洛伦兹单纯形量子引力中的因果结构与光锥奇点
Causal structure and light-conical singularities in Lorentzian simplicial quantum gravity
AI总结：
本文研究洛伦兹Regge引力中的光锥奇点，开发算法采样四维三角剖分构型空间，发现光锥奇点普遍存在，分析其结构及因果不规则性对引力路径积分因果条件选择的启示。
AI中文摘要：
洛伦兹引力路径积分的定义需要明确壳外允许的因果结构，其中光锥奇点尤为重要——这类奇点是光锥结构的余维二不规则性，会使引力作用量变为复数，进而可能导致构型出现指数级增强或抑制。我们研究了洛伦兹Regge引力中这类奇点的普遍性与结构：开发了一种高效算法，用于随机生成满足广义三角不等式的可实现Regge几何，并利用该算法对四维三角剖分的构型空间进行采样。研究发现，光锥奇点是普遍存在的：对于被多个四维单形共享的骨，具有不规则光锥结构的构型占绝对主导，而裤型奇点通常在熵层面更受青睐；对子单形的因果特征施加限制（如要求所有四面体为类空），会显著改变这些统计结果。我们进一步证明，在与简单细化移动相关的无界构型空间区域，既会出现圆顶型奇点，也会在四维空间中出现裤型奇点。除光锥奇点外，我们还更广泛地分析了洛伦兹三角剖分的因果结构，例如引入了成对可嵌入时序拓扑的概念及一组离散因果数据。研究结果表明，因果不规则性是无限制洛伦兹Regge构型空间的固有且丰富特征，这加深了我们需在Regge引力、自旋泡沫模型乃至连续体引力路径积分中施加何种因果条件的疑问。
英文摘要：
The definition of a Lorentzian gravitational path integral requires specifying which causal struc- tures are admitted off shell. Of particular importance are light-conical singularities, codimension- two irregularities of the light cone structure that render the gravitational action complex and can therefore lead to exponential enhancement or suppression of configurations. We investigate the prevalence and structure of such singularities in Lorentzian Regge gravity. We develop an efficient algorithm for randomly generating realizable Regge geometries satisfying the generalized triangle inequalities, and use it to sample the configuration space of four-dimensional triangulations. We find that light-conical singularities are generic: for bones shared by many four-simplices, configurations with irregular light cone structure overwhelmingly dominate, with trouser-type singularities gen- erally entropically favored. Restrictions on the causal character of subsimplices, such as requiring all tetrahedra to be spacelike, can substantially alter these statistics. We further show that both yarmulke- and, in four dimensions, trouser-type singularities can occur on unbounded regions of configuration space associated with simple refinement moves. Beyond light-conical singularities, we analyze the causal structure of Lorentzian triangulations more generally, for example by introducing the notion of pairwise embeddable chrono-topologies and a set of discrete causality data. Our re- sults demonstrate that causal irregularities are an intrinsic and abundant feature of the unrestricted Lorentzian Regge configuration space, sharpening the question of which causality conditions should be imposed in Regge gravity, spin-foam models, and ultimately the continuum gravitational path integral.
