Non-Hermitian quantum mechanics is merely an approximation.

"Non-Hermitian quantum mechanics" is, of course, not an extended theory that denies the unitarity of quantum mechanics. It is a discussion of approximation methods within the scope of standard quantum mechanics.

When I first heard about this when I was a student, I thought, "That's strange." This is because researchers at that time were still seriously trying to consider non-Hermitian Hamiltonians not as an approximation method, but as an extension beyond the framework of conventional quantum mechanics.

In the first place, long before the field of condensed matter physics, non-Hermitian Hamiltonians were used as approximations in the field of elementary particles. For example, an effective Hamiltonian like H=M-iΓ was already commonly used in the analysis of oscillation phenomena between neutral K mesons and their antiparticles. (Γ is the particle decay rate matrix.) This is a well-known fact that appears in every elementary particle textbook.

In fact, H=M-iΓ is a good approximation and explains the experimental data for neutral K meson oscillation phenomena. However, in the field of elementary particles, H=M-iΓ is properly derived and used as an approximation from the original Hermitian Hamiltonian. They never think that there is truly a non-Hermitian Hamiltonian. It is an approximation.

Even if they admit it is an approximation, one can find papers on "open-system non-Hermitian dynamics" with a sloppy stance. They simply declare it to be an open system at the beginning, and then create an arbitrary non-Hermitian Hamiltonian model and perform numerical calculations on it. There is no awareness of the danger of picking up mere artifacts of approximation and presenting them as physical results, which is not physics of open systems at all.

For example, let us consider a Hamiltonian H=p-Ex as a model where charged particles always flow from left to right due to acceleration by a constant electric field. p is the momentum operator, x is the position operator, and E is a positive real number representing the electric field. This H is a proper Hermitian operator.

When approximating this continuous model with a hopping discrete model, assuming that the discrete position x of each site has discrete values, it is possible to consider a non-Hermitian Hamiltonian H'=gΣ |x+1><x|-EΣ x|x><x|. While the original H causes a completely unitary time evolution that preserves total probability, the time evolution of the approximate H' does not. In this non-Hermitian approximation, the total probability undergoes oscillatory time variation and is not conserved.

However, it is strange to treat this temporal oscillation of total probability as a result of genuine open-system physics. The single system described by the original H is not coupled to an external environment and is not an open system. Even if you perform numerical calculations and plot the oscillatory time variation of the total probability, it does not reflect true physics.

Simply writing down a non-Hermitian Hamiltonian model as an "open system" and analyzing it is still meaningless as physics. What kind of approximation does it emerge from? What is the time domain where that approximation is valid? And how large are the correction terms following that approximation? Answering these questions properly is also important for physics.