Relational bundle geometric formulation of non-relativistic quantum mechanics

• URL: http://arxiv.org/abs/2501.02046v1
• Date: Fri, 03 Jan 2025 19:00:00 GMT
• Title: Relational bundle geometric formulation of non-relativistic quantum mechanics
• Authors: J. François, L. Ravera,
• Abstract summary: We present a bundle geometric formulation of non-relativistic many-particles Quantum Mechanics.<n>A wave function is seen to be a $mathbbC$-valued cocyclic tensorial 0-form on configuration space-time.<n>Applying the dressing field method, we obtain a relational reformulation of this geometric non-relativistic QM.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a bundle geometric formulation of non-relativistic many-particles Quantum Mechanics. A wave function is seen to be a $\mathbb{C}$-valued cocyclic tensorial 0-form on configuration space-time seen as a principal bundle, while the Schr\"odinger equation flows from its covariant derivative, with the action functional supplying a (flat) cocyclic connection 1-form on the configuration bundle. In line with the historical motivations of Dirac and Feynman, ours is thus a Lagrangian geometric formulation of QM, in which the Dirac-Feynman path integral arises in a geometrically natural way. Applying the dressing field method, we obtain a relational reformulation of this geometric non-relativistic QM: a relational wave function is realised as a basic cocyclic 0-form on the configuration bundle. In this relational QM, any particle position can be used as a dressing field, i.e. as a "physical reference frame". The dressing field method naturally accounts for the freedom in choosing the dressing field, which is readily understood as a covariance of the relational formulation under changes of physical reference frame.

Related papers

• Symmetry-protected topology and deconfined solitons in a multi-link $\mathbb{Z}_2$ gauge theory [45.88028371034407]
  We study a $mathbbZ$ lattice gauge theory defined on a multi-graph with links that can be visualized as great circles of a spherical shell.<n>We show that this leads to state-dependent tunneling amplitudes underlying a phenomenon analogous to the Peierls instability.<n>By performining a detailed analysis based on matrix product states, we prove that charge deconfinement emerges as a consequence of charge-fractionalization.
  arXiv  Detail & Related papers  (2026-03-02T22:59:25Z)
• A Geometric Approach to Strongly Correlated Bosons: From $N$-Representability to the Generalized BEC Force [12.368400057425]
  Building on recent advances in reduced density matrix theory, we develop a geometric framework for describing strongly correlated lattice bosons.<n>We first establish that translational symmetry, together with a fixed pair interaction, enables an exact functional formulation expressed solely in terms of momentum occupation numbers.
  arXiv  Detail & Related papers  (2026-01-04T20:10:59Z)
• Geometric View of One-Dimensional Quantum Mechanics [0.0]
  We apply De Haro's Geometric View of Theories to one of the simplest quantum systems.<n>The classical phase space M = T*Q is taken as the base of a trivial Hilbert bundle E M x H.<n>The position and momentum representations are realised as different global trivialisations of this bundle.
  arXiv  Detail & Related papers  (2025-12-30T00:48:29Z)
• Geometric foundations of thermodynamics in the quantum regime [0.0]
  We present a comprehensive geometrical formulation of quantum thermodynamics based on contact geometry and principal fiber bundles.<n>A principal fiber bundle over the manifold of density operators distinguishes the quantum state structure from thermodynamic labels.<n>Non-equilibrium extensions, formulated through pseudo-Riemannian metrics and connections on the principal bundle, introduce curvature-induced holonomy that quantifies irreversibility in cyclic processes.
  arXiv  Detail & Related papers  (2025-11-13T09:32:57Z)
• Geometry-induced Coulomb-like potential in helically twisted quantum systems [0.0]
  We investigate the Schr"odinger equation in a three-dimensional helically twisted space characterized by a non-trivial torsion parameter.<n>The intrinsic coupling between angular and longitudinal momenta induced by the torsion gives rise to an attractive Coulomb-like potential term.<n>The interplay between the torsion parameter and the effective Coulomb-like interaction is analyzed in detail, showing how geometric deformation generates novel quantum confinement mechanisms.
  arXiv  Detail & Related papers  (2025-07-06T23:20:26Z)
• Symmetries, Conservation Laws and Entanglement in Non-Hermitian Fermionic Lattices [37.69303106863453]
  Non-Hermitian quantum many-body systems feature steady-state entanglement transitions driven by unitary dynamics and dissipation.<n>We show that the steady state is obtained by filling single-particle right eigenstates with the largest imaginary part of the eigenvalue.<n>We illustrate these principles in the Hatano-Nelson model with periodic boundary conditions and the non-Hermitian Su-Schrieffer-Heeger model.
  arXiv  Detail & Related papers  (2025-04-11T14:06:05Z)
• Quantum geometric tensors from sub-bundle geometry [0.0]
  We use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by a generalization of the Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor, which contains an additional curvature contribution.
  arXiv  Detail & Related papers  (2025-03-21T14:08:06Z)
• A Multi Affine Geometric Framework for Quantum Nonlocality. Unifying Berry Phases, Entanglement, and Coherence [0.0]
  We develop a multi affine geometric framework to unify classical and quantum mechanical laws. We show that divergences in dual affine connections naturally give rise to quantum interference and nonlocal correlations.
  arXiv  Detail & Related papers  (2025-03-05T19:34:58Z)
• Alternative Framework to Quantize Fermionic Fields [0.0]
  We derive the Floreanini-Jackiw representation of the Schr"odinger equation for the wave functional of fermionic fields. We show that the framework can be applied to develop theories of interaction between fermionic fields and other external fields.
  arXiv  Detail & Related papers  (2025-03-04T07:38:03Z)
• Covariant non-perturbative pointer variables for quantum fields [44.99833362998488]
  We derive and renormalize the integro-differential equation that governs the detector pointer-variable dynamics. Our formal solution, expressed in terms of Green's functions, allows for the covariant, and causal analysis of induced observables on the field.
  arXiv  Detail & Related papers  (2025-02-03T11:53:31Z)
• Measurement-induced transitions for interacting fermions [43.04146484262759]
  We develop a field-theoretical framework that provides a unified approach to observables characterizing entanglement and charge fluctuations. Within this framework, we derive a replicated Keldysh non-linear sigma model (NLSM) By using the renormalization-group approach for the NLSM, we determine the phase diagram and the scaling of physical observables.
  arXiv  Detail & Related papers  (2024-10-09T18:00:08Z)
• Interacting chiral fermions on the lattice with matrix product operator norms [37.69303106863453]
  We develop a Hamiltonian formalism for simulating interacting chiral fermions on the lattice. The fermion doubling problem is circumvented by constructing a Fock space endowed with a semi-definite norm. We demonstrate that the scaling limit of the free model recovers the chiral fermion field.
  arXiv  Detail & Related papers  (2024-05-16T17:46:12Z)
• Matter relative to quantum hypersurfaces [44.99833362998488]
  We extend the Page-Wootters formalism to quantum field theory. By treating hypersurfaces as quantum reference frames, we extend quantum frame transformations to changes between classical and nonclassical hypersurfaces.
  arXiv  Detail & Related papers  (2023-08-24T16:39:00Z)
• Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density [0.0]
  We generalize classical mechanics to poly-symplectic fields and recover De Donder-Weyl theory. We provide commutation relations for the classical and quantum fields that generalize the Koopman-von Neumann and Heisenberg algebras.
  arXiv  Detail & Related papers  (2023-05-05T01:30:45Z)
• Stochastic Mechanics and the Unification of Quantum Mechanics with Brownian Motion [0.0]
  We show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a process that is rotated in the complex plane. We then extend this theory to relativistic theories on integrals using the framework of second order geometry.
  arXiv  Detail & Related papers  (2023-01-13T10:40:27Z)
• Geometric phases along quantum trajectories [58.720142291102135]
  We study the distribution function of geometric phases in monitored quantum systems. For the single trajectory exhibiting no quantum jumps, a topological transition in the phase acquired after a cycle. For the same parameters, the density matrix does not show any interference.
  arXiv  Detail & Related papers  (2023-01-10T22:05:18Z)
• Entanglement dynamics of spins using a few complex trajectories [77.34726150561087]
  We consider two spins initially prepared in a product of coherent states and study their entanglement dynamics. We adopt an approach that allowed the derivation of a semiclassical formula for the linear entropy of the reduced density operator.
  arXiv  Detail & Related papers  (2021-08-13T01:44:24Z)
• A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics [0.0]
  We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians.
  arXiv  Detail & Related papers  (2021-07-06T09:27:26Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)
• Quantum Complementarity through Entropic Certainty Principles [0.0]
  Uncertainty relations arise by monotonicity of the relative entropies that participate in the underlying entropic certainty. In general, the entropic certainty principle naturally captures the physics of order/disorder parameters.
  arXiv  Detail & Related papers  (2020-05-04T18:02:47Z)
• Unitary preparation of many body Chern insulators: Adiabatic bulk boundary correspondence [14.4034719868008]
  We prepare an out-of-equilibrium many-body Chern insulator (CI) and associated bulk-boundary correspondence unitarily. We show that a non-linear ramp may work more efficiently in approaching the topological state. We also compute the edge current in the time evolved state of the system under a semi-periodic boundary condition.
  arXiv  Detail & Related papers  (2020-05-04T13:14:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.