The Dirac equation in 1+1 dimensions
Consider the Dirac equation in one space and one time dimension.
Find an explicit form of the \(\gamma^\mu\) matrices satisfying \[\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu}.\]
Find \(\gamma^5\).
Assuming that the fermion is massless, show that one obtains two independent equations of motion, for particles moving to the left and to the right.
Show that the currents are conserved: \[\partial_\mu\left(\overline{\psi}\gamma^\mu\left(\frac{1-\gamma^5}{2}\right)\psi\right)=0, \quad \partial_\mu\left(\overline{\psi}\gamma^\mu\left(\frac{1+\gamma^5}{2}\right)\psi\right)=0.\]
Switch on an electric field \(A^1\) that is constant in space and varies slowly in time. Find the Hamiltonian and its eigenvalues, assuming that the \(x\) axis has a finite length \(L\) and imposing periodic boundary conditions. Plot the energy spectrum for left- and right-handed fermions.
What happens to the energy spectrum if \(A^1\) is slowly changed by \[\Delta A^1 = \frac{2\pi}{eL}?\]