In single-particle quantum mechanics, the probability of finding the particle in all space is conserved due to the hermiticity of the Hamiltonians (and remains equal to unity for all times, if normalized).
But in relativistic quantum field theory, particle numbers are not conserved. For example, in QED, an initial state consisting of an electron-positron pair can annihilate into two photons in the final state. There is no trace of the initial electron and the positron in the final state. Similarly, in $\beta$-decay, an initial neutron is converted into a proton, an electron and an anti-electron neutrino in the final state. There is no neutron after the decay takes place though the Fermi theory Hamiltonian is Hermitian.
So it seems that hermitian Hamiltonian in QFT is not responsible for the conservation of the probability. I seem to have a conceptual glitch here which I would like to be clarified. What is the role of hermitian Hamiltonians in relativistic QFT in the time development of states? There must be some constraining feature of hermitian QFT Hamiltonian. Sorry if the question sounds dumb.
Some more thoughts on this for clarification In single-particle quantum mechanics, the norm of a quantum state is the probability of finding the particle in all of space. Is there a similar probability interpretation of the norm of a state in QFT? Since QFT hamiltonians are hermitian, the norm of the state remains preserved under time development but the state itself can change. This confuses me. To take the example given above the norm of the initial state neutron does not change under time development but the state itself can into other states. How do we interpret this in QFT?