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:2604.04128

Quantization of Lagrangian Descriptors

Javier Jiménez-López, V. J. García-Garrido

math.DSnlin.CDquant-phSubmitted Apr 5, 2026• Announced Apr 7, 2026• 0 citations
arXivHTMLPDF

Abstract

We formulate Lagrangian descriptors (LDs) in the path integral framework. Averaging the classical LD over fluctuations about extremal trajectories defines a quantum LD that incorporates quantum effects. Invariant manifolds, which sharply organize classical transport, become finite-width phase space structures under quantum fluctuations, and their overlap provides a geometric mechanism consistent with tunneling as fluctuation-induced delocalization of transport barriers. We demonstrate this approach for the Hamiltonian saddle, where path integral sampling reveals manifold broadening and barrier penetration. This establishes a geometric framework for studying phase space transport and tunneling beyond the classical regime, while also providing a natural route toward the application of LDs to field theory.

Related papers

Path Integral Quantization and Riemannian-Symplectic Manifolds

Sergei V. Shabanov, John R. Klauder

We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a

•Jan 1, 1998•6 citations

Flux analysis, the correspondence principle, and the structure of quantum phase space

Rex T. Skodje, Henry W. Rohrs, James VanBuskirk

The hydrodynamic analogy for quantum mechanics is developed in a quantum phase-space representation. The key technical step in this development is the formulation of a phase-space flux (or current) density expression. Lagrangian fluid trajectories in phase space are explicitly found as solutions to a set of first-order ordinary differential equations which are formulated without making semiclassical approximation to the quantum dynamics. These fluid trajectories may be used to construct sharp structures in quantum phase space in the same way that one uses classical trajectories in classical mechanics. While these sharp structures do not themselves represent states, their properties ``control'' the dynamics of states in a manner familiar from the theory of first-order flows. The formalism reduces to Liouville dynamics in the classical limit, \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. In addition to several analytic applications to the pure- and mixed-state dynamics, the approach is numerically implemented for tunneling dynamics in a double-well problem. A new picture for quantum tunneling emerges where the tunneling transport occurs when density spirals outward from one well along real-valued fluid trajectories and then into the other well. The tunneling is found to occur through localized portals, or ``flux gates'' in the phase space. We advocate the phase-space hydrodynamic model as a general tool to understand classical quantum correspondence.

Physical Review A•Sep 1, 1989•74 citations

Measure for measure: Covariant skeletonizations of phase space path integrals for systems moving on Riemannian manifolds

Karel V. Kuchař

We define phase space path integrals for systems moving on a Riemannian manifold and subject to a generalized potential by a skeletonization procedure which is manifestly covariant under point transformations. We achieve this goal by introducing a natural analog S(x″,t″‖x′,p′,t′) of the Hamilton principal function with phase space initial data. One class of such functions is based on the parallel transport of momentum, a second class is obtained by a modification of the first class, and a third class is based on the geodesic deviation transport of momentum. The third class of principal functions is geometrically privileged. We skeletonize the canonical action integral by replacing it by a manifestly covariant chain of phase space principal functions. Different functions lead to the same functional as we infinitely refine the skeletonization along a smooth path. Our phase space path integral is always taken with the natural Liouville measure, but the integration over momentum variables brings down a nontrivial measure to the remaining configuration space path integral. Because nondifferentiable rather than smooth paths dominate the path integral, different phase space principal functions generate different configuration space path measures. Such measures lead to quantum propagators which satisfy Schrödinger’s equations with all possible scalar curvature terms ∼ℏ2R. The geometrically privileged phase space principal function (based on the geodesic deviation transport) leads to the Schrödinger equation without any curvature term.

Journal of Mathematical Physics•Aug 1, 1983•37 citations

Hamiltonian mechanics

Belal Ehsan Baaquie

Path integrals are by and large defined directly in terms of the configuration space representation of the quantum entity's degrees of freedom and employ the Lagrangian description of the quantum entity; most of the path integrals in this book follow this approach. The Hamiltonian provides another independent approach for defining path integrals and is discussed in this chapter. Two important path integrals, which are directly based on the Hamiltonian, are the following: (a) one that is defined on the degree of freedom's phase space , defined as the tensor product of the degree of freedom space and its canonical conjugate momentum space; and (b) path integrals using the coherent state basis instead of the coordinate basis. Path integrals defined on phase space, or for coherent states, are both based on the Hamiltonian. To put in the foreground the role of the Hamiltonian in quantum mechanics, the canonical equations connecting the Lagrangian to the Hamiltonian are discussed. A brief review of Hamiltonian mechanics, also called the canonical equations, is given in Section 5.1 and the connection of symmetries with conservation laws is discussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Section 5.3, for both Minkowski and Euclidean time. Phase space path integrals are defined in Section 5.4. Canonical quantization based on the Poisson brackets is discussed in Section 5.5, and Dirac brackets required for quantizing constrained systems are derived in Section 5.7. Coherent states and their path integrals are discussed in Sections 5.10 to 5.14.

Cambridge University Press eBooks•Mar 27, 2014•1 citation

Path Integrals in Noncommutative Quantum Mechanics

Dragovich, B, Rakic, Z

Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.

•Sep 22, 2003•20 citations

Geometric Diagnostics of Scrambling-Related Sensitivity in a Bohmian Preparation Space

Stephen Wiggins

The Out-of-Time-Order Correlator (OTOC) is a standard algebraic diagnostic of quantum information scrambling, but it offers limited direct geometric intuition. In this note, we propose a Bohmian, trajectory-based framework for constructing a geometric diagnostic of scrambling-related sensitivity using Lagrangian Descriptors (LDs). To avoid the uncertainty-principle obstruction to assigning independent initial position and momentum within a single wave function, we evaluate Bohmian dynamics over a two-dimensional preparation space of localized Gaussian wavepackets labeled by their initial center and momentum kick. For the inverted harmonic oscillator, this construction is analytically tractable: the wavepacket-center dynamics and their dependence on preparation parameters can be written explicitly. In particular, away from the equilibrium origin, the exponential growth of the associated preparation-space stability matrix yields an $\mathcal{O}(e^{ωT})$ bound on the sensitivity of the wavepacket-center LDs, motivating a semiclassical comparison with sensitivity structures associated with OTOC growth. In this sense, the LD provides a geometric indicator of scrambling-related sensitivity. We conclude by discussing how this preparation-space picture suggests a program for future work regarding the distinct microcanonical regimes previously reported for the inverted harmonic oscillator.

quant-pharXiv:2603.20803•Mar 24, 2026•0 citations
We formulate Lagrangian descriptors (LDs) in the path integral framework. Averaging the classical LD over fluctuations about extremal trajectories defines a quantum LD that incorporates quantum effects. Invariant manifolds, which sharply organize classical transport, become finite-width phase space structures under quantum fluctuations, and their overlap provides a geometric mechanism consistent with tunneling as fluctuation-induced delocalization of transport barriers. We demonstrate this approach for the Hamiltonian saddle, where path integral sampling reveals manifold broadening and barrier penetration. This establishes a geometric framework for studying phase space transport and tunneling beyond the classical regime, while also providing a natural route toward the application of LDs to field theory.
We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a
The hydrodynamic analogy for quantum mechanics is developed in a quantum phase-space representation. The key technical step in this development is the formulation of a phase-space flux (or current) density expression. Lagrangian fluid trajectories in phase space are explicitly found as solutions to a set of first-order ordinary differential equations which are formulated without making semiclassical approximation to the quantum dynamics. These fluid trajectories may be used to construct sharp structures in quantum phase space in the same way that one uses classical trajectories in classical mechanics. While these sharp structures do not themselves represent states, their properties ``control'' the dynamics of states in a manner familiar from the theory of first-order flows. The formalism reduces to Liouville dynamics in the classical limit, \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. In addition to several analytic applications to the pure- and mixed-state dynamics, the approach is numerically implemented for tunneling dynamics in a double-well problem. A new picture for quantum tunneling emerges where the tunneling transport occurs when density spirals outward from one well along real-valued fluid trajectories and then into the other well. The tunneling is found to occur through localized portals, or ``flux gates'' in the phase space. We advocate the phase-space hydrodynamic model as a general tool to understand classical quantum correspondence.
We define phase space path integrals for systems moving on a Riemannian manifold and subject to a generalized potential by a skeletonization procedure which is manifestly covariant under point transformations. We achieve this goal by introducing a natural analog S(x″,t″‖x′,p′,t′) of the Hamilton principal function with phase space initial data. One class of such functions is based on the parallel transport of momentum, a second class is obtained by a modification of the first class, and a third class is based on the geodesic deviation transport of momentum. The third class of principal functions is geometrically privileged. We skeletonize the canonical action integral by replacing it by a manifestly covariant chain of phase space principal functions. Different functions lead to the same functional as we infinitely refine the skeletonization along a smooth path. Our phase space path integral is always taken with the natural Liouville measure, but the integration over momentum variables brings down a nontrivial measure to the remaining configuration space path integral. Because nondifferentiable rather than smooth paths dominate the path integral, different phase space principal functions generate different configuration space path measures. Such measures lead to quantum propagators which satisfy Schrödinger’s equations with all possible scalar curvature terms ∼ℏ2R. The geometrically privileged phase space principal function (based on the geodesic deviation transport) leads to the Schrödinger equation without any curvature term.
Path integrals are by and large defined directly in terms of the configuration space representation of the quantum entity's degrees of freedom and employ the Lagrangian description of the quantum entity; most of the path integrals in this book follow this approach. The Hamiltonian provides another independent approach for defining path integrals and is discussed in this chapter. Two important path integrals, which are directly based on the Hamiltonian, are the following: (a) one that is defined on the degree of freedom's phase space , defined as the tensor product of the degree of freedom space and its canonical conjugate momentum space; and (b) path integrals using the coherent state basis instead of the coordinate basis. Path integrals defined on phase space, or for coherent states, are both based on the Hamiltonian. To put in the foreground the role of the Hamiltonian in quantum mechanics, the canonical equations connecting the Lagrangian to the Hamiltonian are discussed. A brief review of Hamiltonian mechanics, also called the canonical equations, is given in Section 5.1 and the connection of symmetries with conservation laws is discussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Section 5.3, for both Minkowski and Euclidean time. Phase space path integrals are defined in Section 5.4. Canonical quantization based on the Poisson brackets is discussed in Section 5.5, and Dirac brackets required for quantizing constrained systems are derived in Section 5.7. Coherent states and their path integrals are discussed in Sections 5.10 to 5.14.
Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.
The Out-of-Time-Order Correlator (OTOC) is a standard algebraic diagnostic of quantum information scrambling, but it offers limited direct geometric intuition. In this note, we propose a Bohmian, trajectory-based framework for constructing a geometric diagnostic of scrambling-related sensitivity using Lagrangian Descriptors (LDs). To avoid the uncertainty-principle obstruction to assigning independent initial position and momentum within a single wave function, we evaluate Bohmian dynamics over a two-dimensional preparation space of localized Gaussian wavepackets labeled by their initial center and momentum kick. For the inverted harmonic oscillator, this construction is analytically tractable: the wavepacket-center dynamics and their dependence on preparation parameters can be written explicitly. In particular, away from the equilibrium origin, the exponential growth of the associated preparation-space stability matrix yields an O(eωT)\mathcal{O}(e^{ωT})O(eωT) bound on the sensitivity of the wavepacket-center LDs, motivating a semiclassical comparison with sensitivity structures associated with OTOC growth. In this sense, the LD provides a geometric indicator of scrambling-related sensitivity. We conclude by discussing how this preparation-space picture suggests a program for future work regarding the distinct microcanonical regimes previously reported for the inverted harmonic oscillator.