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

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.

8 pages, 3 tables

References in corpus (11)

• Making Sense of Non-Hermitian Hamiltonians
• Dynamically crossing diabolic points while encircling exceptional curves: A programmable symmetric-asymmetric multimode switch
• General properties of fidelity in non-Hermitian quantum systems with PT symmetry
• Non-Hermitian Heisenberg representation
• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• Exponential sensitivity revival of noisy non-Hermitian quantum sensing with two-photon drives
• Interaction-induced Metal to Topological Insulator Transition
• Entanglement Hamiltonian and effective temperature of non-Hermitian quantum spin ladders
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Heisenberg dynamics for non self-adjoint Hamiltonians: symmetries and derivations
• Non-Hermitian-Hamiltonian-induced unitarity and optional physical inner products in Hilbert space

Cited by in corpus (3)

• Liouvillian and Hamiltonian exceptional points of atomic vapors: The spectral signatures of quantum jumps
• Fidelity and quantum geometry approach to Dirac exceptional points in diamond nitrogen-vacancy centers
• Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics