Non-perturbative Group field theory from combinatorial Dyson-Schwinger equations and their algebraic structure II

基础物理学中最大的理论挑战之一是将联合收割机广义相对论和量子场论结合成一个量子引力理论。最近发现非对易几何的量子场论可以用矩阵论表示法非微扰求解。群场论是这种矩阵场论向更高秩的推广，是引力量子理论的候选者。因此，在群场论中能得到多大程度的非微扰解是一个重要的问题。在这个研究项目中，我们解决这一挑战，利用代数结构的Dyson-Schwinger方程的水平上的重整化。与张量结构相关的量子对称性以及理论的规范不变性允许简化这些方程。这样，我们将发现在哪些条件下群场论可以非微扰地求解，并得到解。控制非微扰区域是一个具有巨大物理意义的开放性问题，因为在这样的量子引力理论中，连续时空的极限与临界轨迹的极限一致。

One of the greatest theoretical challenges in fundamental physics is to combine general relativity and quantum field theory to a quantum theory of gravity. Quantum field theories on non-commutative geometry have recently been found to be solvable non-perturbatively in a matrix-theory representation. Group field theory is a generalization of such matrix field theory to higher rank and is a candidate for a quantum theory of gravity. It is therefore an important question to what extent non-perturbative solutions can be obtained in group field theory as well. In this research project we address this challenge making use of the algebraic structure of renormalization on the level of Dyson-Schwinger equations. Quantum symmetries related both to the tensorial structure as well as the gauge invariance of the theory allow to simplify these equations. In this way we will find under which conditions group field theory can be solved non-perturbatively and derive solutions. Control over the non-perturbative regime is an open issue of huge physical interest since the limit to continuum space-time coincides with the limit to critical loci in such a theory of quantum gravity.