Valleys in quantum mechanics

Aoyama, Hideaki; Kikuchi, Hisashi; Okouchi, Ikuo; Sato, Masatoshi; Wada, Shinya

doi:10.1016/s0370-2693(98)00116-6

Cited by 18 publications

(24 citation statements)

References 17 publications

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“…The others are called the quasi zero modes, corresponding to the relative position and phase of the constituent fractional instantons. To evaluate the integral along such "nearly flat directions" (flat directions in the limit of g → 0), let us define "valley solution" ϕ B (η) (quasi-solution) [131][132][133][134][135] as a bion ansatz satisfying the following properties:…”

Section: A Quasi-moduli Space Of Single Bion Configurationmentioning

confidence: 99%

Bion non-perturbative contributions versus infrared renormalons in two-dimensional ℂPN − 1 models

Fujimori

Kamata

Misumi

et al. 2019

J. High Energ. Phys.

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We derive the semiclassical contributions from the real and complex bions in the twodimensional CP N −1 sigma model on R × S 1 with a twisted boundary condition. The bion configurations are saddle points of the complexified Euclidean action, which can be viewed as bound states of a pair of fractional instantons with opposite topological charges. We first derive the bion solutions by solving the equation of motion in the model with a potential which simulates an interaction induced by fermions in the CP N −1 quantum mechanics. The bion solutions have quasi-moduli parameters corresponding to the relative distance and phase between the constituent fractional instantons. By summing over the Kaluza-Klein modes of the quantum fluctuations around the bion backgrounds, we find that the effective action for the quasi-moduli parameters is renormalized and becomes a function of the dynamical scale (or the renormalized coupling constant). Based on the renormalized effective action, we obtain the semiclassical bion contribution in a weak coupling limit by making use of the Lefschetz thimble method. We find that the non-perturbative contribution vanishes in the supersymmetric case and it has an imaginary ambiguity which is consistent with the expected infrared renormalon ambiguity in non-supersymmetric cases. This is the first explicit result indicating the relation between the complex bion and the infrared renormalon. * Electronic address: toshiaki.fujimori018(at)

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Section: A Quasi-moduli Space Of Single Bion Configurationmentioning

confidence: 99%

Bion non-perturbative contributions versus infrared renormalons in two-dimensional ℂPN − 1 models

Fujimori

Kamata

Misumi

et al. 2019

J. High Energ. Phys.

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“…For ǫ = 0, the valley configurations were found to behave in a qualitatively similar manner [9,11]. There are configurations that smoothly connect one 1 In an earlier paper [9], we have used a (4g 3 q 3 − 3g 4 q 4 ) term in place of gq in the ǫ term of the potential.…”

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confidence: 83%

Valley views: instantons, large order behaviors, and supersymmetry

et al. 1999

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The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. The evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbative calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A 'proof' of Bogomolny's "trick" is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of the perturbation theory is confirmed by explicit calculations, in some cases to the 478-th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schrödinger equation.

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“…(4.26), which looks like the hidden topological angle (HTA) [34]) of n-bion configuration though, the index has very important role: this quantity can be regarded as the intersection number of Lefschetz thimble. 20 Furthermore, comparing the QMI calculation and Gutzwiller's perspective, we can see the new physical meaning of QMI. The QMI calculation is based on the approximation that the cycle is sufficiently large, but from Gutzwiller's point of view, the B cycle is so short that it requires the A cycle to rotate infinite times in order to earn the sufficiently long cycle, and therefore it is considered to be represented in the form of D −2 A B.…”

Section: Comparison To Quasi-moduli Integralmentioning

confidence: 99%

“…the instanton-anti-instanton sector) in a precise way. Many signs of resurgent structure in the perturbative and instanton analysis are already hinted in the old physics literature [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. The renewed interest is due to the precise understanding of the connection between resurgence theory and physical problems in quantum mechanics , matrix models and string theory , and quantum field theory .…”

Section: Jhep12(2020)114mentioning

confidence: 99%

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On exact-WKB analysis, resurgent structure, and quantization conditions

Sueishi

Kamata

Misumi

et al. 2020

J. High Energ. Phys.

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There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller’s analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.

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Valleys in quantum mechanics

Cited by 18 publications

References 17 publications

Bion non-perturbative contributions versus infrared renormalons in two-dimensional ℂPN − 1 models

Bion non-perturbative contributions versus infrared renormalons in two-dimensional ℂPN − 1 models

Valley views: instantons, large order behaviors, and supersymmetry

On exact-WKB analysis, resurgent structure, and quantization conditions

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