Tensor quantum field theories and the large N expansion

这是一个理论粒子物理学的项目。它解决了STFC主题中的问题：什么是基本粒子和场？物理学的基本定律和对称性是什么？量子场论是研究这些问题的答案的框架。我们还远远不能了解所有的一致量子场论，特别是摄动理论之外的一致量子场论，也远远不能理解它们的重整化群不动点结构和相。大N展开是获得某些非摄动结构的一种方法；在模型场论中使用，它使我们能够建立，例如，存在潜在的非平凡共形场论，这些理论没有已知的直接拉格朗日公式。张量场理论，无论是玻色子的还是费米子的，最近都在三维空间的大N极限的背景下进行了研究，在这种情况下，它们通常由所谓的米朗图所主导。它们的直接积对称群导致了具有（在某些情况下非常大的）耦合常数数量的理论，这反过来又导致了具有复杂的相图的复杂的RG结构，由许多不动点和不动点的潜在线组成。因此，它们是鉴定新的CFTs的良好实验室。本项目将研究这些理论的两个方面。在某些情况下，包含具有多指标内部对称的标量场的理论中的大量耦合常数通过截断为更易于管理的自洽子集来处理。然而，目前尚不清楚这个子集是否对全耦合常数空间中额外方向的添加是稳定的，或者是否确实存在其他自洽选择，导致RG结构的本质不同。至少有两种方法可以解决这个问题，本项目将同时采用这两种方法。第一个是蛮力：大量的自由度无法由人工管理，计算必须自动化。然而，即使是自动计算也可以在规模上迅速扩散，并且建立使用传统符号数学系统的最佳基础已经是一个挑战。最近出现的符号数学的深度学习可能最终会变得更强大；目前，该领域的工作主要集中在积分和微分上，但探索高维空间似乎是深度学习的自然选择。解决这个问题的第二种方法是识别隐藏的对称性，这些对称性禁止生成通往额外方向的路径，例如，通过从某些对称是显式的基础理论中生成模型。

This is a project in theoretical particle physics. It addresses questions in the STFC themesWhat are the fundamental particles and fields? and What are the fundamental laws and symmetries of physics? Quantum field theory is the framework within which the answers to these questions can be investigated. We are far from knowing all the consistent quantum field theories, especially beyond perturbation theory, and understanding their renormalisation group fixed-point structure and phases. The large N expansion is a method of accessing some kinds of non-perturbative structure; used in model field theories it enables us to establish, for example, the existence of potentially non-trivial conformal field theories which have no known direct Lagrangian formulation.Tensor field theories, both bosonic and fermionic, have been studied recently in the context of the large N limit in three space-time dimensions where they are typically dominated by so-called melonic diagrams. Their direct product symmetry groups lead to theories with a (very large in some cases) number of coupling constants which in turn leads to a potentially complicated phase diagram with an intricate RG structure consisting of many fixed points and potentially lines of fixed points. They are thus good laboratories for identifying new CFTs. This project will study two aspects of these theories.The large number of coupling constants in theories containing a scalar field with a multiple-index internal symmetry has been dealt with in some cases by truncating to a self-consistent subset of more manageable size. However it is not known whether this subset is stable against addition of extra directions in the full coupling constant space or indeed whether there are other self-consistent choices which lead to substantially different RG structures. There are at least two approaches to this problem and the project will pursue both. The first is brute force: the large number of degrees of freedom cannot be managed by hand and the calculations have to be automated. However even automated calculations can proliferate rapidly in size and establishing the best basis in which to work using conventional symbolic mathematics systems is already a challenge. A speculative line which may turn out to be more powerful ultimately is the recent emergence of deep learning for symbolic mathematics; at present work in this area is on integration and differentiation but exploring a high dimensional space seems a natural candidate for deep learning. The second approach to the problem is to identify hidden symmetries that forbid the generation of paths into the extra directions, for example by generating the model from some underlying theory in which the symmetry is explicit.