The Standard Model as a geometric variational problem

    Project: Research funding

    Project Details

    Abstract

    (1) Wider research context / theoretical framework
    The standard model of elementary particle physics unifies three of the four known forces in nature. It is able to make experimental predictions with enormous precision although its mathematical structure is poorly understood. While physicists aim at quantizing the standard model this project will treat it as a geometric variational problem which can be investigated in a rigorous mathematical manner. In physics it is mostly studied on flat four-dimensional Minkowski space, in this project we will explore the latter on both globally hyperbolic but also on Riemannian manifolds. We expect to find new mathematical structures which will also have implications for physics.

    (2) Hypotheses / research questions / objectives
    The critical points of the action functional of the standard model are given by a coupled system of non-linear partial differential equations. We will derive various existence results for the latter on globally hyperbolic manifolds. In the analysis we will pay special attention to various Dirac-type equations as this field is poorly developed so far. Concerning the standard model on Riemannian manifolds we will establish an existence result for the Dirac-Yang-Mills system using the Atiyah-Singer index theorem. Moreover, we will initiate a rigorous mathematical analysis of the Higgs mechanism, which in physics is responsible for creating masses of elementary particles. The methods used in physics are limited to four-dimensional Minkowski space and it is a challenging mathematical endeavour to extend them to globally hyperbolic manifolds.

    (3) Approach / methods
    To establish existence results on globally hyperbolic manifolds we will try to find a null structure and utilize both the vector field method and derive energy estimates. Also, we will keep in mind that it might be favorable to apply tools from microlocal analysis. In the Riemannian setup we will employ well-established tools such as elliptic regularity and the index theorem. In order to initiate a mathematical treatment of the Higgs mechanism we will employ the concept of a bundle reduction used in mathematical gauge theory.

    (4) Level of originality / innovation
    As the complete standard model has not been analysed in a rigorous mathematical fashion I expect that the results of this project will attract researches from mathematics and physics alike. The analytic and geometric questions that come with the Higgs mechanism have not been taken up by mathematicians and I expect to get many new insights here. Moreover, this project will create new methods for Dirac-type equations on globally hyperbolic manifolds with great potential for future research.

    (5) Primary researchers involved
    The principal investigator of this project will be Volker Branding who will be supported by a postdoc and a PhD student.
    StatusNot started

    Keywords

    • standard model
    • globally hyperbolic manifold
    • Higgs mechanism
    • global existence
    • vanishing results
    • Riemannian manifold