Topics in Theoretical and Mathematical Physics

几个领域的理论和数学物理解决。虽然技术主要是统计和量子力学，但物理应用范围广泛。离散呼吸的量化是一个优先考虑的问题。几年前，掺杂碱卤化物的发光衰减中出现了一个无法解释的异常现象，现在已经发展成为在经典和量子水平上对非线性相互作用（与孤子有关）的一般研究。PI希望更深入地探索的特定量子问题是这些呼吸者对抗量子衰变的稳定性。这已经通过声子基础上的数值对角化和费曼路径积分的使用进行了研究。对于后者，PI利用了路径积分最基本的技术之一，即消除二次自由度（就像费曼对极化子所做的那样），尽管一旦完成了，需要为这种应用开发特定的方法。在非平衡统计力学的研究中，发现了相变与（随机过程的）相变矩阵特征向量之间关系的新结果。一个特定的几何结构，基于有限数量的特征向量，可以用来计算状态空间中任意点到达任何特定吸引力盆地的概率。它也可以用来可视化亚稳相的结构，特别是当它们彼此具有层次关系时（正如人们认为自旋玻璃所获得的那样）。就目前所知，这个建筑是新的。其他关于非平衡系统的工作，例如，研究储层在开放系统中引起复杂性的方式，也在计划中。在物理学的基础领域，PI将继续探索与时间有关的问题。其中一些与他几年前的“反箭”工作有关，另一些则涉及量子跃迁和量子测量理论的实验测试。PI最近关于相互作用系统中波包传播平衡的结果引发了一个普遍的问题，即该结果背后的冯·诺伊曼熵最大化是否也可以用于建立相对于其他自由度的意想不到的低纠缠水平。除了它的科学内容，这项工作在两个主要方面有广泛的影响：对基础问题的公共教育（源于PI最近的“时间之箭”曝光），在另一方面，克拉克森大学的几名本科生在布拉格PI的合作者的光学晶体实验室做过和将要做的研究，他们有独特的文化体验。

Several areas of theoretical and mathematical physics are addressed. The physical applications are broadly ranging, although the techniques are mainly those of statistical and quantum mechanics. The quantization of discrete breathers is a priority. What began some years ago as an unexplained anomaly in the decay of luminescence in doped alkali halides, has blossomed into a general study of nonlinear interactions (related to solitons) both at the classical and quantum levels. The particular quantum issue that the PI expects to explore in yet greater depth, is the stability of these breathers against quantum decay. This has been studied both through numerical diagonalization in a phonon basis and through the use of the Feynman path integral. For the latter, the PI has exploited one of the most basic techniques of the path integral, the elimination of quadratic degrees of freedom (as Feynman did for the polaron), although once that was done, particular methods needed to be developed for this application. In work on nonequilibrium statistical mechanics new results on the relation between phase transitions and the eigenvectors of the transition matrix (for a stochastic process) have been found. A particular geometric construction, based on a limited number of eigenvectors, can be used to compute the probability that an arbitrary point in the state space reaches any particular basin of attraction. It can also be used to visualize the structure of metastable phases, particularly when they bear a hierarchical relation to one another (as is believed to obtain for spin glasses). As far as is known, this construction is new. Other work on nonequilibrium systems, for example, investigating the ways that reservoirs can induce complexity in open systems, is also planned. In fundamental areas of physics the PI will continue to explore time-related issues. Some are related to his 'opposite arrows' work of a few years ago, others deal with quantum transitions and experimental tests of theories of quantum measurement. A recent result of the PI on the equilibration of wave-packet spread in an interacting system, leads to general questions of whether the von Neumann entropy maximization that lies behind that result can also be invoked to established unanticipated low levels of entanglement with respect to other degrees of freedom. Besides its scientific content, this work has broad impact in two principal ways: public education on foundational issues (emanating from the PI's recent 'arrow of time' exposure), and on another front, unique cultural experiences for the several Clarkson undergraduates who have done, and will do, research at the optical crystals laboratory of the PI's collaborators in Prague.