The quantum mechanical worldview is, in many ways, strikingly different from our intuitions about the real world. One such conspicuous difference is that measurable physical quantities in quantum mechanics cannot be represented simply by numbers, but instead need to be considered as mathematical operators, or functions over a space of physical states onto another space of physical states.

However, we usually insist that these operators give back real values, since, in the real world, we associate measurable quantities with real numbers. Consequently, physical operators are normally restricted to being what are called “Hermitian operators”. However, in my article I have explored two scenarios where this dogma can be questioned.

One of the most important operators in quantum mechanics is the Hamiltonian operator, which corresponds to physical energy. It may seem intuitively obvious that the Hamiltonian operator must be Hermitian, but this is, in fact, only true so long as energy is conserved. Now while the energy of the entire universe is certainly conserved, it may not be necessarily so for a part of it, such as a radioactive nuclide. This system loses energy to the outside world whenever it emits an alpha or a beta particle and therefore, does not conserve energy. Such an open system can be described by an effective non-Hermitian Hamiltonian that yields complex energy values corresponding to short-lived “resonant states.” While the eigenfunctions of these states are not normalizable, it simply means that the states extend over the entire universe.

Another instance is where the theory of parity-time (PT) symmetry (symmetry under the combined operations of spatial inversion and time reversal) considers the Hamiltonian of the entire universe as non-Hermitian but operating in a parameter space where its eigenvalues are exclusively real. For example, a system of two closed-shell atoms can be described using a non-Hermitian Hamiltonian that is PT symmetric and yields energy eigenvalues that can be rendered real by parameter tuning. This shows that Hermiticity is only a sufficient condition for real values, not a necessary one.

The non-Hermitian formulation has since been picked up and applied by researchers in various fields, most notably in topological matter and many-body systems. Just as generalizing to complex numbers can often help describe physical systems more elegantly, a non-Hermitian generalization of quantum mechanics can help broaden our perspective of quantum systems.