Quantum Theory for Mathematicians
Quantum Theory for Mathematicians
复制标题
DOI：
10.1007/978-1-4614-7116-5
复制
发表时间：
2013-06
期刊：
影响因子：
--
通讯作者：
B. Hall
中科院分区：
文献类型：
--
作者：
B. Hall
量子物理学的思想在现代数学的许多部分发挥着重要作用。例如，表示论的许多部分都是由量子力学推动的，包括维格纳-麦基诱导表示理论、基里洛夫-科斯坦轨道方法，当然还有量子群。结理论中的琼斯多项式、拓扑中的格罗莫夫-维滕不变量以及代数拓扑中的镜像对称是其他值得注意的例子。 1990年菲尔兹奖授予物理学家埃德·维滕，让人了解量子理论对数学的影响范围。尽管量子力学对数学很重要，但数学家学习这门学科却没有简单的方法。大多数数学家通常不容易理解物理文献中的量子力学书籍。当然，这些书中的数学精确度低于数学家所习惯的水平。此外，有关量子力学的物理书籍假定了数学家通常不具备的经典力学知识。最后，“文化”上存在微妙的差异——术语和符号的差异——这使得阅读物理文献就像为数学家阅读外语一样。很少有书籍试图将量子理论转化为数学家可以理解的术语。本书旨在为之前很少接触过物理学的数学家介绍量子力学。本书的双重目标是（1）用数学家熟悉的语言解释量子力学的物理思想，（2）开发必要的数学工具以严格的方式处理这些思想。我有
Ideas from quantum physics play important roles in many parts of modern mathematics. Many parts of representation theory, for example, are motivated by quantum mechanics, including the Wigner–Mackey theory of induced representations, the Kirillov–Kostant orbit method, and, of course, quantum groups. The Jones polynomial in knot theory, the Gromov–Witten invariants in topology, and mirror symmetry in algebraic topology are other notable examples. The awarding of the 1990 Fields Medal to Ed Witten, a physicist, gives an idea of the scope of the influence of quantum theory in mathematics.Despite the importance of quantum mechanics to mathematics, there is no easy way for mathematicians to learn the subject. Quantum mechanics books in the physics literature are generally not easily understood by most mathematicians. There is, of course, a lower level of mathematical precision in such books than mathematicians are accustomed to. In addition, physics books on quantum mechanics assume knowledge of classical mechanics that mathematicians often do not have. And, finally, there is a subtle difference in “culture”—differences in terminology and notation—that can make reading the physics literature like reading a foreign language for the mathematician. There are few books that attempt to translate quantum theory into terms that mathematicians can understand. This book is intended as an introduction to quantum mechanics for mathematicians with little prior exposure to physics. The twin goals of the book are (1) to explain the physical ideas of quantum mechanics in language mathematicians will be comfortable with, and (2) to develop the necessary mathematical tools to treat those ideas in a rigorous fashion. I have