[Paper Review] Quantization of classical integrable systems. Part III: systems in n-dimensional Euclidean space

Massimo Marino, N. N. Nekhoroshev|arXiv (Cornell University)|Jan 27, 2010

Nonlinear Waves and Solitons17 references3 citations

TL;DR

This paper applies symmetrization-based quantization methods to classical integrable systems in n-dimensional Euclidean space, demonstrating that one-particle systems in central force fields yield quasi-integrable quantum systems via standard symmetrization. For the free rigid body in n=6 dimensions, a modified symmetrization procedure—adding a second-order correction to the Manakov polynomial—is required to restore quantum integrability, with a conjecture that similar modifications may work for n>6.

ABSTRACT

In this paper we give examples of applications of general methods of quantization by symmetrization of classical integrable systems, which have been illustrated in two previous works by the same authors. We consider two classes of systems in n spatial dimensions, which respectively describe a point particle in a central force field and a freely rotating rigid body. In the former case, the application of the general methods to an integrable classical system leads in an almost straightforward way to the quasi-integrability of the corresponding quantum system. In the latter case instead, a modification of the symmetrization procedure is necessary in order to achieve quantum integrability for n=6.

Motivation & Objective

• To extend the general quantization framework for classical integrable systems to n-dimensional Euclidean space.
• To investigate whether standard symmetrization preserves quantum integrability for one-particle systems in central force fields.
• To determine whether the free rigid body in n=6 dimensions requires a modified quantization procedure to achieve quantum integrability.
• To provide a constructive method for modifying symmetrization to restore commutativity with the Hamiltonian in the n=6 case.
• To conjecture that analogous modifications may yield quasi-integrability for n>6 dimensions.

Proposed method

• Apply the general symmetrization procedure from prior work [2] to construct quantum operators from classical integrable sets of functions.
• Use Lie algebraic structures and central subsets to identify integrable sets of classical functions with 2n−k elements, where k is the number of central integrals.
• For the one-particle central force system, show that symmetrization of classical integrals leads directly to a quasi-integrable quantum system.
• For the free rigid body, identify the classical Manakov polynomial of degree four as a central integral in the classical integrable set.
• Demonstrate that standard symmetrization of the Manakov polynomial fails to commute with the Hamiltonian in the n=6 case.
• Introduce a corrected operator by adding a second-order symmetric polynomial in left-invariant momenta, restoring commutativity with both the Hamiltonian and other integrals.

Experimental results

Research questions

• RQ1 Can standard symmetrization quantization preserve integrability for one-particle systems in n-dimensional central force fields?
• RQ2 Why does standard symmetrization fail to produce a quantum integrable system for the free rigid body in n=6 dimensions?
• RQ3 What modification to the symmetrization procedure restores quantum integrability for the n=6 rigid body?
• RQ4 Is there a systematic way to construct quasi-integrable quantum systems for higher-dimensional rigid bodies (n>6)?
• RQ5 How does the number of central integrals in the classical system relate to the dimension n and the generalized moments of inertia?

Key findings

• For one-particle systems in n-dimensional central force fields, standard symmetrization of classical integrals yields a quasi-integrable quantum system with k central operators.
• For the free rigid body in n≤5 dimensions, standard symmetrization leads to a quantum integrable system without modification.
• In the n=6 case, the standard symmetrization of the Manakov polynomial c_{6,2} does not commute with the Hamiltonian, breaking quantum integrability.
• A modified operator C_{6,2} = c_{6,2} + (5/12)∑_{i<j} λ_i²λ_j² (P^L_ij)² restores commutativity with both the Hamiltonian and other integrals.
• The commutator [c_{5,1}, C_{6,2}] vanishes due to a cancellation between terms, verified via direct computation, confirming closure of the quantum integrable set.
• The symbol of the modified operator C_{6,2} matches the classical function c_{6,2}, ensuring functional independence and quasi-independence of the quantum operator set.