perform a Lorentz transformation along the direction of the particle's momentum with a velocity larger than the particle's, changing the direction of the momentum. Since the spin direction remains the same, the helicity of the particle changes.
Spin properties are moreover related to spatial dimension.