Jürgen Tolksdorf

Classification of infinitesimal symmetries in covariant classical mechanics

In the framework of general relativistic classical mechanics on a spacetime with absolute time, we classify the infinitesimal symmetries of the classical structure by means of distinguished Lie subalgebras of the Lie algebra of “special phase function.” These subalgebras are crucial also for the classification of infinitesimal quantum symmetries, which will be analyzed in a forthcoming paper.

Perturbative description of the fermionic projector: Normalization, causality, and Furry's theorem

The causal perturbation expansion of the fermionic projector is performed with a contour integral method. Different normalization conditions are analyzed. It is shown that the corresponding light-cone expansions are causal in the sense that they only involve bounded line integrals. For the resulting loop diagrams we prove a generalized Furry theorem.

Regularity of Solutions of the Nonlinear Sigma Model with Gravitino

We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of Euler–Lagrange equations of the two-dimensional nonlinear sigma model with gravitino is calculated explicitly. The gravitino terms pose additional analytic difficulties to show smoothness of its weak solutions which are overcome using Rivière’s regularity theory and Riesz potential theory.

Bosonic Loop Diagrams as Perturbative Solutions of the Classical Field Equations in ϕ4-Theory

Solutions of the classical ϕ4-theory in Minkowski space-time are analyzed in a perturbation expansion in the nonlinearity. Using the language of Feynman diagrams, the solution of the Cauchy problem is expressed in terms of tree diagrams which involve the retarded Greens function and have one outgoing leg. In order to obtain general tree diagrams, we set up a “classical measurement process” in which a virtual observer of a scattering experiment modifies the field and detects suitable energy differences. By adding a classical stochastic background field, we even obtain all loop diagrams. The expansions are compared with the standard Feynman diagrams of the corresponding quantum field theory.

Gauge Theories of Dirac Type

A specific class of gauge theories is geometrically described in terms of fermions. In particular, it is shown how the geometrical frame presented naturally includes spontaneous symmetry breaking of Yang-Mills gauge theories without making use of a Higgs potential. In more physical terms, it is shown that the Yukawa coupling of fermions, together with gravity, necessarily yields a symmetry reduction provided the fermionic mass is considered as a globally well-defined concept. The structure of this symmetry breaking is shown to be compatible with the symmetry breaking that is induced by the Higgs potential of the minimal Standard Model. As a consequence, it is shown that the fermionic mass has a simple geometrical interpretation in terms of curvature and that the (semiclassical) “fermionic vacuum” determines the intrinsic geometry of space-time. We also discuss the issue of “fermion doubling” in some detail and introduce a specific projection onto the “physical subspace” that is motivated by the Standard M...

Symmetries and conservation laws of a nonlinear sigma model with gravitino

Abstract We study the symmetries and invariances of a version of the action functional of the nonlinear sigma model with gravitino, as considered in Jost et al. (2017). The action is invariant under rescaled conformal transformations, super Weyl transformations, and diffeomorphisms. In particular cases the functional possesses a degenerate supersymmetry. The corresponding conservation laws lead to a geometric interpretation of the energy–momentum tensor and supercurrent as holomorphic sections of appropriate bundles.

The Einstein-Hilbert action with cosmological constant as a functional of generic form

The geometrical underpinnings of a specific class of Dirac operators are discussed. It is demonstrated how this class of Dirac operators allows to relate various geometrical functionals like the Yang-Mills action and the functional of non-linear σ − models (i.e., of (Dirac) harmonic maps). These functionals are shown to be similar to the Einstein-Hilbert action with cosmological constant (EHC). The EHC may thus be regarded as a “generic functional.” As a byproduct, the geometrical setup presented also allows to avoid the issue of “fermion doubling” as usually encountered, for instance, in the geometrical discussion of the Standard Model in terms of Dirac operators. Furthermore, it is demonstrated how the geometrical setup presented allows to derive the cosmological constant term of the EHC from the Einstein-Hilbert functional and the action of a purely gauge coupling Higgs field.

Dirac Type Gauge Theories and the Mass of the Higgs Boson

We discuss the mass of the (physical component of the) Higgs boson in one-loop and top-quark mass approximation. For this the minimal Standard Model is regarded as a specific (parameterized) gauge theory of Dirac type. It is shown that the latter formulation, in contrast to the usual description of the Standard Model, gives a definite value for the Higgs mass. The predicted value for the Higgs mass depends on the value addressed to the top mass mT. We obtain mH = 186 ± 8 GeV for mT = 174 ± 3 GeV (direct observation of top events), resp. mH = 184 ±22 GeV for mT = 172 ±10 GeV (Standard Model electroweak fit). Although the Higgs mass is predicted to be near the upper bound, mH is in full accordance with the range 114 ≤ mH < 193 GeV that is allowed by the Standard Model. We show that the inclusion of (Dirac) massive neutrinos does not alter the results presented. We also briefly discuss how the derived mass values are related to those obtained within the frame of non-commutative geometry.

The Functional of Super Riemann Surfaces – A “Semi-Classical” Survey

This article provides a brief discussion of the functional of super Riemann surfaces from the point of view of classical (i.e., not “super-”) differential geometry. The discussion is based on symmetry considerations and aims to clarify the “borderline” between classical and super differential geometry with respect to the distinguished functional that generalizes the action of harmonic maps and is expected to play a basic role in the discussion of “super Teichmüller space”. The discussion is also motivated by the fact that a geometrical understanding of the functional of super Riemann surfaces from the point of view of super geometry seems to provide serious issues to treat the functional analytically.

The Topology of the Electroweak Interaction

In this paper we show that the Higgs boson of the (minimal) standard model has at most three gauge inequivalent ground states. One of these states is related to ordinary electromagnetism and the other two to electromagnetism within magnetically charged vacua. If space–time is assumed to be rotationally symmetric then the charged electroweak vacua may be identified with Dirac monopoles of magnetic charge g=±1∕2. This offers a physical interpretation of magnetic monopoles and Dirac’s quantization condition of electric charge in terms of the electroweak interaction. Moreover, in the case of the (minimal) standard model the three possible gauge inequivalent ground states of the Higgs boson are shown to fully determine the topological structure of the gauge bundle which underlies the electroweak interaction.