  Despite acute interest in the dynamics of non-Hermitian systems, there is a
lack of consensus in the mathematical formulation of non-Hermitian quantum
mechanics in the community. Different methodologies are used in the literature
to study non-Hermitian dynamics. This ranges from consistent frameworks like
biorthogonal quantum mechanics and metric approach characterized by modified
inner products, to normalization by time-dependent norms inspired by open
quantum systems. In this work, we systematically explore the similarities and
differences among these various methods. Utilizing illustrative models with
exact solutions, we demonstrate that these methods produce not only
quantitatively different results but also distinct physical interpretations.
For dissipative systems where non-Hermiticity arises as an approximation, we
find that the normalization method in the $\mathcal{PT}$-broken regime closely
aligns with the full master equation solutions. In contrast, for quantum
systems where non-Hermiticity can be engineered exactly, incorporating metric
dynamics is crucial for the probabilistic interpretation of quantum mechanics,
necessitating the generalization of unitary symmetry to non-Hermitian systems.
This study lays the groundwork for further exploration of non-Hermitian
Hamiltonians, potentially leveraging generalized symmetries for novel physical
phenomena.
