Heisenberg and Heisenberg-Like Representations via Hilbert Space Bundle Geometry in the Non-Hermitian Regime

TL;DR

The paper addresses the problem of non-Hermitian, time-dependent quantum systems breaking the usual equivalence between Schrödinger and Heisenberg representations. It introduces a metricized Hilbert-space formalism, using a time-dependent metric $G(t)$, to restore consistency and derive a Heisenberg equation of motion with $H_H = U^{-1} H_S U$, ensuring canonical commutation relations remain valid. It further develops a Heisenberg-like representation via a generalized vielbein $E(t)$, where $G(t)=E^(t)^ E(t)$ and observables transform as $O_{HL}= E O_S  E^{-1}$, with a Hermitian generator $H_lat$ that can be set to zero to render states time-independent; this distributes the metric information between states and duals. Collectively, the work provides a physically consistent, geometrically grounded framework that preserves equivalence among Schrödinger, Heisenberg, and Heisenberg-like representations in non-Hermitian QM, and extends naturally to dynamically evolving metrics and time-dependent Hamiltonians.

Abstract

The equivalence between the Schrödinger 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 study, 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 Schrödinger 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.