Quantum Theory, Gravity and Second Order Geometry

• URL: http://arxiv.org/abs/2410.06799v2
• Date: Thu, 24 Oct 2024 11:53:09 GMT
• Title: Quantum Theory, Gravity and Second Order Geometry
• Abstract summary: We argue that a consistent coupling of a quantum theory to gravity requires an extension of ordinary first order' Riemannian geometry. For a 4-dimensional spacetime, tangent spaces become 18-dimensional. We show that, despite the introduction of higher order derivatives, the Ostragradski instability can be avoided.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We argue that a consistent coupling of a quantum theory to gravity requires an extension of ordinary `first order' Riemannian geometry to second order Riemannian geometry, which incorporates both a line element and an area element. This extension results in a misalignment between the dimension of the manifold and the dimension of the tangent spaces. In particular, we find that for a 4-dimensional spacetime, tangent spaces become 18-dimensional. We then discuss the construction of physical theories within this framework, which involves the introduction of terms that are quadratic in derivatives in the action. On a flat spacetime, the quadratic sector is perpendicular to the first order sector and only affects the normalization of the path integral, whereas in a curved spacetime the quadratic sector couples to the first order sector. Moreover, we show that, despite the introduction of higher order derivatives, the Ostragradski instability can be avoided, due to an order mixing of the two sectors. Finally, we comment on extensions to higher order geometry and on relations with non-commutative and generalized geometry.

Related papers

• A Riemannian View on Active Subspaces [0.0]
  Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most.<n>Subspaces are studied in an intrinsically local sense, restricted to mean-centered geodesic-balls.<n>Examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.
  arXiv  Detail & Related papers  (2026-07-28T00:22:24Z)
• Connection Factorization in Constrained Quantum Mechanics [0.0]
  We investigate constrained quantum motion on curves and surfaces using connection factorization methods.<n>We show that Laplace operators admit an exact half-connection factorization generated by connection one-forms.<n>Using orthonormal moving frames, we derive effective geometric contributions for planar curves, spatial curves, and embedded surfaces.
  arXiv  Detail & Related papers  (2026-05-28T02:00:17Z)
• The deep Hilbert space of all-to-all interacting SU(3) atoms: from quantum to classical [45.94400632660844]
  We study the emergence of chaos in multilevel atoms with all-to-all interactions, inspired by cavity QED.<n>Our work contributes to the study of the quantum-classical correspondence in chaotic systems, and uncovers a rich structure in multilevel all-to-all interacting models.
  arXiv  Detail & Related papers  (2025-12-04T19:00:01Z)
• General theory for geometry-dependent non-Hermitian bands [4.330177255999498]
  In non-Hermitian lattices, the spectrum and eigenstates under open boundary conditions depend on the bulk geometry even in the thermodynamic limit.<n>We establish a higher-dimensional non-Bloch band theory based on the concept of "strip generalized Brillouin zones"<n>For the first time, we derive a general criterion for the geometry dependence of non-Hermitian bands.
  arXiv  Detail & Related papers  (2025-06-28T03:37:05Z)
• Moduli space of Conformal Field Theories and non-commutative Riemannian geometry [0.0]
  We discuss the analogy between collapsing Conformal Field Theories and measured Gromov-Hausdorff limit.<n>Motivated by this analogy we propose the notion of non-commutative (quantum)<n>We explain how this structure is related to Connes's spectral triples in the case d=1.
  arXiv  Detail & Related papers  (2025-06-01T08:32:23Z)
• Quantum geometry of the parameter space: a proposal for curved systems [0.0]
  We introduce an equivalent definition that generalizes the Zanardi, et al, formulation of the quantum geometric tensor. The parameter-dependent metric modifies the behavior of both the quantum metric tensor and Berry curvature in a purely geometric way.
  arXiv  Detail & Related papers  (2024-03-14T18:51:26Z)
• Laser-Dressed States on Riemannian Manifolds: A Generalization of the Kramers-Henneberger Transformation [0.0]
  We analytically study the laser-driven nonlinear dynamics of a quantum particle. The geometry of space results in a potential-like term that supports bound states on the manifold. We deduce a Schr"odinger-like equation in the Kramers-Henneberger frame.
  arXiv  Detail & Related papers  (2024-02-16T10:57:34Z)
• Continuous percolation in a Hilbert space for a large system of qubits [58.720142291102135]
  The percolation transition is defined through the appearance of the infinite cluster. We show that the exponentially increasing dimensionality of the Hilbert space makes its covering by finite-size hyperspheres inefficient. Our approach to the percolation transition in compact metric spaces may prove useful for its rigorous treatment in other contexts.
  arXiv  Detail & Related papers  (2022-10-15T13:53:21Z)
• Pointillisme \`a la Signac and Construction of a Pseudo Quantum Phase Space [0.0]
  We construct a quantum-mechanical substitute for the symplectic phase space. The total space of this fiber bundle consists of geometric quantum states. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets.
  arXiv  Detail & Related papers  (2022-07-31T16:43:06Z)
• Geometry Interaction Knowledge Graph Embeddings [153.69745042757066]
  We propose Geometry Interaction knowledge graph Embeddings (GIE), which learns spatial structures interactively between the Euclidean, hyperbolic and hyperspherical spaces. Our proposed GIE can capture a richer set of relational information, model key inference patterns, and enable expressive semantic matching across entities.
  arXiv  Detail & Related papers  (2022-06-24T08:33:43Z)
• A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations [77.86290991564829]
  Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. We study a particular sequence of maps between manifold, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between implementing neural networks of practical interest.
  arXiv  Detail & Related papers  (2021-12-17T11:43:30Z)
• On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization [5.876141028192136]
  This paper proposes a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. We observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization. Results provide a few new theoretical insights to unanswered questions in the literature.
  arXiv  Detail & Related papers  (2021-10-23T03:13:56Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• Geometry of Similarity Comparisons [51.552779977889045]
  We show that the ordinal capacity of a space form is related to its dimension and the sign of its curvature. More importantly, we show that the statistical behavior of the ordinal spread random variables defined on a similarity graph can be used to identify its underlying space form.
  arXiv  Detail & Related papers  (2020-06-17T13:37:42Z)
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder [68.8204255655161]
  We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally defined on an infinite cylinder. We retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting.
  arXiv  Detail & Related papers  (2020-03-16T11:37:23Z)
• Topological hyperbolic lattices [0.0]
  We show how the quantized curvature and edge dominance of hyperbolic geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.
  arXiv  Detail & Related papers  (2020-03-16T03:41:48Z)

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.