Multidimensional moment problems and the quantum-classic divide

想象一下，你得到了一张图表，上面显示了一所学校里孩子们的身高分布。由此你可以很容易地计算出孩子们的平均身高。只要多做一点工作，你就可以计算出平均身高的典型变化。这两个量--均值和方差--是一系列数字中的前两个，称为概率分布的矩。它们给出了有关概率分布图的形状的重要信息(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，但从另一个方向去就不那么简单了。如果一个人可以获得完整的(原则上是无限的)矩集合，那么重构分布是不成问题的。然而，当我们只被给予有限的时刻时，情况又如何呢？关于潜在的分布，我们能说些什么呢？在给定一组矩的情况下，能保证这样的分布甚至存在吗？这些问题属于单变量截断矩问题(TMP)。从几个不同的角度(例如，矩阵理论、算子理论、概率理论和最优化理论)可以很好地理解这一点。然而，TMP的多维类似物，其中给定的有限列表是多重索引的，并且我们担心发生几种不同事情的可能性，事实证明要难以捉摸得多。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(就目前而言，只有在少数有限的环境中才能理解解决方案的具体的充要条件的发现)。到目前为止，多维TMPS所取得的进展促进了函数论、算子论和实代数几何的发展。多维TMPS的应用非常广泛，如概率论、信号处理和理论物理。结合应用，我们发现了多维TMPS与量子理论中的基本问题之间的一个重要而尚未探索的联系。特别是，我们断言，TMP的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。利用算符理论和实代数几何的最新进展，这个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、统一的和形式上严格的框架来研究经典理论和量子理论之间的分歧。

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.