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arXiv:0908.2048 · Russian Journal of Mathematical Physics

Quantum geometry and quantum mechanics of integrable systems

Mihail Vladimirovich Karasev

Published Mar 1, 2009• 3 citations
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Abstract

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ∞) and are invariant under quantum dynamics in the long-time range O(ħ −∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.

DOI
10.1134/s1061920809010051

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Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ∞) and are invariant under quantum dynamics in the long-time range O(ħ −∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.
For a generic quantum integrable system, we describe the asymptotics of the eigenstate density and of the trace of the evolution operator in all orders of the quantization parameter. This is done by using quantum symplectic geometry, which makes the given quantum system to be equivalent to a deformed classical system with arbitrary accuracy with respect to the quantization parameter. The asymptotics is explicitly given via the deformed symplectic form, deformed Liouville-Arnold tori, and deformed Maslov class.
This chapter discusses in some detail the connection between the classical and quantum mechanical equations of motion, largely using examples that are solvable. It recounts how the old quantum theory was formulated by quantizing the action. Choosing irreducible cycles around the tori leads naturally to defining the action and angle variables of motion, and the quantization of action around each cycle. The chapter shows that families of these periodic orbits contribute to the oscillating part of the density of states of an integrable system in the form of a trace formula. It discusses the transition from integrability to chaos in such systems when perturbations are added that couple resonantly to these natural frequencies. The chapter elaborates with a view to develop the semiclassical approximation, and the associated quantization condition. It brings out the similarity between the Schrodinger equation of wave mechanics, and the HJ equation.
Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.
After a brief review of the extensive work done on the theory of invariant tori and their quantization, we show that nevertheless an important connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operators in the energy diagonal representation and Fourier components of the corresponding classical dynamical variables that was the basis for Heisenberg's invention of quantum mechanics. We describe a number of previously unknown or little-known aspects of this relationship, with special emphasis on variational principles and the connection between commutation relations and quantization of action variables. As a single illustration of the utility of these ideas we show that it is possible to obtain approximate solutions to the quantum scheme that are more accurate than the semiclassical approximation with little additional effort compared to the latter.