LEAPS-MPS: Investigating Emergent Gravity in Combinatorial Quantum Systems

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。该奖项资助圣何塞州立大学的Kassahun Betre教授的研究活动。 空间和时间也许是物理实在最基本的方面。 然而，对其性质的更深层次的解释仍然是一个谜。 近年来，许多可能的方法被用来解决这个问题;这些方法通常被归类为弦理论，圈量子引力和因果动力三角测量。 作为该项目的一部分，Betre教授将采用一种补充方法，称为涌现背景独立量子引力（EBIQG）。 这种方法假定空间和时间的构建块是可以用图形形式表示的某些量子力学关系。 在这种方法中，空间和时间只作为宏观平均值出现。 Betre教授将建立模拟图论量子系统的统计和热力学性质的数值工具，并构建二维量子引力中维度和几何出现的模型。 预计该项目还将产生更广泛的重大影响。 圣何塞州立大学是一所为少数民族服务的大学，参加本研究项目的本科生和硕士生不仅将培养分析研究技能，还将培养进行数值模拟的实践技能。 该项目还将为这些学生提供在STEM领域取得成功的经验和信心。 从技术上讲，这项研究将为EBIQG领域带来两个重要的新方向。首先，它突出了空间本身作为起点的统一维度的出现。一个图论的推广的维数的欧氏空间称为归纳维数被用来作为一个动态量的图微观状态的过程中粒到涌现维的光滑流形。其次，本研究项目使用所有的几何图形-曲面三角剖分的图论类似物-而不仅仅是正则格和正则图来建模离散空间。用这种方法，量子引力从图模型中的出现表现为一个简并的零温度基态。因此，这样的研究将有助于铺平道路，从一个非几何图论希尔伯特空间涌现的整数维和几何的具体模型。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).This award funds the research activities of Professor Kassahun Betre at San Jose State University. Space and time are perhaps the most fundamental aspects of physical reality. However, a deeper explanation of their nature is still a mystery. Many possible approaches toward addressing this issue have been pursued in recent years; these are usually grouped under names such as String Theory, Loop Quantum Gravity, and Causal Dynamical Triangulation. As part of this project, Professor Betre will pursue a complementary approach called Emergent Background-Independent Quantum Gravity (EBIQG). This approach posits that the building blocks of space and time are certain quantum-mechanical relationships that can be expressed in the form of graphs. Within such an approach, space and time emerge only as macroscopic averages. Professor Betre will build numerical tools that simulate the statistical and thermodynamic properties of the graph-theoretic quantum systems, and construct a model for the emergence of dimension and geometry in a two-dimensional quantum gravity. This project is also envisioned to have significant broader impacts. San Jose State University is a minority-serving institution, and undergraduate and masters students participating in this research project will develop not only analytical research skills but also practical skills in conducting numerical simulations. This project will also provide these students the experience and confidence to succeed in STEM fields. More technically, this research will bring two important new directions to the field of EBIQG. First, it highlights the emergence of the uniform dimension of space itself as a starting point. A graph-theoretic generalization of the dimension of Euclidean space called inductive dimension is used as a dynamical quantity of the graph microstates that course-grains into the emergent dimension of smooth manifolds. Second, this research project models discretized space using all geometric graphs --- graph-theoretic analogues of triangulations of surfaces --- rather than just regular lattices and regular graphs. With this approach, the emergence of quantum gravity from the graph models appears as a degenerate zero- temperature ground state. Such research will therefore help to pave the way towards a concrete model for emergent integer dimension and geometry from a non-geometric graph-theoretic Hilbert space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.