论文标题
威尔逊地区法律公式，$ \ mathbb {r}^4 $
Wilson Area Law formula on $\mathbb{R}^4$
论文作者
论文摘要
令$ \ mathfrak {g} $为紧凑的半简单量规组的谎言代数。对于$ \ Mathfrak {g} $ - 值1形$ a $，请考虑Yang-Mills Action \ begin {equination} s _ {\ rm ym}（a）= \ int = \ int {\ int {\ mathbb {r}^4}^4}^4}^4}^4} \ weft | da + da + da + a \ wedge a \ wedge a \ riant | $ t \ mathbb {r}^4 $上的Euclidean Metric。我们要理解以下路径积分，\ begin {equination} {\ rm tr} \ \ \ int_ {a \ in \ mathcal {a} _ {\ mathbb {\ mathbb {r}^4，\ mathfrak {g}}}}}}}}}}} /\ mathcal {g}} e^{ - \ frac {1} {2} s _ {\ rm ym}（a）} \ da，\ nonumber \ end \ end {qore}，其中$ da $是$ \ mathfrak {g} $ valued 1-forms，modulo gaume，modulo gaume，modulo paremage $ da $ da $ $ \ mathcal {a} _ {\ mathbb {r}^4，\ mathfrak {g}} /\ Mathcal {g} $。在这里，$ s $是一些紧凑的平坦矩形表面。 使用抽象的Wiener空间，我们可以为紧凑的半简单量规组定义严格的Yang-Mills路径积分。随后，我们将使用重新归一化技术和渐近自由来从定义中得出威尔逊地区法律公式。 区域定律公式最重要的应用之一是解释为什么在夸克和古夸克之间测量的电位是其距离的线性函数。
Let $\mathfrak{g}$ be the Lie Algebra of a compact semi-simple gauge group. For a $\mathfrak{g}$-valued 1-form $A$, consider the Yang-Mills action \begin{equation} S_{\rm YM}(A) = \int_{\mathbb{R}^4} \left|dA + A \wedge A \right|^2\ dω, \nonumber \end{equation} using the Euclidean metric on $T\mathbb{R}^4$. We want to make sense of the following path integral, \begin{equation} {\rm Tr}\ \int_{A \in \mathcal{A}_{\mathbb{R}^4, \mathfrak{g}} /\mathcal{G}} \exp \left[ c\int_{S} dA\right] e^{-\frac{1}{2}S_{\rm YM}(A)}\ DA, \nonumber \end{equation} whereby $DA$ is some Lebesgue type of measure on the space of $\mathfrak{g}$-valued 1-forms, modulo gauge transformations $\mathcal{A}_{\mathbb{R}^4, \mathfrak{g}} /\mathcal{G}$. Here, $S$ is some compact flat rectangular surface. Using an Abstract Wiener space, we can define a Yang-Mills path integral rigorously, for a compact semi-simple gauge group. Subsequently, we will then derive the Wilson area law formula from the definition, using renormalization techniques and asymptotic freedom. One of the most important applications of the Area Law formula will be to explain why the potential measured between a quark and antiquark is a linear function of its distance.
