扩展体传播子
Extended Body Propagators
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明三种不同来源的扩展体传播子均为Polyakov路径积分的实现，明确了Ansoldi传播子的概率密度函数，构造了适用于二维单纯复形的面积测度，还将管状世界面测度归结为三角形测度。
AI中文摘要：
Ansoldi等人、Erbin等人和Stanford等人设计的三种看似不同的扩展体传播子分别出现在完全不同的场景中，不过每种都可视为Polyakov路径积分的一种实现。本文证明，Ansoldi传播子对应环境维度D下概率密度函数为$f(A)=C\\,A\\,(\alpha^{2}+A^{2})^{-D/2}\mathbf{1}_{A>0}$的随机变量的特征函数，其中尺度$\alpha=\sqrt{b/2}$。该指数由维度决定，与Ansoldi/几何比较中的其他因素无关；Erbin等人的在壳两点振幅与JT引力在下文明确的意义上与该分布的D=4成员一致。经过广泛的文献综述，作者认为该观察结果及构造的面积测度是新颖的。不过，该面积测度将成为将早期非扩展体路径空间测度扩展到二维单纯复形的自然候选。我们进一步证明，本文的面积测度恰好是三角形面积的概率法则，该三角形的两条边来自配套手稿中一维Polyakov积分得到的测度。最后，我们以部分严格的论证将任意两条边界曲线之间的管状世界面测度归结为该三角形的测度。
英文摘要：
Three seemingly distinct extended-body propagators devised by Ansoldi et al., Erbin et al., and Stanford et al.\ respectively arise in completely different settings, though each can be considered a realization of the Polyakov path integral. This paper proves that the Ansoldi propagator corresponds to the characteristic function of the random variable with the probability density function $f(A)=C\,A\,(α^{2}+A^{2})^{-D/2}\mathbf{1}_{A>0}$ in ambient dimension $D$, of scale $α=\sqrt{b/2}$. The exponent is fixed by the dimension and by nothing else in the Ansoldi/geometric comparison; the on-shell two-point amplitude of Erbin et al. and JT gravity agree with its $D=4$ member in the senses made precise below. After an extensive literature review, the author believes that this observation and constructed area measure is novel. The area measure will, however, emerge as the natural candidate for the extension of the earlier non-extended body path space measures to two dimensional simplicial complexes. We further show that the area measure of this paper is exactly the law of the area of a triangle, two of whose sides are drawn from the measure obtained in the companion manuscripts for the one-dimensional Polyakov integral. We close with an argument, rigorous in part, which reduces the measure of tubular worldsheets between two arbitrary boundary curves to that of this triangle.
