Heisenberg and Heisenberg-like representations via Hilbert-space-bundle geometry in the non-Hermitian regime

The equivalence between the Schr"odinger and Heisenberg representations is a cornerstone of quantum mechanics. However, this relationship remains unclear in the non-Hermitian regime, particularly when the Hamiltonian is time dependent. In this paper, we address this gap by establishing the connection between the two representations, incorporating the metric of the Hilbert space bundle. We not only demonstrate the consistency between the Schr"odinger and Heisenberg representations but also present a Heisenberg-like representation grounded in the generalized vielbein formalism, which provides a clear and intuitive geometric interpretation. Unlike the standard Heisenberg representation, where the metric of the Hilbert space is encoded solely in the dual states, the Heisenberg-like representation distributes the metric information between both the states and the dual states. Despite this distinction, it retains the same Heisenberg equation of motion for operators. Within this formalism, the Hamiltonian is replaced by a Hermitian counterpart, while the ``non-Hermiticity'' is transferred to the operators. Moreover, this approach extends to regimes with a dynamical metric (beyond the pseudo-Hermitian framework) and to systems governed by time-dependent Hamiltonians.