Parity-time-(PT-)symmetric, non-Hermitian quantum many-body systems are realized as open quantum systems with PT symmetry and compatible couplings to the environment. I review the methodology to theoretically investigate such systems. Taking a mathematical physics perspective, the corresponding Hamiltonians are studied on one hand using the formalisms of PT-symmetric and biorthogonal quantum mechanics, both of which are strongly related. In these frameworks, the notion of observables is altered compared to the standard formalism of Hermitian quantum mechanics. Furthermore, the inner product is modified by including a metric operator. On the other hand, the formalism of Hermitian quantum mechanics is employed. Considering several models ranging from a toy model to one exhibiting quantum critical behavior, I show that the standard formalism always leads to reasonable results while PT-symmetric and biorthogonal quantum mechanics produces results that cannot be reconciled with our physical intuition. The ancilla approach provides a justification for using the standard formalism independent of any specific models. I discuss that within the standard formalism useful methods of quantum many-body theory such as linear response theory and functional integrals cannot easily be extended to non-Hermitian systems. Ironically, this is straightforward within PT-symmetric and biorthogonal quantum mechanics.
