Pérez-Sánchez Carlos I
Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany.
Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland.
Lett Math Phys. 2022;112(3):58. doi: 10.1007/s11005-022-01546-x. Epub 2022 Jun 11.
We focus on functional renormalization for ensembles of several (say ) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form for certain noncommutative polynomials in the matrices. This article shows how the "algebra of functional renormalization"-that is, the structure that makes the renormalization flow equation computable-is derived from ribbon graphs, only by requiring the one-loop structure that such equation (due to Wetterich) is expected to have. Whenever it is possible to compute the renormalization flow in terms of -invariants, the structure gained is the matrix algebra with entries in , being the free algebra generated by the Hermitian matrices of size (the flowing random variables) with multiplication of homogeneous elements in given, for each , by which, together with the condition for each complex , fully define the symbol .
我们关注几个（比如说(N)个）随机矩阵系综的泛函重整化，其势函数包含多迹项，也就是说，概率测度包含形如(\exp(-N^2\mathrm{Tr}[V(M)]))的因子，其中(V(M))是矩阵(M)中的某些非交换多项式。本文展示了“泛函重整化代数”——即使得重整化流方程可计算的结构——是如何仅通过要求该方程（由于韦特里希）预期具有的单圈结构从带状图推导出来的。只要有可能根据(\beta)-不变量来计算重整化流，所得到的结构就是矩阵代数(\mathcal{M}N(\mathbb{C}))，其元素属于(\mathbb{C})，(\mathcal{M}N(\mathbb{C}))是由(N\times N)大小的(N^2)个厄米矩阵（流动的随机变量）生成的自由代数，对于每个(k)，齐次元素在(\mathcal{M}N(\mathbb{C}))中的乘法由([X_k,X_l]=i\theta{kl})给出，其中(\theta{kl})对于每个复数(\theta)，与条件([X_k,X_l]=i\theta{kl})一起完全定义了符号(\mathcal{M}_N(\mathbb{C}))。
