**Electroweak interactions and, in particular, neutrinos**

Purely leptonic interactions (muon and tau decays, neutrino-electron reactions, etc) as well as semi-leptonic decays (precision studies of pion and kaon decays, inverse beta decay, etc.) have been prime sources for our understanding of the structure of weak interactions ever since their discovery. Precision studies and detailed analyses provide stringent tests for the minimal standard model of electroweak interactions.

The mysteries of the neutrino sector add much to the fascination of this branch of elementary particle physics. Knowing that at least some of the neutrinos have finite masses, the challenge is to understand the pattern of their masses in relation to their state mixing as observed in neutrino oscillations.

(For contributions by me and my former group see list of publications.)

**Quark properties and quark mixing**

Lately we were preoccupied by the question to which extent the quark masses of the three quark (flavour) generations can be reconstructed from their mass eigenvalues and from their observed mixing. While the mass matrices fix the mixing pattern uniquely, the converse, obviously, is not true: A set of given masses and (CKM)-mixing matrix are compatible with an infinity of mass matrices. Nevertheless we succeeded in clarifying completely the reconstruction problem by “sweeping” the space of admissible mass matrices from a given input of data.

**Noncommutative geometry and the standard model**

Noncommutative geometry, a recent branch of mathematics originally inspired by the Heisenberg algebra in quantum mechanics, has various relationships to modern physics (fractals, quantum field theory, standard model, strings). Together with R. Coquereaux and his group in Marseille-Luminy we developed the Mainz-Marseille variant of the (minimal) standard model in the frame of noncommutative geometry. In essence, this provides a generalization of gauge theories which adds more, (unlike grand unified theories) not less, structure to the standard model. Among other successes the model assigns a natural role to the Higgs fields and predicts spontaneous symmetry breaking.

More generally, geometry (commutative or noncommutative) is at work in many aspects of quantum field theory, from anomalies whose geometric origin we helped to clarify, to Kaluza-Klein theories where geometric mechanisms are at work in the compactification of dimensions beyond the four space-time dimensions. Also the spin-statistics relationship (theorem by Fierz, Pauli) bears interesting geometric properties.

(For more on these topics see list of publications.)

**More information**