Multidimensional moment problems and the quantum-classic divide
多维矩问题和量子经典鸿沟
基本信息
- 批准号:EP/W024500/1
- 负责人:
- 金额:$ 12.56万
- 依托单位:
- 依托单位国家:英国
- 项目类别:Research Grant
- 财政年份:2022
- 资助国家:英国
- 起止时间:2022 至 无数据
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Imagine you are given a graph that gives you the distribution of child heights in a school. From it you could easily calculate the average height of the children. With a bit more work you could calculate the typical variation of heights about that average. These two quantities -- the mean and the variance -- are the first two in a sequence of numbers, known as the moments of the probability distribution. They give important information about the shape of the probability distribution graph (e.g., where its "middle" is in the the case of the mean).Whilst going from a probability distribution to the collection of moments is trivial, going in the other direction is not so simple. If one has access to the complete (in principle infinite) set of moments, then reconstructing the distribution is no problem. However, what about the case when we are only given a finite set of moments? What can we say about the underlying distribution? Is there any guarantee that, given a set of moments, that such a distribution even exists?These questions come under the title of the univariate truncated moment problem (TMP). And this is well understood from several different points of view (e.g., matrix theory, operator theory, probability theory and optimisation theory). However, multidimensional analogues of the TMP, where the given finite list is multiply indexed and we are worried about the probability of several different things happening, have proven to be much more elusive. Standard approaches that are known to work for in the univariate setting are inadequate in the multidimensional setting (insofar, as the discovery of concrete necessary and sufficient conditions for a solution are only understood in a few limited settings). The progress so far made with multidimensional TMPs has stimulated advances in function theory, operator theory and real algebraic geometry. Applications for multidimensional TMPs are in full abundance, e.g., probability theory, signal processing and theoretical physics.With applications in mind, we have identified an important and as-yet unexplored connection between multidimensional TMPs and foundational questions in quantum theory. In particular, we assert that the question of the existence of solutions to the TMP where the measure is supported on a given finite set is fundamental to understanding whether a set of observations is consistent with quantum or classical mechanics. Utilising recent advances in operator theory and real algebraic geometry, this project will develop new tools for solving these moment problems which will enable us to create a novel, unified and formally-rigorous framework for investigating the divide between classical and quantum theories.
想象一下,给你一张图表,上面是学校里孩子身高的分布情况。由此你可以很容易地计算出孩子们的平均身高。再多做一点工作,你就可以计算出平均高度的典型变化。这两个量——均值和方差——是数列的前两个,被称为概率分布的矩量。它们提供了关于概率分布图形状的重要信息(例如,在平均值的情况下,它的“中间”在哪里)。虽然从概率分布到矩的集合是微不足道的,但从另一个方向去做就不那么简单了。如果一个人能够获得完整的(原则上是无限的)矩集,那么重构分布是没有问题的。但是,如果只给我们一个有限的矩集呢?关于底层分布我们能说些什么呢?有什么能保证,给定一组矩,这样的分布存在吗?这些问题被称为单变量截断矩问题(TMP)。从几个不同的角度(例如,矩阵理论,算子理论,概率论和优化理论)可以很好地理解这一点。然而,TMP的多维类比,其中给定的有限列表是多重索引的,我们担心发生几种不同事情的概率,已被证明是更加难以捉摸的。已知在单变量环境中有效的标准方法在多维环境中是不够的(到目前为止,因为解决方案的具体必要条件和充分条件的发现仅在少数有限的环境中被理解)。迄今为止,多维TMPs的进展刺激了函数理论、算子理论和实代数几何的发展。在概率论、信号处理和理论物理等领域,多维tmp的应用十分丰富。考虑到应用,我们已经确定了多维tmp与量子理论基础问题之间的重要且尚未探索的联系。特别是,我们断言在给定有限集合上支持测度的TMP解的存在性问题是理解一组观测是否与量子力学或经典力学一致的基础。利用算子理论和实际代数几何的最新进展,该项目将开发新的工具来解决这些矩问题,这将使我们能够创建一个新的、统一的、形式上严格的框架来研究经典理论和量子理论之间的鸿沟。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
数据更新时间:{{ journalArticles.updateTime }}
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
数据更新时间:{{ journalArticles.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ monograph.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ sciAawards.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ conferencePapers.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ patent.updateTime }}
David Patrick Kimsey其他文献
David Patrick Kimsey的其他文献
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
相似国自然基金
Probing matter-antimatter asymmetry with the muon electric dipole moment
- 批准号:
- 批准年份:2020
- 资助金额:30 万元
- 项目类别:
组合序列的moment问题研究
- 批准号:12001301
- 批准年份:2020
- 资助金额:24.0 万元
- 项目类别:青年科学基金项目
Fano流形的moment-weight等式及Mabuchi度量的存在性
- 批准号:11801156
- 批准年份:2018
- 资助金额:20.0 万元
- 项目类别:青年科学基金项目
与Catalan-like数相关的组合问题研究
- 批准号:11701249
- 批准年份:2017
- 资助金额:25.0 万元
- 项目类别:青年科学基金项目
加速器中微子振荡实验MOMENT的优化及其唯象学探究
- 批准号:11505301
- 批准年份:2015
- 资助金额:20.0 万元
- 项目类别:青年科学基金项目
关于Moment-Angle流形的几个问题
- 批准号:11401118
- 批准年份:2014
- 资助金额:23.0 万元
- 项目类别:青年科学基金项目
稳定同伦中的无限降阶法与moment-angle流形
- 批准号:11471167
- 批准年份:2014
- 资助金额:65.0 万元
- 项目类别:面上项目
可压缩多介质ALE框架下的MOF界面重构方法研究
- 批准号:11001026
- 批准年份:2010
- 资助金额:17.0 万元
- 项目类别:青年科学基金项目
相似海外基金
Noncommutative Szego Theory, Moment Problems, and Related Problems in Noncommutative Analysis
非交换 Szego 理论、矩问题以及非交换分析中的相关问题
- 批准号:
2751175 - 财政年份:2022
- 资助金额:
$ 12.56万 - 项目类别:
Studentship
Development of complex moment-based methods and mathematical risk avoidance techniques for infinite dimensional eigenvalue problems
针对无限维特征值问题开发复杂的基于矩的方法和数学风险规避技术
- 批准号:
21H03451 - 财政年份:2021
- 资助金额:
$ 12.56万 - 项目类别:
Grant-in-Aid for Scientific Research (B)
Semidefinite Programming Methods for Moment and Optimization Problems
矩量和优化问题的半定规划方法
- 批准号:
1417985 - 财政年份:2014
- 资助金额:
$ 12.56万 - 项目类别:
Standard Grant
Classical Realizability and Quantum Representability: Truncated Moment Problems in Statistical Physics and Quantum Chemistry
经典可实现性和量子可表示性:统计物理和量子化学中的截断矩问题
- 批准号:
EP/H022767/1 - 财政年份:2010
- 资助金额:
$ 12.56万 - 项目类别:
Research Grant
Discrete Moment Problems and Applications
离散矩问题及应用
- 批准号:
0856663 - 财政年份:2009
- 资助金额:
$ 12.56万 - 项目类别:
Standard Grant
RUI: Truncated Multivariable Moment Problems & Applications: An Operator Theoretic Approach
RUI:截断多变量矩问题
- 批准号:
0758378 - 财政年份:2008
- 资助金额:
$ 12.56万 - 项目类别:
Standard Grant
Fourier analysis and moment problems
傅里叶分析和矩问题
- 批准号:
36534-2002 - 财政年份:2007
- 资助金额:
$ 12.56万 - 项目类别:
Discovery Grants Program - Individual
Toward a Unified, Moment-Based Treatment of Multi-Variate, Interacting Population Balance Problems - Development/Incorporation of Realistic Rate Laws
对多变量、相互作用的人口平衡问题进行统一的、基于时刻的处理——现实利率法的开发/结合
- 批准号:
0522944 - 财政年份:2006
- 资助金额:
$ 12.56万 - 项目类别:
Continuing Grant
Fourier analysis and moment problems
傅里叶分析和矩问题
- 批准号:
36534-2002 - 财政年份:2005
- 资助金额:
$ 12.56万 - 项目类别:
Discovery Grants Program - Individual
RUI: Truncated Multivariable Moment Problems & Applications: An Operator Theoretic Approach
RUI:截断多变量矩问题
- 批准号:
0457138 - 财政年份:2005
- 资助金额:
$ 12.56万 - 项目类别:
Standard Grant