Multidimensional moment problems and the quantum-classic divide

想象一下，你有一张图表，上面显示了一所学校里孩子的身高分布。由此，你可以很容易地计算出孩子们的平均身高。再多做点工作，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--是一个数字序列中的前两个，称为概率分布的矩。它们给出了关于概率分布图形状的重要信息（例如，虽然从概率分布到矩的集合是微不足道的，但在另一个方向上就不那么简单了。如果一个人有机会获得完整的（原则上是无限的）矩集，那么重建分布是没有问题的。然而，当我们只得到一组有限的时刻时，情况又如何呢？关于潜在的分布，我们能说些什么呢？在给定一组时刻的情况下，是否有任何保证，这样的分布甚至存在？这些问题的标题下的单变量截断矩问题（TMP）。从几个不同的观点来看，这是很好理解的（例如，矩阵理论、算子理论、概率论和最优化理论）。然而，TMP的多维类似物，其中给定的有限列表是多重索引的，我们担心几个不同的事情发生的概率，已经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（因为只有在少数有限的环境中才能理解对解的具体必要和充分条件的发现）。迄今为止的进展与多维TMP刺激了进步的功能理论，算子理论和真实的代数几何。多维TMP的应用非常丰富，例如，概率论，信号处理和理论物理。考虑到应用，我们已经确定了多维TMP和量子理论中的基础问题之间的重要且尚未探索的联系。特别是，我们断言，在一个给定的有限集上的措施是支持TMP的解决方案的存在性问题是根本的理解是否一组观察是符合量子或经典力学。利用算子理论和真实的代数几何的最新进展，该项目将开发新的工具来解决这些时刻的问题，这将使我们能够创建一个新的，统一的和形式严格的框架，调查经典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.