T7: Field theory and randomness

Quantum field theory (QFT) is the fundamental framework to describe matter at its smallest length scales. QFT has motivated groundbreaking developments in many different mathematical fields including the theory of operator algebras, probability theory, representation theory, complex algebraic geometry and others.

The construction of a non-linear QFT in the physical case of 4 space-time dimensions is an outstanding open problem and part of the Yang–Mills Clay Millenium Prize Problem. In 3 space-time dimensions, mathematical models have been constructed with different techniques. All known constructions are based on a perturbative analysis around linear theories. This is not expected to work directly in 4-dimensional theories because these theories are critical: Linear and non-linear terms have the same scaling behaviour and are therefore expected to both contribute on all scales.

The key goals of this topic centre around the construction of a critical (bosonic) theory. We will focus on scalar QFTs on non-commutative geometries building on the one hand on fundamental results by Wulkenhaar who found explicit formulae for the leading terms in the topological expansion of solutions and on the other hand on recent breakthrough in the probabilistic construction of QFTs by the stochastic analysis community.

  • Mathematical fields

    • Arithmetic geometry and representation theory
    • Operator algebras and mathematical physics
    • Differential geometry
    • Applied analysis and theory of PDEs
    • Stochastic analysis
    • Theory of stochastic processes
    • Numerical analysis, machine learning and scientific computing

  • Collaborations with other Topics

  • Selected publications and preprints

    since 2019

    $\bullet $ Rodrigo Bazaes, Isabel Lammers, and Chiranjib Mukherjee. Subcritical Gaussian multiplicative chaos in the Wiener space: construction, moments and volume decay. Probability Theory and Related Fields, April 2024. doi:10.1007/s00440-024-01271-7.

    $\bullet $ R. Bazaes, C. Mukherjee, M. Sellke, and S.R.S. Varadhan. Effective mass of the fröhlich polaron and the landau-pekar-spohn conjecture. arXiv e-prints, February 2024. arXiv:2307.13058.

    $\bullet $ Sebastian Becker, Arnulf Jentzen, Marvin S. Müller, and Philippe von Wurstemberger. Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing. Mathematical Finance, 34(1):90–150, January 2024. doi:10.1111/mafi.12405.

    $\bullet $ Yifei Zhao. Half-integral levels. arXiv e-prints, December 2023. arXiv:2312.11058.

    $\bullet $ Jörg Schürmann and Raimar Wulkenhaar. An algebraic approach to a quartic analogue of the Kontsevich model. Mathematical Proceedings of the Cambridge Philosophical Society, 174(3):471–495, May 2023. doi:10.1017/S0305004122000366.

    $\bullet $ Ajay Chandra, Augustin Moinat, and Hendrik Weber. A priori bounds for the $\Phi ^4$ equation in the full sub-critical regime. Archive for Rational Mechanics and Analysis, 247(3):48, May 2023. doi:10.1007/s00205-023-01876-7.

    $\bullet $ Paolo Grazieschi, Konstantin Matetski, and Hendrik Weber. The dynamical Ising-Kac model in 3D converges to $\Phi ^4_3$. arXiv e-prints, March 2023. arXiv:2303.10242.

    $\bullet $ Johannes Branahl, Alexander Hock, and Raimar Wulkenhaar. Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures. Commun. Math. Phys., 393(3):1529–1582, August 2022. doi:10.1007/s00220-022-04392-z.

    $\bullet $ Jins de Jong, Alexander Hock, and Raimar Wulkenhaar. Nested Catalan tables and a recurrence relation in noncommutative quantum field theory. Ann. Inst. Henri Poincaré D, 9(1):47–72, April 2022. doi:10.4171/aihpd/113.

    $\bullet $ Giuseppe Da Prato, Arnulf Jentzen, and Michael Röckner. A mild Itô formula for SPDEs. Trans. Amer. Math. Soc., 372:3755–3807, June 2019. doi:10.1090/tran/7165.

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  • Recent publications and preprints

    since 2023

    $\bullet $ Sonja Cox, Arnulf Jentzen, and Felix Lindner. Weak convergence rates for temporal numerical approximations of the semilinear stochastic wave equation with multiplicative noise. Numer. Math., 156(6):2131–2177, December 2024. doi:10.1007/s00211-024-01425-8.

    $\bullet $ Sonja Cox, Arnulf Jentzen, and Felix Lindner. Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise. Numer. Math., September 2024. arXiv:1901.05535.

    $\bullet $ Paolo Grazieschi, Konstantin Matetski, and Hendrik Weber. Martingale-driven integrals and singular SPDEs. Probability Theory and Related Fields, August 2024. arXiv:2303.10245, doi:10.1007/s00440-024-01311-2.

    $\bullet $ Rodrigo Bazaes, Isabel Lammers, and Chiranjib Mukherjee. Subcritical Gaussian multiplicative chaos in the Wiener space: construction, moments and volume decay. Probability Theory and Related Fields, April 2024. doi:10.1007/s00440-024-01271-7.

    $\bullet $ R. Bazaes, C. Mukherjee, M. Sellke, and S.R.S. Varadhan. Effective mass of the fröhlich polaron and the landau-pekar-spohn conjecture. arXiv e-prints, February 2024. arXiv:2307.13058.

    $\bullet $ Ajay Chandra, Guilherme de Lima Feltes, and Hendrik Weber. A priori bounds for 2-d generalised parabolic Anderson model. arXiv e-prints, February 2024. arXiv:2402.05544.

    $\bullet $ Sebastian Becker, Arnulf Jentzen, Marvin S. Müller, and Philippe von Wurstemberger. Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing. Mathematical Finance, 34(1):90–150, January 2024. doi:10.1111/mafi.12405.

    $\bullet $ Guilherme de Lima Feltes and Hendrik Weber. Brownian particle in the curl of 2-d stochastic heat equations. Journal of Statistical Physics, 191(2):16, January 2024. doi:10.1007/s10955-023-03224-1.

    $\bullet $ Yifei Zhao. Half-integral levels. arXiv e-prints, December 2023. arXiv:2312.11058.

    $\bullet $ Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, and Ariel Neufeld. An efficient Monte Carlo scheme for Zakai equations. Communications in Nonlinear Science and Numerical Simulation, 126:107438, November 2023. doi:10.1016/j.cnsns.2023.107438.

    $\bullet $ Leonardo Tolomeo and Hendrik Weber. Phase transition for invariant measures of the focusing Schrödinger equation. arXiv e-prints, June 2023. arXiv:2306.07697.

    $\bullet $ Jörg Schürmann and Raimar Wulkenhaar. An algebraic approach to a quartic analogue of the Kontsevich model. Mathematical Proceedings of the Cambridge Philosophical Society, 174(3):471–495, May 2023. doi:10.1017/S0305004122000366.

    $\bullet $ Ajay Chandra, Augustin Moinat, and Hendrik Weber. A priori bounds for the $\Phi ^4$ equation in the full sub-critical regime. Archive for Rational Mechanics and Analysis, 247(3):48, May 2023. doi:10.1007/s00205-023-01876-7.

    $\bullet $ Olivier Biquard and Hans-Joachim Hein. The renormalized volume of a 4-dimensional Ricci-flat ALE space. Journal of Differential Geometry, 123(3):411–429, March 2023. doi:10.4310/jdg/1683307004.

    $\bullet $ Paolo Grazieschi, Konstantin Matetski, and Hendrik Weber. The dynamical Ising-Kac model in 3D converges to $\Phi ^4_3$. arXiv e-prints, March 2023. arXiv:2303.10242.

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    further publications

Back to research programme

Integrability

© MM/vl

Investigators: Hein, Holzegel, Schürmann, Wulkenhaar, Zhao

This unit proceeds from Wulkenhaar’s recursive solution of the quartic Kontsevich model and connections established in the first funding period to integrable structures in mathematical physics more generally. Examples include moduli spaces of curves and special solutions of the Einstein equations,

Stochastic analysis of quantum fields

© MM/vl

Investigators: Jentzen, Kabluchko, Löwe, Weber, Wulkenhaar

This unit addresses the construction of a non-commutative QFT combining exact solvability and stochastic analysis as described above. In this unit, we also plan to analyse numerical algorithms for QFT computations, combining the latest developments in stochastic analysis with machine learning methods developed in Topic 10.

Effective theories and disorder

Investigators: Löwe, Mukherjee, Weber

This unit addresses questions at the interface of field theory and statistical mechanics of disordered systems. We focus on probabilistic projects that have a close link to quantum mechanics, such as the polaron problem or stochastic PDEs (SPDEs) arising as scaling limits or supercritical SPDEs.