on}\gamma^\mu\lambda&=&
-\overline{\lambda}\gamma^\mu\epsilon&=&
-\left(\overline{\epsilon}\gamma^\mu\lambda\right)^\dagger\ ,\\
\overline{\epsilon}\gamma^\mu\gamma_5\lambda&=&
\overline{\lambda}\gamma^\mu\gamma_5\epsilon&=&
\left(\overline{\epsilon}\gamma^\mu\gamma_5\lambda\right)^\dagger\ ,\\
\overline{\epsilon}\gamma^\mu\gamma^\nu\lambda&=&
-\overline{\lambda}\gamma^\mu\gamma^\nu\epsilon&=&
\left(\overline{\epsilon}\gamma^\mu\gamma^\nu\lambda\right)^\dagger\ .
\end{array}$$ It is a useful exercise to establish any of these identities.

The Dirac Equation {#Sec3.5}
------------------

Let us now consider the dynamics of a single free Dirac spinor field, thus described, at the classical level, by complex valued Grassmann odd variables forming a 4-component Dirac spinor $\psi(x^\mu)$. The action principle for a such a system is given by the Lorentz invariant quantity $$S\left[\psi,\overline{\psi}\right]=\int d^4x^\mu\,
{\mathcal L}\left(\psi,\partial_\mu\psi\right)\ ,$$ with the Lagrangian density[^19] $${\mathcal L}=\frac{1}{2}i\left[\,\overline{\psi}\gamma^\mu\partial_\mu\psi\,-\,
\partial_\mu\overline{\psi}\gamma^\mu\psi\,\right]\,-\,m\overline{\psi}\psi\ .
\label{eq:DiracL}$$ Through the variational principle, the associated equation of motion is the celebrated Dirac equation, $$\left[i\gamma^\mu\partial_\mu\,-\,m\right]\,\psi(x^\mu)=0\ .
\label{eq:Dirac}$$

A few remarks are in order. Given the relations in (\[eq:chiral1\]) and (\[eq:chiral2\]), it is clear that the kinetic term $\overline{\psi}\gamma^\mu\partial_\mu\psi$ couples the chiral components of the Dirac spinor by preserving their chirality, while the coupling $m\overline{\psi}\psi$ switches between the two chirality components. As will become clear hereafter, since the real parameter $m\ge 0$ in fact determines the mass of the particle quanta associated to such a field, a massless Dirac particle propagates without flipping its chirality, whereas a massive particle sees both its chiral components contribute to its spacetime dynamics.

The term $m\o