Constraint Phase Space Formulations for Finite-State Quantum Systems: The Relation between Commutator Variables and Complex Stiefel Manifolds (2503.16062v1)

Abstract: We have recently developed the \textit{constraint} coordinate-momentum \textit{phase space} (CPS) formulation for finite-state quantum systems. It has been implemented for the electronic subsystem in nonadiabatic transition dynamics to develop practical trajectory-based approaches. In the generalized CPS formulation for the mapping Hamiltonian of the classical mapping model with commutator variables (CMMcv) method [\textit{J. Phys. Chem. A} \textbf{2021}, 125, 6845-6863], each {connected} component of the generalized CPS is the \textit{complex Stiefel manifold} labeled by the eigenvalue set of the mapping kernel. Such a phase space structure allows for exact trajectory-based dynamics for pure discrete (electronic) degrees of freedom (DOFs), where the equations of motion of each trajectory are isomorphic to the time-dependent Schr\"odinger equation. We employ covariant kernels {within the generalized CPS framework} to develop two approaches that naturally yield exact evaluation of time correlation functions (TCFs) for pure discrete (electronic) DOFs. In addition, we briefly discuss the phase space mapping formalisms where the contribution of each trajectory to the integral expression of the {TCF} of population dynamics is strictly positive semi-definite. The generalized CPS formulation also indicates that the equations of motion in phase space mapping model I of our previous work [\textit{J. Chem. Phys.} \textbf{2016}, 145, 204105; \textbf{2017}, 146, 024110; \textbf{2019}, 151, 024105] lead to a complex Stiefel manifold $\mathrm{U}(F)/\mathrm{U}(F-2)$. It is expected that the generalized CPS formulation has implications for simulations of both nonadiabatic transition dynamics and many-body quantum dynamics for spins/bosons/fermions.

An Advanced Framework for Quantum Dynamics on Complex Stiefel Manifolds

Recent advancements in the phase space formulation for finite-state quantum systems have been proposed through the comprehensive work by Shang, Cheng, and Liu. The paper introduces a generalized constraint phase space (CPS) formulation, extending beyond traditional mappings to leverage the mathematical structure of complex Stiefel manifolds, specifically $\mathrm{U}(F)/\mathrm{U}(F-r)$, to describe finite-state quantum systems. This development allows for an exact, trajectory-based dynamic description for systems with discrete electronic degrees of freedom.

The paper establishes a direct relation between commutator variables in quantum mechanics and complex Stiefel manifolds, providing a significant theoretical expansion to the field of phase space methods. The proposed framework is rooted in affording a rigorous bridge between quantum and classical dynamics, a conceptual innovation lying in the generalized CPS formulation employing covariant kernels. This design facilitates the exact evaluation of time correlation functions (TCFs) in finite-state systems, a notable advancement for accurately simulating electronic population dynamics in quantum chemistry.

Strategically, the paper approaches the mapping from finite-state quantum systems to phase space by using complex Stiefel manifolds. This approach uniquely differentiates itself from earlier methods that were constrained by singularities and nonlinear complexities, especially when the number of states involved is large. By incorporating the commutator matrix $\mathbf{\Gamma}$, the work introduces a more comprehensive phase space structure, unveiling the potential of these manifolds in simulating pure discrete dynamics while maintaining linearity and excluding singularities.

In practical terms, the authors demonstrate two methods within the generalized CPS framework that naturally accomplish the exact dynamics for pure discrete systems by employing covariant kernels: these approaches yield exact trajectory-based dynamics, thereby offering significant implications for nonadiabatic dynamics simulation and many-body quantum dynamics involving spins, bosons, or fermions. This conceptual shift also aligns with the necessity to model composite systems in quantum physics, where the novel phase space is now structured around the unified trajectory dynamics via the complex Stiefel manifolds.

The implications of the generalized CPS formulation are profound in both theoretical and computational chemistry. It opens pathways for precise trajectory-based simulations in composite systems that intertwine nuclear and electronic coordinates, enhancing our capacity to preview quantum mechanical behavior with classical analogs. By avoiding traditional constraints, such as those encountered with $\mathrm{SU}(F)$ or $\mathrm{SU}(2)$ representations, this formulation presents a versatile and potentially more accurate methodological complement to existing quantum mechanics interpretative frameworks.

Looking into the future, the generalized CPS formulation suggests a paradigm shift for the simulation of complex quantum systems in computational chemistry and materials science. Its theoretical underpinnings offer strategies to map quantum dynamics onto trajectory-based models that mirror classical systems -- a critical need for advancing the accuracy and efficiency of simulations in contemporary research fields. As computational techniques further assimilate these high-fidelity mathematical constructs, we can anticipate marked improvements in the modeling of dynamic quantum phenomena across disciplines.