Euclidean formulation of relativistic quantum theory

Modern particle physics is built upon two fundamental pillars. They are 1. Quantum mechanics and 2. The theory of special relativity.

The goal of my research is to incorporate these two theories in a Euclidean representation. The condition that enables us to do that is known as reflection positivity. The basic principle of reflection positivity (RP) is often named after Robert Osterwalder and Konrad Schrader. The general structure of reflection positive kernels and operators on Euclidean representations of Hilbert spaces are discussed in my thesis.

In its original form, the Osterwalder-Schrader idea served as a useful link between problems in quantum physics and in mathematics. More precisely, one links Euclidean field theory (math) to relativistic quantum field theory (physics). The problems are interesting in both incarnations, both for the setting of relativistic fields, and the other side, Euclidean fields. Tools and solutions available on one side often yield insights on the other side. The basic symmetry groups are different, i.e., the Poincaré group vs the Euclidian group. Hence the associated harmonic analysis and spectral theory are different of course.

As a general principle, the reflection positivity (RP) correspondence has proved useful in mathematics and in many neighboring areas. It successfully combines powerful tools from analysis, from geometry, from representation theory, and from physics. Up to the present; for example, the RP-correspondence has served to link abelian (commutative) properties of Gaussian processes/fields in the Euclidean setting, to the context of non-commutativity in the study of quantum fields. Moreover, since its inception, reflection positivity has been generalized and extended in many new and diverse directions. And by now, it has further become a powerful tool in noncommutative harmonic analysis, and in the theory of unitary representations of Lie groups. And there are many other powerful uses in yet other directions, for example, from RP in mathematics to the study of interactions and scattering in physics; the latter the focus of the present thesis.