Autumn · project 13 of 20

Spectral flow and the chiral anomaly in 1+1

Topic: Symmetries and Noether's theorem

Goal: Understand the origin of the chiral anomaly as a geometric effect related to the spectral flow of energy levels.

  1. Consider the massless Dirac equation in 1+1, \[i\gamma^\mu(\partial_\mu+ieA_\mu)\psi=0,\] on a ring of length \(L\).

  2. Impose periodic boundary conditions and find the energy spectrum \(E_n(A_1)\).

  3. Change \(A_1(t)\) slowly from \(0\) to \[\Delta A_1=\frac{2\pi}{eL}.\]

  4. Plot the evolution of the energy levels and show that the levels cross zero.

  5. Show that the numbers of right- and left-handed states change, even though the chiral current is formally conserved.

  6. Discuss why this cannot be explained without taking the vacuum into account, even in a single-particle setting.

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