Quantum and thermal fluctuations in quantum mechanics and field theories from a new version of semiclassical theory

Escobar-Ruiz, A. M.; Shuryak, Edward; Turbiner, A. V.

doi:10.1103/physrevd.93.105039

Cited by 23 publications

(52 citation statements)

References 30 publications

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“…It allows us to reach, at least, 12 d.d. in the energies of (0, 0), (0, 1), (0, 2) states at D = 1, 2, 3, 6, see Tables I -III denoted as E (2) 0 . Making analysis of numerical results suggests that, when E 2 is evaluated, the E (2) 0 provides 12 -13 correct d.d., at least.…”

Section: The Approximant and Variational Calculationsmentioning

confidence: 99%

“…Making analysis of numerical results suggests that, when E 2 is evaluated, the E (2) 0 provides 12 -13 correct d.d., at least. It implies that, once E 2 is taken into account, all digits of E (2) 0 printed in Tables I -IV are exact. These highly accurate results are confirmed independently by the calculating the second correction of the variational energy E 3 which is always ≤ 10 −12 for all D and g 2 that we have studied.…”

Section: The Approximant and Variational Calculationsmentioning

confidence: 99%

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Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

Valle

Turbiner

2020

Int. J. Mod. Phys. A

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In our previous paper I (del Valle-Turbiner, 2019) it was developed the formalism to study the general D-dimensional radial anharmonic oscillator with potential V (r) = 1 g 2V (gr). It was based on the Perturbation Theory (PT) in powers of g (weak coupling regime) and in inverse, fractional powers of g (strong coupling regime) in both r-space and in (gr)-space, respectively. As the result it was introduced -the Approximant -a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials V (r) = r 2 + g 2(m−1) r 2m , m = 2, 3, respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8-12 figures for any D = 1, 2, 3 . . . and g ≥ 0, while the relative deviation of the Approximant from the exact eigenfunction is less than 10 −6 for any r ≥ 0.

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Section: The Approximant and Variational Calculationsmentioning

confidence: 99%

Section: The Approximant and Variational Calculationsmentioning

confidence: 99%

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

Valle

Turbiner

2020

Int. J. Mod. Phys. A

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“…(29), and all transcendental contributions are again cancelled out, and the answer is the meromorphic function of X DW P only [1].…”

Section: Dwpmentioning

confidence: 99%

“…In our previous paper [1] we used Feynman diagrams to calculate one and two-loop corrections to classical flucton action for DWP potential. For two other famous quantum-mechanical problems, the AHO and SGP, we had only presented the derivation of the Green functions and the (one-loop) determinants.…”

Section: Arxiv:170506159v1 [Hep-th] 17 May 2017mentioning

confidence: 99%

“…In our previous paper [1], to be referred below simply as I, we introduced a different version of the semiclassical approach, based on Feynman's path integral representation of the density matrix [2,3] analytically continued to imaginary (Euclidean) time. It corresponds to a transition from quantum mechanics to statistical mechanics: If the time is defined as a periodic variable, with period β = /T , the density matrix corresponds to quantum ensemble at nonzero temperature T .…”

Section: Introductionmentioning

confidence: 99%

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Fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II.

2017

Self Cite

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This is the second paper on semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding endpoints. The classical path, interpolating between this point and the classical vacuum, called "flucton", plus systematic one-and two-loop corrections, has been calculated in the first paper [1] for double-well potential and now extended for a number of quantum-mechanical problems (anharmonic oscillator, sine-Gordon potential). The method is based on systematic expansion in Feynman diagrams and thus can be extended to QFTs. We show that the loop expansion in QM reminds the leading log-approximations in QFT. In this sequel we present complete set of results obtained using this method in unified way. Alternatively, starting from the Schrödinger equation we derive a generalized Bloch equation which semiclassicallike, iterative solution generates the loop expansion. We re-derive two loop expansions for all three above potentials and now extend it to three loops, which has not yet been done via Feynman diagrams. All results for both methods are fully consistent with each other. Asymmetric (tilted) double-well potential (non-degenerate minima) is also studied using the second method.

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A practical method in calculating one loop quantum fluctuations to the energy of the non-topological soliton

Shu

2017

Sci. China Phys. Mech. Astron.

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I have used a practical method to calculate the one-loop quantum correction to the energy of the non-topological soliton in Friedberg-Lee model. The quantum effects which come from the quarks of the Dirac sea scattering with the soliton bag are calculated by a summation of the discrete and continuum energy spectrum of the Dirac equation in the background field of soliton. The phase shift of the continuum spectrum is numerically calculated in an efficient way and all the divergences are removed by the same renormalization procedure.PACS numbers: 11.10. Gh, 11.15.Kc, 11.27.+d, 12.39.Ba Non-topological soliton models which are effective models inspired from the underlying QCD theory are phenomenologically successful in describing the low energy nuclear physics. However, the main calculation methods in these models are based on mean field approximation, in other words treating the fields classically [1][2][3][4]. The quantum corrections in the background fields of spatially non-trivial configurations are very difficult to calculate. This is partly due to the fact that these calculations are nonlocal. During the past decades different calculation methods and approximate schemes have been developed on this problem [5][6][7][8][9][10][11][12]. As the calculation of quantum corrections of solitons is much more complex than those usual calculations of quantum loop corrections of trivial background fields, most studies on this problem are based on the derivative expansion method [5][6][7][8][9]. The renormalization in this method is a very nontrivial task. One remarkable calculation method was that developed by Farhi, Graham, Haagensen and Jaffe [13]. It is a systematic and efficient scheme for calculating the quantum corrections about static field configuration in renormalizable field theories, in which all the divergences are removed by the same renormalization procedure. As originally this method was applied in the Higggs like models and the main interest was focused on studying solitons in the standard electroweak models [13,14], there are no applications of this method, as far as I know, in strong interaction hadronic models, like the FriedbergLee(FL) model, the linear sigma model and other QCD effective models. In recent years topological solitons in strong interaction QCD theory have drawn lots of attentions [15,16]. One needs an efficient method to calculate the quantum correction of the soliton in effective QCD theories [17]. So in this paper as the first small step I want to introduce this method to calculate the one loop quantum fluctuation of the non-topological soliton in the FL model. In this method one makes the energy level summation by calculating the discrete and continuous energy spectrum and the continuum contribution is determined through evaluating scattering phase shift in a concise way. The renormalization of the field configuration energy could be done in a manner consistent with on-shell mass and coupling constant renormalization in the perturbative sector. Comparing to the precedent calculati...

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Quantum and thermal fluctuations in quantum mechanics and field theories from a new version of semiclassical theory

Cited by 23 publications

References 30 publications

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

Fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II.

A practical method in calculating one loop quantum fluctuations to the energy of the non-topological soliton

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