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Nonlinear Gibbs measures and quantum field theory

(Reading seminar Summer 2024)

Phan Thành Nam, Moodle (ID: 33922, pass: Gibbs%24)

General Information

Description: The nonlinear Gibbs measure has played a central role in constructive quantum field theory in the 1970s, and it has become a popular tool to study nonlinear dispersive equations with low-regular data as well as nonlinear stochastic PDEs with noise. In this seminar, we will discuss the construction of the nonlinear Gibbs measure for the cubic nonlinear Schrödinger equation and the connection to interacting Bose gases at high temperature.

Audience : Bachelor and Master students of Mathematics.

Credits: 3 ECTS.

Language: English.

Time and place: Friday, 14:15-16:00, B251.

References:
  • J. Bourgain. Periodic nonlinear Schrödinger equation and invariant measures. Comm. Math. Phys. 166 (1994), pp. 1-26.
  • J. Bourgain. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Comm. Math. Phys. 176 (1996), pp. 421–445.
  • M. Lewin, P.T. Nam, and N. Rougerie. Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits. J. Math. Phys. 59 (2018), 041901.
  • M. Lewin, P.T. Nam, and N. Rougerie. Classical field theory limit of many-body quantum Gibbs states in 2D and 3D. Invent. Math. 224 (2021), 315–444.
  • J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger. A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation. Adv. Math. 353 (2019), 67-115.
  • J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger. The mean-field limit of quantum Bose gases at positive temperature. J. Amer. Math. Soc. 35 (2022), 955-1030.
  • J. Dolbeault, P. Felmer, M. Loss, E. Paturel. Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems. J. Funct. Anal. 238 (2006), 193-220.
  • M. Christandl, R. König, G. Mitchison, and R. Renner. One-and-a-Half Quantum de Finetti Theorems. Commun. Math. Phys. 273 (2007), 473–498.
  • M. Lewin, P.T. Nam, and N. Rougerie. Remarks on the quantum de Finetti theorem for bosonic systems. Appl. Math. Res. Express (AMRX) 1 (2015), 48-63.

Schedule:

26.4.2024. Introduction and distribution of the reading material.

17.5. Construction of the Gaussian measure.

7.6. Construction of the nonlinear Gibbs measure Phi^4_1.

21.6. Applications of the Gibbs measure to the Cauchy problem in NLS.

12.7. Derivation of the Gibbs measure from many-body quantum mechanics.