Interacting and open quantum systems can be formulated in terms of an effective non-Hermitian (NH) Hamiltonian; however, there are important constraints that must be satisfied by the effective action and the associated Green's functions. One common approach to many-body NH systems is to incorporate the anti-Hermitian part of the Hamiltonian directly in the Matsubara Green's function. Here, we show that such an approach is incompatible with the standard framework for systems with interactions. Furthermore, we furnish a consistent physical description for such systems by determining their distinction from conventional interacting physics, and find that they are described by pseudo-Hermitian quantum mechanics. In addition, we characterize the zero-temperature distribution functions within several frameworks for NH systems. As an application of our results, we consider the electromagnetic response of a NH quasiparticle Hamiltonian based on the ($1+1$)-dimensional NH Dirac model subject to various physical descriptions.

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• Dissipative dynamics
• Electrical conductivity
• Electronic structure
• Fermions
• Lifetimes & widths