Quantum Riemannian Geometry
暫譯: 量子黎曼幾何
Beggs, Edwin J., Majid, Shahn
- 出版商: Springer
- 出版日期: 2020-02-01
- 售價: $3,730
- 貴賓價: 9.5 折 $3,544
- 語言: 英文
- 頁數: 809
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 3030302938
- ISBN-13: 9783030302931
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相關分類:
量子 Quantum
海外代購書籍(需單獨結帳)
商品描述
This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points.
Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a bottom up' one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum Levi-Civita' bimodule connection, geometric Laplacians and, in some cases, Dirac operators. The book also covers elements of Connes' approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules.
A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.
商品描述(中文翻譯)
這本書提供了現代微分幾何的一個全面概述，其中坐標不必交換。這需要重新發明微分幾何，僅參考坐標代數，現在可能是非交換的，而不是實際的點。
這樣的理論對於 Hopf 代數或量子群的幾何是必要的，這些提供了關鍵的例子，同時在物理學中用於建模量子重力效應，形成量子時空。這種數學形式主義可以應用於任何代數，並包括圖形幾何和有限群的李理論。甚至 2 x 2 矩陣的代數也顯示出豐富的量子黎曼幾何模態。所採取的方法是「自下而上」，不同層次的幾何依次建立，從微分形式開始，逐步推進到量子 Levi-Civita 雙模連接、幾何拉普拉斯算子，以及在某些情況下的 Dirac 算子。這本書還涵蓋了 Connes 來自循環上同調和光譜三元組的主題元素。其他主題包括各種其他上同調理論、全純結構和非交換 D-模塊。
這本書的一個獨特特點是其建設性的方法以及從大量數學物理文獻中提取的豐富例子，現在建立在堅實的代數基礎上。包括帶有解答的練習題，它可以用作高級課程的教科書，也可以作為研究人員的參考資料。
作者簡介
Edwin J. Beggs studied mathematics at Churchill college Cambridge, moving to St Catherine's college Oxford to study for a DPhil under the supervision of Graeme Segal, finishing in 1988. He became a research assistant working with David Evans on operator algebras (giving a formula for the real rank of matrix valued functions) in Swansea and was appointed to a lectureship there. He has worked with Peter Johnson, finding the inverse scattering method for solitons in affine Toda field theory. He has worked with various coauthors on noncommutative differential geometry, introducing noncommutative sheaf theory, noncommutative complex structures and bar categories as well as working on bimodule connections and quantum Riemannian geometry. He also works on physics and computation in computer science.
Shahn Majid graduated from Cambridge, including Part III of the mathematics tripos, followed by a PhD at Harvard in 1988. After a year in Swansea, he spent ten years in DAMTP in Cambridge before moving to Queen Mary. He was one of the pioneers of the modern theory of Hopf algebras or quantum groups, introducing in his PhD thesis one of the two main classes at the time, the bicrossproduct ones associated to Lie group factorisations. Other results include the earliest models of quantum spacetime with quantum symmetry, the theory of Hopf algebras in braided categories and the dual/centre of a monoidal category. He was one of the coauthors of the theory of quantum principal bundles and introduced a frame bundle approach to quantum Riemannian geometry. In recent years he has been working on the bimodule approach with a view to quantum gravity.
作者簡介(中文翻譯)
埃德溫·J·貝格斯在劍橋的丘吉爾學院學習數學，隨後轉到牛津的聖凱瑟琳學院在格雷姆·西格爾的指導下攻讀DPhil，於1988年完成學位。他成為研究助理，與大衛·埃文斯在斯旺西合作研究算子代數（為矩陣值函數的實秩提供公式），並在那裡獲得講師職位。他與彼得·約翰遜合作，發現了仿射Toda場論中孤子逆散射方法。他與多位合著者合作研究非交換微分幾何，介紹了非交換層理論、非交換複結構和條形類別，並研究了雙模塊連接和量子黎曼幾何。他還在計算機科學中研究物理學和計算。
沙恩·馬吉德畢業於劍橋，包括數學三部曲的第三部分，隨後於1988年在哈佛獲得博士學位。在斯旺西待了一年後，他在劍橋的DAMTP工作了十年，然後轉到女王瑪麗大學。他是現代霍普夫代數或量子群理論的先驅之一，在他的博士論文中介紹了當時的兩個主要類別之一，即與李群分解相關的雙交叉積類別。其他成果包括具有量子對稱的量子時空的最早模型、編織類別中的霍普夫代數理論以及單元類別的對偶/中心。他是量子主束理論的共同作者之一，並引入了量子黎曼幾何的框架束方法。近年來，他一直在研究雙模塊方法，以期望解決量子重力問題。
