四维Yang-Mills量子场论中的费米化、收敛微扰理论和关联
Fermionization, Convergent Perturbation Theory, and Correlations in the Yang-Mills Quantum Field Theory in Four Dimensions
摘要：
我们表明，杨米尔斯量子场论的动量和时空截止在四个欧几里德维是等价的，在一个适当的图解微扰理论的条款，一项一项的费米理论与非局部相互作用条款。当施加进一步的动量截断时，这个费米子理论有收敛的微扰展开。在这种微扰展开式的零阶中，连接对的通用分量的相关函数$E(x,y)$由显式的有限维积分公式给出，我们推测它将表现为$$E(x,y) \sim |x - y|^{-2 - 2 d_G},$$ \noindent对于$|x-y|>>0,$，其中$d_G$是取决于规范群$G.$的正整数。在$G=SU(n),$的情况下，我们推测$$d_G = {\rm dim}SU(n) - {\rm dim}S(U(n-1) \times U(1)),$$ \noindent，使得相关性的衰减速率随着$n \to \infty.$
英文摘要：
We show that the Yang-Mills quantum field theory with momentum and spacetime cutoffs in four Euclidean dimensions is equivalent, term by term in an appropriately resummed perturbation theory, to a Fermionic theory with nonlocal interaction terms. When a further momentum cutoff is imposed, this Fermionic theory has a convergent perturbation expansion. To zeroth order in this perturbation expansion, the correlation function $E(x,y)$ of generic components of pairs of connections is given by an explicit, finite-dimensional integral formula, which we conjecture will behave as $$E(x,y) \sim |x - y|^{-2 - 2 d_G},$$ \noindent for $|x-y|>>0,$ where $d_G$ is a positive integer depending on the gauge group $G.$ In the case where $G=SU(n),$ we conjecture that $$d_G = {\rm dim}SU(n) - {\rm dim}S(U(n-1) \times U(1)),$$ \noindent so that the rate of decay of correlations increases as $n \to \infty.$
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