Fermionic entropy in 2D black hole

The Dirac equation for a Dirac field in a 2D black hole with a Schwarzschild-like metric is solved. Contrary to the bosonic eigensolution, the fermionic one is found to be divergent on the horizon, but it is not too singular and can be normalized in the inner product. ʹt Hooftʹs boundary condition, which is suitable for scalar bosons, is found not to be applicable to the fermionic eigensolutions. In order to discretize the spectrum, a “quasi-periodic” boundary condition is first proposed, and the fermionic effective Hamiltonian operator HF is shown to be self-adjoint with this “quasi-periodic” boundary condition. The result shows that the divergence in the fermionic entropy has the same form as that in the bosonic one, except that the coefficient is different. Thus the original divergence in the bosonic entropy cannot be cancelled by a supersymmetric Bose-Fermi cancellation.