# Quantum integrability of quadratic Killing tensors

@article{Duval2005QuantumIO, title={Quantum integrability of quadratic Killing tensors}, author={Christian Duval and Galliano Valent}, journal={Journal of Mathematical Physics}, year={2005}, volume={46}, pages={053516} }

Quantum integrability of classical integrable systems given by quadratic Killing tensors on curved configuration spaces is investigated. It is proven that, using a “minimal” quantization scheme, quantum integrability is ensured for a large class of classic examples.

## 46 Citations

### QUANTUM INTEGRABILITY FOR THE BELTRAMI-LAPLACE OPERATORS OF PROJECTIVELY EQUIVALENT METRICS OF ARBITRARY SIGNATURES

- Mathematics, Computer Science
- 2020

It is shown that in all signatures the Killing tensors constructed by projectively equivalent metrics correspond to commuting differential operators.

### Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures

- Mathematics
- 2019

We generalize the result of [31] to all signatures.

### Killing tensors and canonical geometry

- Physics
- 2014

The systematic derivation of constants of the motion, based on Killing tensors and the gauge covariant approach, is outlined. Quantum dots are shown to support second-, third- and fourth-rank Killing…

### Geodesic flows of c-projectively equivalent metrics are quantum integrable

- Mathematics
- 2021

Given two c-projectively equivalent metrics on a Kähler manifold we show that the canoncially constructed, Poisson-commuting integrals of motion of the geodesic flow, linear and quadratic in momenta,…

### Natural star-products on symplectic manifolds and related quantum mechanical operators

- Mathematics
- 2014

### Position Representation of Quantum Mechanics over Riemannian Configuration Space

- PhysicsQuantum versus Classical Mechanics and Integrability Problems
- 2019

The last chapter of the book is devoted to two very important issues of the developed quantum theory. The first one is related with systematic construction of the so called position representation of…

## References

SHOWING 1-10 OF 42 REFERENCES

### Projectively Equivariant Quantization and Symbol Calculus: Noncommutative Hypergeometric Functions

- Mathematics
- 2001

We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative…

### Projectively Equivariant Quantization Map

- Mathematics
- 2000

Let M be a manifold endowed with a symmetric affine connection Γ. The aim of this Letter is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T* M…

### Integrability Versus Separability for the Multi-Centre Metrics

- Mathematics
- 2004

The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra…

### Integrability and the Kerr-(A)dS black hole in five dimensions

- Mathematics
- 2005

In this paper we prove that the Hamilton-Jacobi equation for a particle in the five dimensional Kerr-(A)dS black hole is separable, for arbitrary rotation parameters. As a result we find an…

### CLASSICAL AND QUANTUM INTEGRABLE SYSTEMS IN GL(2)+* AND SEPARATION OF VARIABLES

- Mathematics
- 1995

Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra gI + *(2, IR) are integrated by…

### Various Quantum Mechanical Aspects of Quadratic Forms

- Mathematics
- 1995

Two of the oldest known classical integrable systems are: (i) n-decoupled harmonic oscillators, constrained to the surface of the n-sphere (i.e., the classical C. Neumann system), and (ii) geodesic…

### Geometry of Quadrics and Spectral Theory

- Mathematics
- 1980

In this paper we are concerned with integrable Hamiltonian systems. This concept goes back to classical analytical dynamics of the last century. Briefly these are nonlinear systems of ordinary…

### Quantum Mechanics of Yano tensors: Dirac equation in curved spacetime

- Physics
- 2004

In spacetimes admitting Yano tensors the classical theory of the spinning particle possesses enhanced worldline supersymmetry. Quantum mechanically generators of extra supersymmetries correspond to…