Topics in Quantum Field Theory

本文研究了最近发现的代数结构，这些结构增强了我们对量子场论和重整化群的理解。Kreimer将研究量子场论的数论方面及其与Hopf代数结构的相互作用，重点是在非摄动结果中的应用。Rosenberg将研究微分几何和量子场论之间的联系，正如重整化理论中的Birkhoff分解及其相应的平面联系所建议的那样。Jaffe将根据这些最新的发展扩展建设性场论的方法。总的来说，这项工作研究了量子场论（QFT）的数学结构和重整化群。这些理论概念是研究物质相变（如沸腾、冻结）以及最高能量物理的基础。虽然经过五十年的研究，QFT背后的数学仍然没有得到很好的理解，但在实验室中，QFT的预测能力因其准确性而显着。Kreimer和他的合作者已经发现了基本QFT计算的代数和组合数学结构，这大大简化了这些计算。在本研究中，我们将进一步加深对这些概念的理论理解，并探索新的应用。

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.