• Research Areas

    • Matrix Models
    • Exactly solvable qunatum field theories
    • Topological recursion

  • Doctoral Thesis

    Matrix Field Theory

    Betreuer
    Professor Dr. Raimar Wulkenhaar
    Doctoral Subject
    Mathematik
    Doctoral Degree
    Dr. rer. nat.
    Awarded by
    Department 10 – Mathematics and Computer Science
    This thesis studies matrix field theories, which are a special type of matrix models. First, the different types of applications are pointed out, from (noncommutative) quantum field theory over 2-dimensional quantum gravity up to algebraic geometry with explicit computation of intersection numbers on the moduli space of complex curves.
    \\
    The Kontsevich model, which has proved the Witten conjecture, is the simplest example of a matrix field theory. Generalisations of this model will be studied, where different potentials and the spectral dimension are introduced. Because they are naturally embedded into a Riemann surface, the correlation functions are graded by the genus and thenumber of boundary components. The renormalisation procedure of quantum field theory leads to finite UV-limit.
    \\
    We provide a method to determine closed Schwinger-Dyson equations with the usage of Ward-Takahashi identities in the continuum limit. The cubic (Kontsevich model) and the quartic (Grosse-Wulkenhaar model) potentials are studied separately.
    \\
    The cubic model is solved completely for any spectral dimension < 8, i.e. all correlation functions are derived explicitly. Inspired by topological recursion, we propose creation and annihilation operators by differential and residue operators. The exact results are confirmed by perturbative computations with Feynman graphs renormalised by Zimmermann’s forest formula. The number and the amplitudes of the graphs grow factorially, which is known as renormalon problem. However, these series are convergent since the exact results are provided. A further differential operator is derived to determine allfree energies. Additionally, by the theorem of Kontsevich, the intersection numbers of the moduli space of complex curves $\bar{\mathcal{M}}_{g,b}$ are found.
    \\
    For the quartic model, the 2-point function is derived for any spectral dimension < 6 explicitly. The first step is to derive an angle function which is, after analytic continuation, interpreted as an effective measure. On the 4-dimensional noncommutative Moyal space, the effective measure is given by a hypergeometric function. Its asymptotic behaviour changes the spectral dimension effectively to $4 - 2 \frac{\arcsin(\lambda \pi)}{\pi}$ for $\lambda| < \frac{1}{\pi}$. This dimension drop prevents the quantum field theoretical 4-dimensional $\Phi^4$-model on the Moyal spacefrom the triviality problem. After combinatorial analysis, an explicit (not recursive) formula for any planar N -point function is provided.
    \\
    The evident difference between the cubic and the quartic model is of algebraic-geometric nature. Computing correlation functions via topological recursion needs the spectral curve as initial data. This algebraic curve has for the cubic model only onebranch point which coincides with the pole of the stable correlation functions. However, the quartic model has a spectral curve which admits infinitely many branch points in the continuum limit.

  • CV

    Education

    PhD studies in mathematics, University of Münster
    PhD studies in mathematics, Tokyo University of Science
    Master studies in physics, University of Münster
    Bachelor studies in physics, University of Münster

    External Functions

    Member of the Cluster of Excellence "Mathematics Münster"
    Member of the RTG "Strong and Weak Interactions - from Hadrons to Dark Matter"
    Member of the German Physical Society

  • Teaching

    • Übung: Tutorial Mathematics for Physicists II [102206]
      (in cooperation with Prof. Dr. Raimar Wulkenhaar)

    • Übung: Tutorial Mathematics for Physicists I [100204]
      (in cooperation with Prof. Dr. Raimar Wulkenhaar)

  • Publications

  • Scientific Talks

    • Hock, Alexander (): ‘Solution of φ^4_4 on the Moyal Space’. Algebraic Structures in Perturbative Quantum Field Theory, IHES, Paris, France, .
    • Hock, Alexander (): ‘From the quartic analogue of the Kontsevich model to the Hermitian 2-Matrix Model’. Séminaire Tensor Journal Club, France (remote talk), .

    • Hock, Alexander (): „Topological Recusion of the Kontsevich Model and its Quartic Analogue“. Research Seminar: Algebraic Geometry, Tübingen, .
    • Hock, Alexander (): ‘Solution of the noncommutative φ^4_4 model’. LQP, Foundations and Constructive Aspects of QFT, Göttingen, .
    • Hock, Alexander (): ‘Matrix Field Theory’. Research Seminar: Differential Geometry, Tokyo University of Science, .

    • Hock, Alexander (): ‘Schwinger-Dyson equations and Ward identities in NC QFT on the example of scalar models’. LQP, Foundations and Constructive Aspects of QFT, Wuppertal, .