Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physica Scripta

Geodesic evolution of quantum state in the method of flow equations

Yuichi Itto

Published Sep 1, 2010• 0 citations
Publisher

Abstract

Generalizing Wegner's method of flow equations for successfully diagonalizing a Hamiltonian, a condition is found under which the flow of a quantum state becomes geodesic in a submanifold of the projective Hilbert space.
DOI
10.1088/0031-8949/2010/t140/014024

Related papers

Geometry of projective Hilbert space

Alexander N. Grigorenko

Geodesic equations determined by the metric in a projective Hilbert space openP are found. It is shown that connections in the tangent bundle over openP consistent with these geodesics can be introduced. A connection with zero torsion is proposed. Quantum-system dynamics follows the geodesics of another connection which is found for the case of the constant Hamiltonian. A set of geometrical quantities is defined, conserved by a quantum evolution.

Physical Review A•Dec 1, 1992•14 citations

Nonlinear dynamics from linear quantum evolutions

Florio M. Ciaglia, Fabio Di Cosmo, Armando Figueroa +5 more

Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group $\phi_t$ of transformations on a Hilbert space, $\mathcal{H}$, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schr\"{o}dinger equation to a submanifold invariant under the flow $\phi_t$. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow $\phi_t$. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.

•Jan 1, 2019•3 citations

Geometry of quantum mechanics

Abhay Ashtekar, Troy A. Schilling

Quantum mechanics is formulated on the true space of physical states—the projective Hilbert space. It is found that the postulates of quantum mechanics assume an intrinsically geometric form. In particular, observables are represented by real‐valued functions, the Lie bracket on the space of observables is given by a Poisson structure, and the Schrödinger evolution is equivalent to the flow along a Hamiltonian vector field.

AIP conference proceedings•Jan 1, 1995•111 citations

Geometry of the complex extension of Wigner's theorem

Dorje C. Brody

Wigner’s theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalization of Wigner’s theorem: a holomorphically projective (complex geodesic curves preserving) transformation on a quantum state space must be generated by a Hamiltonian that is not necessarily Hermitian.

arXiv:1305.0658•Journal of Physics A Mathematical and Theoretical•Sep 9, 2013•4 citations

Emergent Integrability in a Wegner-type Flow Equation

Yuta Agawa

Continuous unitary transformations, or flow equations, provide a powerful non-perturbative framework for diagonalizing quantum many-body Hamiltonians. This work presents a first-principles proof that the complex, high-dimensional dynamics generated by a Wegner-type flow equation can, under specific and well-defined conditions, rigorously collapse to an exactly integrable second-order ordinary differential equation. We demonstrate that for a one-dimensional spinless fermion model in the non-interacting limit (Δ₀=0), the flow equations for the coupling constants can be derived exactly, revealing a hidden mathematical structure. We term this phenomenon "Emergent Integrability," where the diagonalization flow itself possesses an integrable submanifold, even if the initial Hamiltonian is not integrable. This discovery, established through rigorous and unabridged derivations, reveals that the process of diagonalization can be fundamentally simpler than the initial problem, shedding new light on the intrinsic mathematical structure of renormalization group-like transformations.

•Jul 31, 2025•0 citations

Geometric Hamiltonian quantum mechanics and applications

Davide Pastorello

Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.

International Journal of Geometric Methods in Modern Physics•Aug 10, 2016•7 citations
Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group ϕt\phi_tϕt​ of transformations on a Hilbert space, H\mathcal{H}H, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schr\"{o}dinger equation to a submanifold invariant under the flow ϕt\phi_tϕt​. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow ϕt\phi_tϕt​. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.
Continuous unitary transformations, or flow equations, provide a powerful non-perturbative framework for diagonalizing quantum many-body Hamiltonians. This work presents a first-principles proof that the complex, high-dimensional dynamics generated by a Wegner-type flow equation can, under specific and well-defined conditions, rigorously collapse to an exactly integrable second-order ordinary differential equation. We demonstrate that for a one-dimensional spinless fermion model in the non-interacting limit (Δ₀=0), the flow equations for the coupling constants can be derived exactly, revealing a hidden mathematical structure. We term this phenomenon "Emergent Integrability," where the diagonalization flow itself possesses an integrable submanifold, even if the initial Hamiltonian is not integrable. This discovery, established through rigorous and unabridged derivations, reveals that the process of diagonalization can be fundamentally simpler than the initial problem, shedding new light on the intrinsic mathematical structure of renormalization group-like transformations.
Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.