Classical Realizability and Quantum Representability: Truncated Moment Problems in Statistical Physics and Quantum Chemistry

复杂的系统，如由分子组成的液体，由原子组成的大分子，由草组成的草坪等，是不可能完全描述的。事实上，这样的描述甚至是不可取的，因为人们会被无法解释的信息所淹没。通常，系统的一些特性，如密度分布、物体间距离的相对频率，是非常重要的。处理这类复杂系统的一个有效方法是集中研究这些特征的性质。在这种方法中，描述特性的方程系统以某种特别的方式导出。那么问题就来了：这些方程的解是否仍然与原来考虑的复杂系统相容？换句话说，复杂系统的状态是否存在，这将导致这些特征？例如，如果有效方程预测粒子密度为负，那么答案将是“否”。对于更复杂的特征或特征的集合，人们不能期望它们之间的关系（通常以不等式的形式）如此明显。可实现性和可表示性问题是为了确定这些条件，并确定哪些假定的特征实际上可以通过底层系统的状态来实现，可实现性和可表示性在不同的领域反复出现，因此它们似乎是一个非常有前途的复杂系统的观点。这也是及时的攻击这些问题，由于最近的兴趣，这些问题在许多不同领域的统计力学，如干扰，随机包装，最佳包装在高维，异质材料，以及在量子化学。由于对这些问题的基本数学结构缺乏了解，这两个问题都可以解释为高维截断矩问题，因此阻碍了进展。即使是二维的情况下，已经知道是非常困难的。理想情况下，人们将得到一种方法，它允许人们从宏观测量导出微观相互作用，人们可以给出一个理论描述的所有不等式的假定相关函数的特征实现的基础上的一般方法来自截断矩问题。不幸的是，这种描述是如此间接，以至于只有少数条件是明确已知的。以显式的方式表示进一步的条件是一个非常困难的问题。除了它的实际重要性，这最后一个问题提供了一个重要的项目之间的联系和领域的纯数学。

Complex systems, like liquids made out of molecules, large molecules made out of atoms, lawn made out of grass, etc. are impossible to describe fully. In fact, such a description it is not even desirable, as one would be overwhelmed by information impossible to interpret. Typically, a few characteristics of the system, like density profiles, relative frequencies of inter-object distances, are of great importance. An effective way of treating such complex systems is to concentrate on the properties of these characteristics. In such an approach a system of equations describing the characteristics is derived in some ad hoc manner. The question is then: are the solutions of these equations still compatible with the originally considered complex system? In other words, do states of the complex system exist, which would give rise to these characteristics? As an example, if the effective equations predicted a negative density of particles, then the answer would be 'no'. For more complicated characteristics or collections of characteristics, one cannot expect the relations between them, which are usually in the form of inequalities, to be so obvious. The realizability and representability problems are to identify these conditions and to determine which putative characteristics can in fact be realized by a state of the underlying system.Realizability and representability arise repeatedly in different areas, thus they seem to be a very promising viewpoint on complex systems. It is also timely to attack these problems, due to a recent interest in these problems as in many different areas of statistical mechanics, like jamming, random packing, optimal packing in high dimensions, and heterogeneous materials, as well as in quantum chemistry. Progress is hindered by a lack of understanding of the underlying mathematical structure of these problems, both of which can be interpreted as high-dimensional truncated moment problems. Even the two dimensional case is already known to be very difficult. Ideally, one would obtain an approach which permits one to derive the microscopic interactions from macroscopic measurements.One can give a theoretical description of all inequalities for putative correlation functions characterizing realizability based on a general approach coming from the theory of truncated moment problems. This description is unfortunately so indirect that only a few conditions are known explicitly. It is a very hard problem to express further conditions in an explicit manner. Beside its practical importance this last question provides an important connection between the project and areas of pure mathematics.