Quantum Monodromy in Integrable Systems

Ngọc, San Vũ

doi:10.1007/s002200050621

Cited by 85 publications

(84 citation statements)

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“…We refer the reader to Refs. [Ngoc99,NSZ06] for a rigorous definition of this semi-classical point of view. Here, we consider only the graphical representations of semi-classical monodromy as a pictorial illustration of classical monodromy.…”

Section: Resultsmentioning

confidence: 99%

“…A non trivial monodromy can be expected if the set of regular values of the image of the energy-momentum map is not simply connected. Hamiltonian monodromy has profound implications both in classical and quantum mechanics [Ngoc99] since it is the simplest topological obstruction to the existence of global action-angle variables [Dui80] and thus of global good quantum numbers [Ngoc99]. The phenomenon of Hamiltonian monodromy has been exhibited in a large variety of physical systems both in classical and quantum mechanics [AKE04, SC00, KR03, CWT99, SZ99, WJD03, GCSZ04,EJS04].…”

Section: Introductionmentioning

confidence: 99%

“…The formal extension of this action to the whole group leads to monodromy matrices with fractional coefficients and to the denomination fractional monodromy. One of the main motivations for the introduction of this new concept is given by the quantum manifestation of monodromy in the discrete joint spectrum of the energy-momentum map [Ngoc99,NSZ06]. This spectrum can be represented as a lattice of points in R 2 .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fractional Hamiltonian monodromy from a Gauss–Manin monodromy

Sugny

Mardešić

Pelletier

et al. 2008

Journal of Mathematical Physics

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Abstract. Fractional Hamiltonian Monodromy is a generalization of the notion of Hamiltonian Monodromy, recently introduced by N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií for energy-momentum maps whose image has a particular type of non-isolated singularities. In this paper, we analyze the notion of Fractional Hamiltonian Monodromy in terms of the Gauss-Manin Monodromy of a Riemann surface constructed from the energy-momentum map and associated to a loop in complex space which bypasses the line of singularities. We also prove some propositions on Fractional Hamiltonian Monodromy for 1 : −n and m : −n resonant systems.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fractional Hamiltonian monodromy from a Gauss–Manin monodromy

Sugny

Mardešić

Pelletier

et al. 2008

Journal of Mathematical Physics

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“…The appearance of theta functions in this context is quite natural from a mechanical point of view: briefly, this goes as follows. Given a basis of cycles on a Liouville torus, constructed as in [27,28] -see also Section 4 below -a natural complex structure thereon is determined upon setting…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian monodromy via geometric quantization and theta functions

Sansonetto

Spera

2010

Journal of Geometry and Physics

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In this paper, Hamiltonian monodromy is addressed from the point of view of geometric quantization, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections. In the case of completely integrable Hamiltonian systems with two degrees of freedom, a link is established between monodromy and (2-level) theta functions, by resorting to the by now classical differential geometric intepretation of the latter as covariantly constant sections of a flat connection, via the heat equation. Furthermore, it is shown that monodromy is tied to the braiding of the Weiestraß roots pertaining to a Lagrangian torus, when endowed with a natural complex structure (making it an elliptic curve) manufactured from a natural basis of cycles thereon. Finally, a new derivation of the monodromy of the spherical pendulum is provided.

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“…A number of simple classical mechanical systems, including the spherical pendulum, the Lagrange top, and others [2] are known to have this property. Quantum monodromy [3,4] was discovered in many fundamental quantum systems, such as the hydrogen atom in external fields [5][6][7], the H + 2 molecular ion [8], (a) E-mail: jbdelo@wm.edu systems with coupled angular momenta [9], rotating dipolar molecules in external electric field [10,11], rovibrational structure in molecules [12][13][14][15][16][17], collective vibrations of nuclei [18], or of Bose condensates [19] -to name af e w .…”

mentioning

confidence: 99%