时空的小尺度结构：vanVleck行列式和等测地线曲面
Small scale structure of spacetime: van Vleck determinant and equi-geodesic surfaces
摘要：
最近有人认为，如果时空$\mathcal M$在小尺度下具有非平凡结构，那么对它的适当的半经典描述应该基于非定域双张量而不是诸如度规$g_{ab}$的定域张量。两个最相关的双张量是辛格的世界函数$Ω(p,p_0)$和范弗莱克行列式（VVD）$Δ(p,p_0)$，因为它们编码了时空的度量性质和测地线的聚焦行为。它们还描述了d 'Alembartian $_{p_0} \square_p$的两个点函数的主导短距离行为。我们首先讨论由$Ω(p,p_0)=constant$定义的等测地曲面$Σ_{G,p_0}$在测地凸邻域内的内在和外在几何性质，并着重讨论了与这些曲面的几何性质有关的VVD的一些初等恒等式.作为一个旁白，我们还评论的贡献$Σ_{G,p_0}$的表面项中的Einstein-Hilbert（EH）的行动，并表明，它可以写为体积积分的$\square \ln Δ$。然后，我们研究的小尺度结构的时空中存在的洛伦兹不变的短程截止$\ell_0$使用$Ω$和$Δ$，根据最近发展的一些想法。我们导出了一个二阶双张量$q_{ab}$，它自然地产生从下面有界的测地区间，并提出了一个一般的和数学上严格的分析的短距离结构的时空的基础上（a）的几何$Σ_{G,p_0}$，（b）的结构的非本地的d 'Alembartian与$q_{ab}$，和（c）的性质的VVD。特别是，我们表明，里奇双标量的$q_{ab}$是完全由$Σ_{G,p_0}$，潮汐张量，前两个衍生物的范Vleck行列式，并有一个非平凡的“经典”限制（常数）$R_{ab} q^a q^b$（见正文）。
英文摘要：
It has recently been argued that if spacetime $\mathcal M$ possesses non-trivial structure at small scales, an appropriate semi-classical description of it should be based on non-local bi-tensors instead of local tensors such as the metric $g_{ab}$. Two most relevant bi-tensors in this context are Synge's World function $Ω(p,p_0)$ and the van Vleck determinant (VVD) $Δ(p,p_0)$, as they encode the metric properties of spacetime and (de)focussing behaviour of geodesics. They also characterize the leading short distance behavior of two point functions of the d'Alembartian $_{p_0} \square_p$. We begin by discussing the intrinsic and extrinsic geometry of equi-geodesic surfaces $Σ_{G,p_0}$ defined by $Ω(p,p_0)=constant$ in a geodesically convex neighbourhood of an event $p_0$, and highlight some elementary identities relating the VVD with geometry of these surfaces. As an aside, we also comment on the contribution of $Σ_{G,p_0}$ to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of $\square \ln Δ$. We then study the small scale structure of spacetime in presence of a Lorentz invariant short distance cut-off $\ell_0$ using $Ω$ and $Δ$, based on some recently developed ideas. We derive a 2nd rank bi-tensor $q_{ab}$ which naturally yields geodesic intervals bounded from below, and present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of $Σ_{G,p_0}$, (b) structure of the non local d'Alembartian associated with $q_{ab}$, and (c) properties of the VVD. In particular, we show that the Ricci bi-scalar of $q_{ab}$ is completely determined by $Σ_{G,p_0}$, the tidal tensor, and first two derivatives of the van Vleck determinant, and has a non-trivial "classical" limit given by (constant) $R_{ab} q^a q^b$ (see text).
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