Topics in Theoretical and Mathematical Physics

理论和数学物理的几个领域得到解决。物理应用是广泛的，虽然技术主要是统计和量子力学。离散呼吸子的量化是一个优先事项。几年前开始的掺杂碱金属卤化物发光衰减中的一种无法解释的异常现象，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期望更深入地探索的特定量子问题是这些呼吸子对量子衰变的稳定性。这已经研究了通过数值对角化在声子的基础上，并通过使用费曼路径积分。对于后者，PI利用了路径积分的最基本技术之一，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，需要为这个应用开发特定的方法。在非平衡统计力学的工作中，人们发现了相变与转移矩阵（随机过程）的特征向量之间关系的新结果。基于有限数量的特征向量，一种特殊的几何构造可以用来计算状态空间中任意点到达任何特定吸引域的概率。它也可以用于可视化亚稳相的结构，特别是当它们彼此具有层次关系时（如认为获得自旋玻璃）。据我们所知，这座建筑是新的。其他工作的非平衡系统，例如，调查的方式，水库可以诱导开放系统的复杂性，也计划。在物理学的基础领域，PI将继续探索与时间相关的问题。有些与他几年前的“相反的箭头”工作有关，其他则涉及量子跃迁和量子测量理论的实验测试。最近的结果PI的波包传播在相互作用系统中的平衡，导致了一般的问题，是否冯诺依曼熵最大化，这背后的结果也可以调用建立未预料到的低水平的纠缠相对于其他自由度。除了它的科学内容，这项工作有两个主要方面的广泛影响：基础问题的公共教育（从PI最近的'时间之箭'曝光），并在另一个方面，独特的文化经验，为几个克拉克森本科生谁已经做了，并将做，在PI的合作者在布拉格的光学晶体实验室的研究。

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.