Topics in Quantum Field Theory

Trustees Of Boston University, Boston

Investigators

Abstract

This proposal investigates recently discovered algebraic structures which enhance our understanding of quantum field theory and the renormalization group. Kreimer will investigate number theoretic aspects of quantum field theory and their interplay with Hopf algebra structures, with an emphasis on applications to non-perturbative results. Rosenberg will work on the connection between differential geometry and quantum field theory, as suggested by the Birkhoff decomposition in renormalization theory and their corresponding flat connections. Jaffe will extend the methods of constructive field theory in light of these recent developments. In general, this work investigates the mathematical structure of quantum field theory (QFT) and the renormalization group. These are theoretical concepts which underlie the study of matter in phase transitions (e.g. boiling, freezing) as well as physics at the highest energies. While the mathematics behind QFT is still not well understood after fifty years of research, the predicative power of QFT in the laboratory is remarkable for its accuracy. Kreimer and his collaborators have discovered algebraic and combinatorial mathematical structures underlying basic QFT computations, which substantially simplify these calculations. In this research, we will both further the theoretical understanding of these concepts as well as investigate new applications.