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

arXiv:quant-ph/9612002

Borel Quantization: Kinematics and Dynamics

H. D. Doebner, P. Nattermann

Published Nov 29, 1996• 5 citations
arXivHTMLPDF

Abstract

In this contribution we review results on the kinematics of a quantum system localized on a connected configuration manifold and compatible dynamics for the quantum system including external fields and leading to non-linear Schrödinger equations for pure states.
DOI
10.48550/arxiv.quant-ph/9612002

Related papers

Nonlinear quantum mechanics: Results and open questions

Gerald A. Goldin

About 15 years ago, we (Heinz-Dietrich Doebner and I) proposed a special type of nonlinear modification of the usual Schrödinger time-evolution equation in quantum mechanics. Our equation was motivated by certain unitary representations of the group of diffeomorphisms of physical space, in the framework of either nonrelativistic local current algebra or quantum Borel kinematics. Subsequently, we developed this and related approaches to nonlinearity in quantum mechanics considerably further, to incorporate theories of measurement, groups of nonlinear gauge transformations, symmetry and invariance properties, unification of a large family of nonlinear perturbations, and possible physical contexts for quantum nonlinearity. Some of our results and highlights of some open questions are summarized.

Physics of Atomic Nuclei•May 1, 2008•7 citations

Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics

Saeid Molladavoudi

In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of information-theoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function. In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same information-theoretic discussion of the Chapter 3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,with the Fisher information metric replaced by the Jensen-Shannon distance information. Furthermore, we show that the non-linear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of three-qubit pure states with a specific (shifted) spectra of single-particle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations.

•Jun 1, 2013•0 citations

Schrödinger and related equations as hamiltonian systems, manifolds of second-order tensors and new ideas of nonlinearity in quantum mechanics

Jan Jerzy Sławianowski, Vasyl Kovalchuk

arXiv:0812.5055•Reports on Mathematical Physics•Jan 1, 2010•13 citations

Borel quantum kinematics of rank k on smooth manifolds

Ute A. Mueller, H. D. Doebner

The authors propose a model for quantizing a nonrelativistic physical system that admits a finite number of internal degrees of freedom. They assume that the configuration space of the given system carries the structure of a smooth manifold M. The kinematics of the given system is assumed to be describable by a flow model. Under certain technical restrictions momentum and position operators are seen to act on a Hilbert space of square-integrable sections of a Hermitian vector bundle over M. They determine their necessary shape up to isometric isomorphism and distinguish two types. In particular, they show that the number of inequivalent quantizations depends on the topology of the underlying configuration space.

Journal of Physics A Mathematical and General•Feb 7, 1993•11 citations

Quantum mechanics on Riemannian manifold in Schwinger's quantization approach II

N.M. Chepilko, A.V. Romanenko

arXiv:hep-th/0102115•The European Physical Journal C•Jul 1, 2001•3 citations

Viscous Dynamics on the Almost Hermitian Manifold: Deriving Quantum Mechanics from Non-Integrable Geometry

Yousif Basim Raad

We present a rigorous geometric framework that unifies spacetime topology with quantum dynamics by relaxing the Riemannian axiom of a torsion-free connection. We define the physical vacuum as an Almost Hermitian Manifold $\mathcal{M} \cong \mathbb{C}^4$ governed by the Chern connection, which preserves the non-integrable complex structure $J$ at the cost of non-vanishing Torsion. We demonstrate that the Schrödinger Equation emerges naturally as the linear hydrodynamic limit of a stochastic diffusion process driven by the imaginary sector of the Torsion tensor (the Lee Form). Furthermore, we prove via the Cayley-Dickson construction and Adams' Theorem on Parallelizability that the maximal dimensionality of a causal, unitary vacuum is constrained to 8 real dimensions ($\mathcal{V} \cong \mathbb{O}$), with 4-dimensional spacetime emerging as a Lagrangian submanifold. This framework resolves the "measurement problem" by identifying $\hbar$ with the kinematic viscosity of the symplectic manifold, rendering quantum non-locality a consequence of symplectic rigidity (Gromov's Theorem).

•Jan 1, 2026•0 citations
About 15 years ago, we (Heinz-Dietrich Doebner and I) proposed a special type of nonlinear modification of the usual Schrödinger time-evolution equation in quantum mechanics. Our equation was motivated by certain unitary representations of the group of diffeomorphisms of physical space, in the framework of either nonrelativistic local current algebra or quantum Borel kinematics. Subsequently, we developed this and related approaches to nonlinearity in quantum mechanics considerably further, to incorporate theories of measurement, groups of nonlinear gauge transformations, symmetry and invariance properties, unification of a large family of nonlinear perturbations, and possible physical contexts for quantum nonlinearity. Some of our results and highlights of some open questions are summarized.
In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of information-theoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function. In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same information-theoretic discussion of the Chapter 3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,with the Fisher information metric replaced by the Jensen-Shannon distance information. Furthermore, we show that the non-linear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of three-qubit pure states with a specific (shifted) spectra of single-particle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations.
The authors propose a model for quantizing a nonrelativistic physical system that admits a finite number of internal degrees of freedom. They assume that the configuration space of the given system carries the structure of a smooth manifold M. The kinematics of the given system is assumed to be describable by a flow model. Under certain technical restrictions momentum and position operators are seen to act on a Hilbert space of square-integrable sections of a Hermitian vector bundle over M. They determine their necessary shape up to isometric isomorphism and distinguish two types. In particular, they show that the number of inequivalent quantizations depends on the topology of the underlying configuration space.
We present a rigorous geometric framework that unifies spacetime topology with quantum dynamics by relaxing the Riemannian axiom of a torsion-free connection. We define the physical vacuum as an Almost Hermitian Manifold M≅C4\mathcal{M} \cong \mathbb{C}^4M≅C4 governed by the Chern connection, which preserves the non-integrable complex structure JJJ at the cost of non-vanishing Torsion. We demonstrate that the Schrödinger Equation emerges naturally as the linear hydrodynamic limit of a stochastic diffusion process driven by the imaginary sector of the Torsion tensor (the Lee Form). Furthermore, we prove via the Cayley-Dickson construction and Adams' Theorem on Parallelizability that the maximal dimensionality of a causal, unitary vacuum is constrained to 8 real dimensions (V≅O\mathcal{V} \cong \mathbb{O}V≅O), with 4-dimensional spacetime emerging as a Lagrangian submanifold. This framework resolves the "measurement problem" by identifying ℏ\hbarℏ with the kinematic viscosity of the symplectic manifold, rendering quantum non-locality a consequence of symplectic rigidity (Gromov's Theorem).