International Research Fellowship Program: Introduction of Field Theory into the Causal Set Context

国际研究奖学金计划：将场论引入因果集背景

• 批准号: 0853079
• 负责人: Roman Sverdlov
• 金额: $ 0.5万
• 依托单位: Sverdlov, Roman M
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0853079&HistoricalAwards=false
• 关键词: International; Research; Fellowship; Program; Introduction

0853079SverdlovThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.This award will support a twenty-four-month research fellowship by Dr. Roman M. Sverdlov to work with Dr. Sumati Surya at the Raman Research Institute in Bangalore, India.Causal Sets, originally proposed by Rafael Sorkin, is one model of discrete space time intended to develop quantum theory of gravity. According to this model, the space time we live in can be described (both macroscopically and microscopically) as a locally finite set of points and their causal relations. On a macroscopical level, a causal relation between the two points is an answer to the question as to whether or not it is possible to travel between them without going faster than the speed of light. It has been shown by Hawking that if the volumes of regions are specified which, in discrete setting, can be done by assuming each point takes up the same volume, one can deduce metric from the causal structure. Thus, causal structure can be viewed as a generalization of a metric for a microscopic setting, where manifold-like geometry breaks down due to quantum fluctuations. One unexplored area of the theory, however, is to show how the approximate manifold structure is restored on a larger scale. Scientists focus on addressing topological questions in a toy model of absolute vacuum, and have postponed introducing particles until topology is under control (although there are few exceptions, such as the work of Johnston and Jacobsen). On the first glance, this makes sense: geometry is needed in order for particles to propagate. This, however, is a point in which the PI?s research deviates from the norm substantially. Since geometry is identified with gravity, which is a field, it is not possible to have geometry literally without matter. This also means that the manifold-like topology on a large scale might well be a consequence of the dynamics matter is subjected to (in particular, gravitational field). Thus, he proposes to reverse the steps of the program: first introduce fields and Lagrangians in non-manifoldlike causal set, and later explore the large scale geometry once the non-manifoldlike dynamics is defined. However, in light of the fact that the prediction for a manifold-like scenario is the ultimate verification of a theory, he often considers ?special cases? where manifold structure is being put by hand; typically, these cases involve Poisson distribution of points on Lorentzian manifold. In his dissertation he proposed a way to define the known fields and their Lagrangians for a general causal set. However, the next step of going from Lagrangian to propagator is very problematic, due to the fact that the number of degrees of freedom of path integration is equal to the number of points (or even worse, pairs of points) of the entire universe. In regular quantum field theory this is done by imagining a regular cubic lattice, in which case it is possible to perform different integrals ?all at once?. However, due to the fact that the difference between the edges of the lattice and its diagonals violate relativity, such structure cannot be used, so in order to obtain some results one has to think of another method. This can be somewhat justified by analogy with Einstein?s equation, which does not have an exact solutions either. However, in order for this analogy to work, there has to be some simple cases that do have exact solution. The goal of his project is to find such cases.One direction of research is a toy model of a causal set consisting of two points A and B on a manifold singled out beforehand, together with random sprinkling of n other points. Then causal set based propagator is computed between points A and B, where all geometry is ignored, except for causal relations between relevant points. That result is compared to the propagator between A and B computed by ordinary methods of quantum field theory, where selection of n points is ignored. The above work can realistically be done by numeric simulations of toy models of causal sets consisting of very few points (10 points, 100 points, etc). However, he hopes to analytically show that for very large number of points the causal set based propagator satisfies some form of differential equation, which would allow the application of the theory to many point scenario.

0853079斯维尔德洛夫该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。国际研究奖学金计划使美国科学家和工程师能够在国外进行9到24个月的研究。 该计划的奖项提供了联合研究的机会，以及使用独特或互补的设施，专业知识和国外的实验条件。Sverdlov与印度班加罗尔的拉曼研究所的Sumati Surya博士一起工作。因果集最初由Rafael Sorkin提出，是一种离散时空模型，旨在发展量子引力理论。 根据这个模型，我们生活的时空可以被描述（宏观和微观）为一个局部有限的点集及其因果关系。在宏观层面上，这两点之间的因果关系是对是否有可能在它们之间旅行而不超过光速的问题的答案。霍金已经证明，如果区域的体积被指定，在离散的情况下，可以通过假设每个点占据相同的体积来完成，那么人们可以从因果结构中推导出度规。因此，因果结构可以被看作是微观环境中度量的推广，其中流形状几何由于量子涨落而分解。 然而，该理论的一个未探索的领域是如何在更大的尺度上恢复近似流形结构。 科学家专注于在绝对真空的玩具模型中解决拓扑问题，并推迟引入粒子，直到拓扑结构得到控制（尽管有一些例外，如约翰斯顿和雅各布森的工作）。 乍一看，这是有道理的：粒子传播需要几何。然而，这是一个点，其中PI？的研究大大偏离了常规。由于几何学等同于引力，而引力是一种场，因此不可能有字面上没有物质的几何学。这也意味着大尺度上的流形拓扑结构很可能是物质所受动力学（特别是引力场）的结果。 因此，他建议颠倒程序的步骤：首先在非流形因果集中引入场和拉格朗日，然后在定义了非流形动力学之后探索大尺度几何。 然而，鉴于这一事实，即预测一个流形样的情况是最终验证的理论，他经常认为？特殊情况？其中流形结构是手工放置的;通常，这些情况涉及洛伦兹流形上点的泊松分布。 在他的论文中，他提出了一种方法来定义已知领域和拉格朗日一般因果集。然而，从拉格朗日到传播算子的下一步是非常有问题的，因为路径积分的自由度的数量等于整个宇宙的点（甚至更糟，点对）的数量。在规则量子场论中，这是通过想象一个规则的立方晶格来完成的，在这种情况下，可以执行不同的积分？一下子？然而，由于格的边与其对角线之间的差异违反相对性的事实，这种结构不能使用，因此为了获得某些结果，必须考虑另一种方法。 这可以通过与爱因斯坦的类比来证明吗？的方程，也没有精确解。然而，为了使这个类比起作用，必须有一些简单的情况，确实有精确的解决方案。他的项目的目标就是找到这样的案例，其中一个研究方向是建立一个因果集的玩具模型，这个因果集由事先挑选出来的流形上的两个点A和B组成，再加上随机散布的n个点。然后在点A和B之间计算基于因果集的传播算子，其中忽略所有几何形状，除了相关点之间的因果关系。将该结果与通过量子场论的普通方法计算的A和B之间的传播子进行比较，其中忽略n个点的选择。上述工作可以通过由非常少的点（10点、100点等）组成的因果集的玩具模型的数值模拟来现实地完成。然而，他希望通过分析表明，对于非常大量的点，基于因果集的传播子满足某种形式的微分方程，这将允许将理论应用于许多点场景。