Chirality, helicity and spin
Definition: Chirality
A Dirac spinor is a doublet of the two component left- and right-handed Weyl spinors $$\psi=\begin{pmatrix}\psi_L\\ \psi_R\end{pmatrix}$$ The two components are called left-handed and right-handed and refer to the $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ representations of the Lorentz group. The handedness of a spinor is called chirality
The left-handed and right-handed states do not mix under Lorentz transformations, since they live in different representations of the Lorentz group.
One can project out the left- and right-handed components of a dirac spinor using the $\gamma^5$ matrix:
$\gamma^5$ matrix
The $\gamma^5$ matrix is defined as $$\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$$ In the Weyl representation $$\gamma^5=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}$$
Since the components $\psi_L$ and $\psi_R$ are eigenvectors with eigenvalues $\pm 1$ we can define projection operators $$P_R = \frac{1+\gamma^5}{2},\quad P_L=\frac{1-\gamma^5}{2},$$ which give e.g. $$P_R\psi = \begin{pmatrix}0\\ \psi_R\end{pmatrix}.$$ Properties of $\gamma^5$:
- $(\gamma^5)^2=1$
- ${\gamma^5,\gamma^\mu}=0$
- So $\gamma^5$ also fulfills the property of the clifford algebra
The Dirac equation can be written as $$\begin{pmatrix}-m & i\sigma^\mu\partial_\mu\i\bar{\sigma}^\mu\partial_\mu & -m\end{pmatrix}\begin{pmatrix}\psi_L \\ \psi_R\end{pmatrix}=0.$$ In momentum space (by Fourier transformation) this gives $$\sigma^\mu p_\mu\psi_R = (E-\vec{\sigma}\cdot\vec{p})\psi_R=m\psi_L$$ $$\bar{\sigma}^\mu p_\mu\psi_L = (E+\vec{\sigma}\cdot\vec{p})\psi_L=m\psi_R$$ In the massless case $m=0$, we see:
- $\psi_R$ and $\psi_L$ are eigenvectors of the operator $h=\frac{\vec{\sigma}\cdot\vec{p}}{|\vec{p}|}$ with eigenvalues $\pm 1$.
- The equations for $\psi_R$ and $\psi_L$ decouple.
Definition: Helicity
Helicity is the projection of Spin on the direction of momentum given by the operator $$h=\frac{\vec{\sigma}\cdot\vec{p}}{|\vec{p}|}.$$ In the massless theory, left- and right-handed spinors $\psi_L$ and $\psi_R$ are eigenstates of the helicity operator and have opposite helicity. In the masless limit, helicity and chirality eigenstates are the same
Helicity can also be defined for particles with different Spin than $1/2$, in general $$\vec{S}\cdot \vec{p},\Psi_s,$$ where $\vec{S}$ are the generators for rotations. For Spin $1/2$ we have $\vec{S}=\frac{1}{2}\vec{\sigma}$.
In the massive case $m\neq 0$ helicity is still conserved, but the helicity eigenstates are different from the chirality eigenstates $\psi_L$ and $\psi_R$.
A chiral field theory denores a theory which is not symmetric under the exchange of $(A,B)$ and $(B,A)$ representations.
Sometimes one also writes $\psi_{L(R)}$ as a Dirac spinor as $\gamma_5\psi_L=-\psi_L$ or $\gamma_5\psi_R=\psi_R$. If a Dirac or Weyl spinor is ment will be apperant from the context.
In the free massless case, the spin eigenstates are also helicity and chirality eigenstates. This means that the Hamiltonian commutes with the operator for chiality, $\gamma_5$, the operator for helicity, $\frac{\vec{S}\cdot\vec{p}}{E}$, and the Spin oprators, $\vec{S}$.
The QED interacion $\bar{\psi}\cancel{A}\psi=\bar{\psi}_L\cancel{A}\psi_L+\bar{\psi}_R\cancel{A}\psi_R$ is non-chiral, meaning that chirality is conserved ($\psi_L$ and $\psi_R$ are treated symmetrically), but it does not preserve Helicity.