(t_1) \rangle$ at time $t_1$ and the same state $| \psi (t_2) \rangle$ at time $t_2$ within the Schrödinger picture, i.e.

$$| \psi (t_2) \rangle=U(t_2,t_1) | \psi (t_1) \rangle.
\label{92}$$

If the time evolution operator $U(t_1,t_2)$ is unitary this results in a constant overall probability. But Eq. $(\ref{91})$ tells us that this is only the case because of the hermiticity of the Hamiltonian. In the case of a non-hermitean Hamiltonian unitary breaks down and the norm of the state can change. In our case this is due to the fact that the singularity at the centre of the black hole leads to an instability of the states resulting in a probability current through the event horizon. Before drawing our attention to this phenomenon again we will first show this non-hermiticity of the Hamiltonian of Eq. $(\ref{88})$.

Hermiticity implies that the expectation value of an operator is real

$$\langle \Psi | H | \Psi \rangle = \langle \Psi | H | \Psi \rangle^{*}.
\label{81}$$

In coordinate representation the inner product of Dirac Spinors is defined as

$$\langle \Phi | \Psi \rangle = \int\limits_{-\infty}^{\infty}\phi^{\dagger}\gamma^0 \psi\ d^3r = \int\limits_{-\infty}^{\infty}\bar \phi \psi\ d^3r.
\label{82}$$

As the free Dirac Hamiltonian $H_{Dirac}$ is hermitian, we have 