Math 2.0

Quantum theory Definition

Quantum chaos

Builds on Laplace–Beltrami operatorGeodesic flowErgodic transformationRiemannian manifoldSpectrum of the Laplace–Beltrami operator

Definition

Let (M,g)(M, g) be a compact connected Riemannian manifold without boundary, and let Δg\Delta_g be its Laplace–Beltrami operator. The quantum system is −Δg-\Delta_g on L2(M)L^2(M), whose classical limit is the geodesic flow on the unit cotangent bundle S∗MS^*M. The system is called quantum chaotic when this geodesic flow is chaotic. Here chaotic means ergodic for the Liouville measure, or a stronger property such as being Anosov. Quantum chaos is the study of the spectrum and the eigenfunctions of such systems in the high-energy limit. More generally it studies semiclassical operators such as −ℏ2Δ+V-\hbar^2 \Delta + V as ℏ→0\hbar \to 0 when the classical Hamiltonian flow is chaotic.

Remarks

Example: a compact hyperbolic surface, of constant curvature −1-1, has an Anosov and ergodic geodesic flow, so the Laplacian on it is quantum chaotic. Quantum ergodicity holds for it, and for arithmetic surfaces quantum unique ergodicity has been proved for the joint eigenfunctions of the Laplacian and the Hecke operators.

Non-example: on the flat torus ℝ2/2πℤ2\R^2 / 2\pi\Z^2 the geodesic flow keeps the direction of motion fixed, so it is not ergodic on S∗MS^*M. Its eigenfunctions ei(kx+ly)e^{i(kx + ly)} are not chaotic, and its eigenvalues k2+l2k^2 + l^2 have high multiplicities.

The Bohigas–Giannoni–Schmit conjecture predicts that the local spacings of the eigenvalues of chaotic systems follow the statistics of random matrices. The Berry–Tabor conjecture predicts Poisson statistics for generic integrable systems. Both are supported by extensive numerical evidence and remain open in general.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Ergodic transformation, concept, no evidence levelErgodic transformationconceptGeodesic, concept, no evidence levelGeodesicconceptLaplace–Beltrami operator, concept, no evidence levelLaplace–Beltrami oper…conceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptRiemannian metric, concept, no evidence levelRiemannian metricconceptSectional curvature, concept, no evidence levelSectional curvatureconceptSmooth function, concept, no evidence levelSmooth functionconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptSpectrum of the Laplace–Beltrami operator, concept, no evidence levelSpectrum of the Lapla…conceptRiemannian manifold, definition, no evidence levelRiemannian manifolddefinitionTangent bundle, definition, no evidence levelTangent bundledefinitionGeodesic flow, example, no evidence levelGeodesic flowexampleQuantum chaos, definition, no evidence levelQuantum chaosdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Ergodic transformation concept
  2. Geodesic concept
  3. Laplace–Beltrami operator concept
  4. Riemannian manifold concept
  5. Riemannian metric concept
  6. Sectional curvature concept
  7. Smooth function concept
  8. Smooth manifold concept
  9. Spectrum of the Laplace–Beltrami operator concept
  10. Riemannian manifold definition
  11. Tangent bundle definition
  12. Geodesic flow example
  13. Quantum chaos definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_chaos
About this page
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Kinddefinition
WritingW0 stub
MSC81Q50, 58J51, 37D40
DefinesQuantum chaos
UsesLaplace–Beltrami operator, Geodesic flow, Ergodic transformation, Riemannian manifold, Spectrum of the Laplace–Beltrami operator
Changed3 November 2026

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