Phase-space analysis of self-interacting Quantum Field Theories

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Sujet de Thèse

• Titre (en anglais): Phase-space analysis of self-interacting Quantum Field Theories

• Unité de recherche : IRMAR, UMR-6625

• Thème : PDE and Mathematical physics

• Mots clefs : Microlocal and semi-classical analysis, quantum eld theory, spectral analysis

• Directeurs de thèse :

Zied Ammari ([email protected]) IRMAR- Université Rennes 1

Maher Zerzeri ([email protected]) LAGA- Université Paris 13

Objectif de la thèse (en anglais)

Quantum eld theory is one of the topics of mathematical physics that raises the most prominent challenges to mathematics, see [6, 7]. On the other hand the subject is experiencing a renewed interest, particularly in relation to the recent trends of employing probabilistic tools and stochastic quantization to the analysis of nonlinear PDEs. In this context, we aim to introduce a new approach to QFT based on phase-space analysis in innite dimensional spaces and to show that this point of view leads to a better understanding of some non-perturbative aspects of quantum eld theories. The old results of the sixties and seventies provide us for instance with elegant constructions of the ϕ⁴ and ϕ²ⁿ non-linear quantum elds in two and three-dimensional space-times. Moreover, pseudo-dierential operators in innite-dimensional spaces have been introduced and studied since at least the nineties, see for instance the work of Sergio Albeverio and Alexei Daletskii [1]. In the recent years, it has become clear that several ideas inspired by micolocal analysis are quite fruitful in quantum electrodynamics and in many-body theory, see [4, 3, 5] and [2]. We therefore propose in this thesis a formal extension of microlocal analysis to innite dimensional phase spaces and a study of the ϕ⁴ and ϕ²ⁿ quantum eld models with these new tools. Propagation of singularities and semi-classical expansions of ground states energies would be among the main issues to be addressed.

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References

[1] Sergio Albeverio and Alexei Daletskii. Algebras of pseudodierential operators in L2 given by smooth measures on Hilbert spaces. Math.

Nachr., 192:522, 1998.

[2] Zied Ammari and Marco Falconi. Bohr's correspondence principle for the renormalized Nelson model. SIAM J. Math. Anal., 49(6):50315095, 2017.

[3] Zied Ammari and Francis Nier. Mean eld limit for bosons and innite dimensional phase-space analysis. Ann. Henri Poincaré, 9(8):15031574, 2008.

[4] Zied Ammari and Maher Zerzeri. On the classical limit of self-interacting quantum eld Hamiltonians with cutos. Hokkaido Math. J., 43(3):385 425, 2014.

[5] Laurent Amour, Lisette Jager, and Jean Nourrigat. On bounded pseudodierential operators in Wiener spaces. J. Funct. Anal., 269(9):2747 2812, 2015.

[6] Arthur Jae. Constructive quantum eld theory. In Mathematical physics 2000, pages 111127. Imp. Coll. Press, London, 2000.

[7] Arthur Jae and Edward Witten. Quantum Yang-Mills theory. In The millennium prize problems, pages 129152. Clay Math. Inst., Cambridge, MA, 2006.

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