Effective Field Theories and PDE

在理论物理学的许多问题中，利用“重”和“轻”激发之间的等级尺度是可能取得进展的。当考虑轻场的激发时，有一个很好理解的过程，从模型中正式删除重场，留下由无限扩展-有效场论扩展-其项通常具有越来越高的导数的形式给出的轻激发的场方程。这就提出了一个难题，因为天真地截断这样一个系列会引起病态。最近在严格理解这种情况方面取得了进展。这个项目将带来现代偏微分方程分析的技术，以了解如何有效场理论可以作为经典偏微分方程的限制。特别是，这将涉及非线性波动方程的平均技术的详细检查，在广义相对论和宇宙学的应用。这福尔斯自然属于EPSRC的数学物理和数学分析研究领域的职权范围。

In many problems in theoretical physics it is possible to make progress by exploiting a hierarchy of scales between 'heavy' and 'light' excitations. When considering excitations of the light fields, there is a well-understood process to formally remove the heavy fields from the model, leaving field equations for the light excitations which are formally given by an infinite expansion - the Effective Field Theory expansion - whose terms typically have higher and higher derivatives. This has presented a puzzle, since naively truncating such a series gives rise to pathologies. Recently progress has been made in understanding such situations rigorously. This project will bring the techniques of modern PDE analysis to bear on understanding how Effective Field Theories can arise as limits of classical PDE. In particular this will involve a detailed examination of averaging techniques for nonlinear wave equations, with applications in General Relativity and Cosmology.This falls naturally within the remit of EPSRC's Mathematical Physics and Mathematical Analysis Research Areas.