Introduction to M(atrix) theory and noncommutative geometry
Prototype Theory
Rosch------Prototype Category Theory
• 语言学家Labov 和Rosch 先后发表了他们 对于自然范畴的试验研究结果，以此证明 维氏的“家族相似性”原理适用于描述自 然界中的许多范畴，即许多自然范畴都具 有维氏所说的“家族相似性”。Labov 和 Rosch 把具有“家族相似性”的这些自然 范畴称为“原型范畴”（prototype category），即具有原型（prototype, 即范 畴的典型成员）的范畴。从而建立了现代 范畴理论，原型范畴理论。
Definitions of Categorization
F.Ungerer ＆ H.J.Schmid: The mental process of classification. And its products are the cognitive categories. Categorization is
E.g.
• furniture / chair/ table/ lamp/ kitchen chair/living-room chair /kitchen table/dining room table/floor lamp/desk lamp
Category
Definitions
赵艳芳：严格来讲，范畴是事物在认知中的归类。 F.Ungerer＆H.J.Schmid: The product of the mental process of classification. They can be understood as mental concepts
• Definitions of Prototype • Basic contents of Prototype Theory • Factors that influence the Prototype
L16.1德里达——其他文献引述
文献资料库：其他文献引述16.1德里达其他文献引述：1，德里达论延异延异（本文译自雅克·德里达《哲学的边缘》<芝加哥大学出版社，1982年。
>）[法国]雅克·德里达汪民安译我就此要谈谈一个字母。
如果字母表,以及对字母表所作的众多的冒险式的思考是可信的话,我就来谈谈第一个字母。
因此,我要谈谈字母a。
这第一个字母,很明显地有必要悄悄地从各处潜入difference(差异)一词的书写中。
这种潜入要发生在关于文字的书写(Writing on Writing)过程中,在文字内部的书写过程中。
这种文字的不同轨迹就此发现它们自身在某个明确的点上,同一种粗略的拼写错误相交叉、同规范书写并使书写得体的规章法度中的差错相交叉。
人们总是能够,在实际上或判断上抹去或化减差错,(根据每次分析过的情景,尽管这种情景大体一致)发现它是严重的和不得体的,也即是说——根据最诚挚的设想——好笑的,这样,即使人们试图默默地忽略这样的逾规行为,人们对这种逾规行为的兴趣还是可以被事先看作和指定为缄默的反语法、无声的误置的文字耗损的规定。
人们似乎总是认为这点无关紧要。
此时此地,我应当声明,今天的演讲不是对拼写中的缄默差错进行辩护,更不是为此而认错,而是对某种持续加剧的嬉戏进行辩解。
另一方面,如果我谈及了——至少是暗暗地——我以前莽撞地发表的文本,我请求原谅。
这的确是因为——在某种程度上,即使在理论上,归根结底,这是不可能的——我意图在一束(sheaf)中重组不同的方向,根据这些方向,我能够利用我临时性称作词的或概念的:延异(diff·rance),或者更恰当地让它将它本身强加于我身上,尽管如我们所见,延异从字面上而言,既非一个词,也非一个概念。
我强调束这一词有两个原因:一方面,如我可能的那样,我不会全神贯注地文本接文本,语境接语境去描述一个历史,去讲述它的发展步骤,去表明每次施加书写混乱的经济;相反,我将关注这种经济的普遍系统;另一方面,束这个词似乎更适宜于表明这一点:被提议的这种组合有复杂的编织结构和交错结构,这种交错容许不同的线团、不同的意义或力量之线从各个方向爆破,犹如它时时准备着将自己和他者绑扎在一起。
光笔式三坐标测量机的蛙跳算法
( sin!sin cos + cos sin ) + ( - sin!sin s in + cos co s ) + ( - cos!sin ) = 1
2 2 2 2
( - sin !cos cos + sin sin ) + ( sin!co s sin + sin co s ) + ( cos!cos ) = 1
图 2 蛙跳式坐标变换原理图
第 4期
光笔式三坐标测量机的蛙跳算法
21
在式 ( 3 ) 中 Con, Vx , Vy 和 Vz 是未知量, 可以根 据个 Con 各 元 素 的关 系 求 出这 些 量。设 坐 标系 co s!cos Con = sin!sin cos + cos sin - sin!cos co s + sin sin 在式 ( 4 ) 中 , 每一行的三个量的平方和为 1 ,即 ( cos!cos ) + ( - cos!sin ) + sin ! = 1
1 1 1
x y z
2 2 2
x y z
3 3 3
。
y3 和 y
设新坐标系相对于原来坐标系的平移向量为 ( Vx , Vy , Vz )、 旋转矩阵为 Con, 则 x1 y1 z1 x2 y2 z2 x3 y 3 = Con z3 x y
1 1
x2 y2 z2
x y z
3 3 3
Vx + Vy Vz
光笔式三坐标测量机的蛙跳算法
赵呈浩 刘书桂 邢科斌
(天津大学精密测试技术与仪器国家重点实验室 , 天津 300072) 摘 要
在用光笔坐标机进行测量时 , 为了达 到要求 , 经 常要采 用蛙跳法 进行测 量。建立了 相应的 数学模
Introductiontothetheoryofcomputation课程设计
Introduction to the Theory of Computation 课程设计一、课程介绍计算机科学中的基础课程之一是自动机理论，它是算法和编程语言设计的基础。
一种能够处理自动机模型的工具是正则表达式，可以用于文本搜索和验证输入的正确性。
本课程探讨自动机模型和正则表达式的基础，并向同学介绍扩展的自动机模型和编译器的数据结构。
本课程是一门面向计算机科学专业的本科课程，全球的计算机科学相关专业都包括这一课程。
本课程要求学生掌握正则表达式和自动机，以及它们在计算机科学中的应用。
二、主要内容本课程的主要内容包括：1. Automata（自动机）自动机是检测、接受和处理诸如自然语言、计算机程序和网络协议等输入的最常用模型之一。
本课程会介绍有限自动机和正则表达式的概念和应用，以及用于对自然语言进行识别和处理的有限状态机。
2. Regular Expressions（正则表达式）正则表达式是描述字符串模式的计算机语言，其主要用途是在文本编辑器和字符查找功能中进行字符串匹配和替换。
本课程会讨论正则表达式在输入验证机制和编译器设计中的应用。
3. Context-free grammars（上下文无关文法）上下文无关文法是描述计算机语言的形式语言之一，它为编译器和自然语言处理提供了重要的支持。
本课程会重点介绍上下文无关文法的一些优雅且适用的应用，例如利用本文中的语法使解析器更加高效、利用上下文无关文法构建编译器的结果优化策略。
4. Finite-State Transducers（有限状态转换器）有限状态转换器是自动机的扩展模型，主要用于在计算机科学中语言处理领域的翻译、翻译活动和语音识别等方面的开发。
本课程会介绍有限状态转换器的概念和应用。
5. Compiler Data Structures（编译器数据结构）编译器数据结构是与编译过程相关的数据结构，例如符号表、语法分析树和中间代码。
本课程会介绍编译器中常用的特定数据结构，例如词法分析器和解析器。
introduction to management science中文版 -回复
introduction to management science中文版-回复什么是《Introduction to Management Science》（《管理科学导论》）？《Introduction to Management Science》是一本著名的管理科学教材，由美国管理科学教授Frederick S. Hillier和Mark S. Hillier合著。
本书是管理科学领域的经典之作，被广泛应用于大学本科和研究生的管理科学教学中。
它旨在通过数学建模和优化技术的应用，帮助管理者和决策者解决各种管理问题。
本书内容全面而系统，包含了管理科学的基本理论和实践应用。
它介绍了决策分析、线性规划、网络优化、整数规划、非线性规划、决策树等各种优化方法，并提供了大量的实例和案例分析，以帮助读者理解和运用这些方法。
此外，本书还介绍了风险分析、项目管理、供应链管理、市场营销等课题，为读者展示了管理科学在各种实际情景下的应用。
《Introduction to Management Science》的特点在于它将数学建模和优化技术与实际管理问题相结合，突出了实用性。
书中的案例涉及到不同行业和组织，包括制造业、物流公司、零售业、医疗机构等，使读者能够将理论应用于不同领域的管理问题中。
此外，本书注重解决问题的过程，强调决策者在制定和实施方案时的思考和判断能力。
本书不仅适用于管理科学专业的学生，也适合其他管理相关专业的学生，以及对管理科学感兴趣的读者。
它不仅提供了学术理论知识，也给予读者实际运用的指导。
无论是在学术研究中，还是在实际工作中，都可以从中获益匪浅。
在今天的社会中，管理者面临着越来越多的复杂和困难的决策问题。
《Introduction to Management Science》提供了一种科学的方法来解决这些问题，帮助管理者做出可行的和可持续的决策。
通过学习本书，读者可以了解到管理科学的基本原理和方法，提升自己的决策分析能力。
introduction to management science中文版 -回复
introduction to management science中文版-回复Introduction to Management Science（管理科学导论）中文版是一本经典的管理学教材。
本文将为您一步一步回答以该书为主题的文章。
第一步：简介（Introduction）在这一部分，我们将介绍《Introduction to Management Science （管理科学导论）》这本书的基本概况。
介绍书的作者、出版日期、主要内容和目的。
《Introduction to Management Science（管理科学导论）》是由美国管理学家Hillier和Hill撰写的。
该书的第一版于1977年出版，目前已经出版到第13版。
它广泛应用于管理科学和运筹学课程，并被全球许多大学的工商管理专业采用。
该书的主要内容包括线性规划、整数规划、网络分析、对策分析和决策分析等方面的基础方法和技术。
它的目的是引导学生学习如何使用管理科学方法来解决实际问题，开发决策模型和优化解决方案。
第二步：线性规划（Linear Programming）在这一部分，我们将介绍《Introduction to Management Science （管理科学导论）》中关于线性规划的内容。
解释线性规划的概念和原理，并举例说明其在实际中的应用。
线性规划是一种优化方法，用于在各种约束条件下最大化或最小化一个线性目标函数。
它在商业和工业中有广泛的应用，包括生产计划、物流运输问题和资源分配等。
在《Introduction to Management Science（管理科学导论）》中，线性规划被详细讲解。
从问题建模开始，到线性规划模型的构建和解决方法的讲解，学生可以学习如何使用线性规划方法来解决现实世界中的问题。
例如，一个公司在有限的生产能力下，如何最大化利润？通过线性规划，可以将生产过程中的约束条件和目标转化为数学模型，从而找到最优的解决方案。
毕业论文目录英文版
Contents Abstract．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．1 Key words．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．1 Introduction．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．1一、History of evolution of Confucianism’s people-oriented thoughts ．．．．．．．．．2（一）Embryonic stage．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．21、Social background．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．22、Character's view．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3（二）Formation stage．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．31、Social background．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．32、Character's view．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3（三）Development and improving stage．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．41、Social background．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．42、Character's view．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．4（四）Maturity and transformation．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．51、Social background．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．52、Character's view．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．5二、The content and essence of Confucianism’s people-oriented thoughts．．．．．5（一）Main content of Confucianism’s people-oriented thoughts．．．．．．．．．．．．．．51、Supporting and enriching people．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．62、Respecting people．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．63、Educating people．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．74、Trusting people．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．7 （二）Essence of the Confucian ism’s people-oriented thoughts．．．．．．．．．．．．．．．8 1、Respecting the monarch and esteeming the people and emphasis on the idea that people are the foundation of a country．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．8 2、The strategy to Maintain autocratic monarchy ．．．．．．．．．．．．．．．．．．．．．．．．．．83、The moral identity to Strengthening autocratic monarchy ．．．．．．．．．．．．．．．．94、The parochialism a nd bestowal of Confu cianism’s people-oriented thoughts．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．9 三、Influences and enlightenment of Confucianism’s people-oriented thoughts．10（一）P ositive influences and negative influences of Confucianism’s people-oriented thoughts．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．101、Positive influences．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．102、Negative influences ．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．11 （二）The enlightenment of Confucianism’s people-oriented thoughts in modern society．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．13 1、Keeping economic development as the central task and improving people’s living standard．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．13 2、Esteeming people and ensuring that people are the real masters of their own country．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．133、Maintaining the close tie of party and people and solving the problems concerning people’s interest．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．144、Conforming to public opinion, caring for the masses and adhering to people-oriented thought ．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．145、Coordinating the interests of all parties and promoting justice and harmony．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．15 Conclusion．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．15 Bibliography．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．16 Acknowledgements．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．18 Author Introduction．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．18 Statement．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．19。
英文原版量子论科普
英文原版量子论科普If you're interested in reading about quantum theory in English, here are some recommendations:1. "Quantum Mechanics: The Theoretical Minimum" by Leonard Susskind and Art Friedman. This textbook provides a thorough and accessible introduction to quantum mechanics, starting from the basics and building up to more advanced concepts. It's written in a lively and engaging style, making it suitable for self-study or as a classroom textbook.2. "Quantum Physics: A First Encounter" by John Taylor. This book is aimed at undergraduates and covers the key ideas of quantum theory, including wavefunctions, operators, measurement, and entanglement. It provides plenty of examples and exercises to help readers understand and apply the theory.3. "Quantum Computation and Quantum Information" by Michael Nielsen and Isaac Chuang. This textbook provides a comprehensive introduction to the field of quantum information science, covering quantum computing, quantum algorithms, quantum error correction, and quantum cryptography. It's suitable for graduate students and researchers in the field.4. "The Quantum World" by Christopher French and Carlo Michelli. This book provides a broad overview of modern quantum theory and its applications, including quantum computing, quantum cryptography, and quantum metrology. It's written in a clear and accessible style, making it suitable for non-experts who want to understand the basics of quantum theory.Remember that reading about quantum theory can be challenging because it involves concepts that are counterintuitive and打破常识。
范畴论入门教材
范畴论入门教材范畴论（Category Theory）是一门数学分支，研究具有一定结构和关系的数学对象之间的映射关系。
它广泛应用于许多领域，包括计算机科学、物理学和逻辑学等。
以下是一些范畴论入门教材的相关参考内容，帮助读者对范畴论有更深入的了解。
1.《范畴论入门》（Introduction to Category Theory）- Harold Simmons这本书是范畴论入门的经典教材之一。
它从基础概念开始介绍范畴论的基本原理和主要理论，如范畴、箭头、态射和伴随等。
此外，它还涵盖了范畴论在不同领域的应用，如代数学、拓扑学和计算机科学等。
2.《范畴论导论》（An Introduction to Category Theory）- Harold Simmons这本书是范畴论的入门教材之一。
它通过简洁的语言和丰富的例子介绍了范畴论的基本概念和理论。
它还展示了范畴论的应用领域，在解决问题和推理中的重要性。
3.《范畴论：简明导引》（Category Theory: A Gentle Introduction）- Peter Smith这本书是范畴论的入门教材。
它以简明易懂的方式介绍了范畴论的基础概念和基本原理。
此外，它还提供了一些实际的范畴论应用案例，帮助读者更好地理解范畴论的概念和理论。
4.《范畴论导引》（Category Theory in Context）- Emily Riehl这本书以范畴论为中心，通过作者独特的视角介绍了范畴论的基本概念和原理。
它提供了详细的解释和丰富的例子，以帮助读者更好地理解范畴论的核心思想和应用领域。
5.《范畴论简介》（A Course in Category Theory）- Steve Awodey这本书是范畴论的入门教材之一。
它通过详细的讲解和丰富的例子，系统地介绍了范畴论的基本概念和主要理论。
它还包含了一些实际的应用案例，帮助读者将范畴论的概念应用到实际问题中。
APS复习笔记_实验心理学
Trichromatic Theory 三原⾊理论 and Antagonism theory 拮抗理论1.Trichronmatic theory regards that Red, Green and Blue are the three primary colors which can form all colors.Another supplementary theory, named Atagonism theory, suggests that all colors come from three subsystems, andeach system contains two opposing components: red and green, yellow and blue, and white and black, which explainshow color blindness workds. For instance, the classification of color blindness includes red-green color blindness,blue-yellow color blindness and total color blindness.Auditory2.Sound is transmitted by sound waves, the amplitude and frequency of which determine the loudness and pitch ofsound respectively. For instance, how people distinguish music and noise is that music consists of a mixture of soundwaves with a fixed and periodic waveform, while noise has an irregular waveform and period.Perception3.Perception is produced when the stimulus acts directly on the senses, and it is the reflection of the whole of things inpeople's cognitive activities.1) Integrity 整体性 :The tendency to form gestalts (完形的), the whole is greater than the sum of its parts2) Constancy 恒常性: Size constancy, lightness constancy, shape constancy (e.g. mice)3) Comprehension 理解性: The interpretation of ambiguous figures (e.g. young girl vs. old women, frog vs horse)4) Selectivity 选择性: Involved attentionSubjective Contour 主观轮廓In the process of perception, the incomplete graph is regarded as the completed graph, because the past knowledgeand experience provide supplementary information to the current perceptual activityGestalt 格式塔法则4.(1) Law of proximity 接近法则: objects that are close in space and time are easily perceived as a whole(2) Law of similarity 相似法则: objects with similar physical properties (strength, color, size, shape, etc.) are easilyperceived as a whole(3) Law of continuity 连续法则: objects characterized by continuity or common direction of motion are easilyperceived as the same whole(4) Law of closure 封闭法则: people tend to supplement the contour(轮廓) of the defect so that perception becomes acomplete closed figurePerception Processes5.(1) Bottom-up processing: Closely related to empirical facts, It starts with sensory information from the outside world,in other words, it is a process by data driven. (e.g. someone is touching you)(2) Top-down processing: involves in cognitive factors past perceptual experience, knowledge, motivation andcultural background, It begins with the concept storage in memory, so it is a concept processing. (e.g. The skin is softand the hands are small, so you know there might be a child touching you)Three-level memory system6.Ebbinghaus forgetting curve7.⽆标题附件40 KBbasic forms of long-term memory8.Reasoning9.De fference between Deductive and Inductive reasoning:If the premises are correct, the conclusion of a deductive argument is certain;in contrast, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.Heuristic10.Experimental Design11.Reaction Time 反应时11.Reaction time is the most important response variable in psychological research, It has the characteristics ofobjectivity and operation. There are two basic types of reaction time, which are simple reaction time and selectivereaction time. The simple reaction time asks subjects to perform a single response to a single stimulus, whileselective reation time demands that subjects are given more than one different stimulus and responded differently todifferent stimuli, which means subjects are required to identify the stimulus before responding. In Donders's theory ofsubtracting reaction time hypothesized that the reaction time represents the sequential mental activities processing,because different mental processes are continuous, and the sum of the occupied time is the total reaction time, sothe reaction time of complex mental process minus the reaction time of simple mental process can get the reactiontime of mental process that may exist between the two.Possible effects of the psychological experiment12.(1) Expectation Effect (期望效应): When a subject expectsto have a certain kind of outcome they will, without realizingit, they will change their actions and behaviours to actually get this exact outcome.(2) Placebo Effect (安慰剂效应): A placebo is an inert treatment or substance that has no known effects. Researchersmight utilize a placebo control group, which is a group of participants who are exposed to the placebo or fakeindependent variable. To test whether the manipulation of independent variables is effective.Sensory memory Short-term memor y Long-term memory1s~4s 15s~30sFrom different sensory pathways Experienced early stage cognitive processing Transformed from short-term memorythrough rehearsalLarge capacity but short retention time 1.Cannot be retrieved once disturbed 2. 1.Old information is quickly overwritten by new information if not rehearsal Keeping lifetime and unlimited capacity1.The forgetting of long-term memory isregarded as the failure of informationretrieval, rather than the loss ofinformation itself2.Semantic vs. Episodic Prospective vs. Retrospective Declarative vs.ProceduralSemantic: Mem ory of vocabulary, concepts and abstract concepts, "language system" , is necessary for language activitiesProspective:Memories of future events/plans that have not yet occurred Declarative: Memory ofverbally describableinformation andknowledge about theworldEpisodic: memory of specific events, situations Retrospective: Memory of past events/information Procedural:Unspeakable motorskills Deductive Reasoning 演绎推理Inductive Reasoning归纳推理Premise 1: Every teacher teaches his course in the classroom Premise 2: Classrooms are in the school Conclusion: Teachers work in theschoolsChoose concepts that do notbelong to the same category :“Cat ”, “Dog ”, “Bird ”, “Building"Choosing"Building"Deductive is a mental process of drawing deductive inference.1.An inference is deductively valid if its conclusion follows logically from its premises 2.Inductive reasoning is a methodof reasoning in which a body ofobservations is considered toderive a general principle.1.It consists of making broadgeneralizations based onspecific observations.2.Within Subject Design Between Subject Design Mixed DesignAll subjects received all levels of treatment of the independent variables Single independent variable The subjects were treated with only one level of independent variable, and subjects are radomly assigned to different experiment group which accapts only one experimental manipulation Single independent variableThe subjects received differentexperimental design treatmentson different independentvariables, e.g. participant acceptwithin subject treatment on factorA, and accept between subjecttreatment in factor Bmultiple independent variablesReduce the number of subjects recruited-Is an economical way to reduce the cost of subject recruitment 1.Reduce the impact of individual differences 2.It can balance out the random error caused by experimental operation - participant are randomly accept one of the experimental manipulation 1.Do not be affected by differentlevels of the same experimentalconditions2.It can be conducted whenthere are more than twoindependent variables1.acceptable for complexexperimental design2.。
The BRS invariance of noncommutative U(N) Yang-Mills theory at the one-loop level
a r X i v :h e p -t h /0012024v 3 25 J a n 2001FT/UCM–60–2000The BRS invariance of noncommutative U (N )Yang-Mills theoryat the one-loop levelC.P.Mart´ın*andD.S´a nchez-Ruiz †Departamento de F´ısica Te´o rica I,Universidad Complutense,28040Madrid,SpainWe show that U (N )Yang-Mills theory on noncommutative Minkowski space-timecan be renormalized,in a BRS invariant way,at the one-loop level,by multiplicative dimensional renormalization of its coupling constant,its gauge parameter and its fields.It is shown that the Slavnov-Taylor equation,the gauge-fixing equation and the ghost equation hold,up to order ¯h ,for the MS renormalized noncommutative U (N )Yang-Mills theory.We give the value of the pole part of every 1PI diagram which is UV divergent.1.-IntroductionNoncommutative field theories occur both in the ordinary (commutative space-time)field theory setting and in the realm of string theory.The study of the large N limit of ordinary field theories naturally leads to field theories over noncommutative space [1,2].General relativity and Heisenberg’s uncertainty principles give rise,when strong gravita-tional fields are on,to space-times defined by noncommuting operators [3],whereupon it arises the need to define quantum field theories over noncommutative space-times.Su-per Yang-Mills theories on noncommutative tori occur in compactifications of M(atrix)-theory [4],M(atrix)-theory on noncommutative tori being the subject of a good many papers [5].Theories of strings ending on D-branes in the presence of a NS-NS B-field lead to noncommutative space-times;their infinite tension limit being –if unitarity allows it–certain noncommutative field theories [6].It is therefore no wonder that a sizeable amount of work has been put in understanding,either in the continuum [7,8]or on the lattice [9],whether quantum field theories make sense on noncommutative space-times.Applications to collider physics and Cosmology have just begun to come up [10].Quantum field theories on noncommutative space-time present a characteristic con-nection between UV and IR scales:the virtual high-momenta modes contributing to a given Green function yield,when moving around a planar loop,an UV divergence,butgive rise to an IR divergence-even if the classical Lagrangian has only massivefields-as they propagate along a nonplanar loop.This is the UV/IR mixing unveiled in ref.[11], which has been further investigated in refs.[12].The new–as regards to quantumfield theory on commutative space-time–IR divergences that occur in noncommutativefield the-ories makes it impossible[13],beyond a few loops,that these theories be renormalizable ´a la Bologiubov-Parasiuk-Hepp-Zimmerman[14],if supersymmetry is not called in[15]. Besides,they lack a Wilsonian action[11],which puts in jeopardy the implementation in noncommutativefield theories of Wilson’s renormalization group program[16];and,hence, the existence of a continuum limit for these theories.The existence of a continuum limit for noncommutativefield theories has been studied in ref.[17].It is well known[7]that,at the one-loop level,only if a diagram is planar it can be UV divergent and that the momentum structure of this divergence,should it exist,is the product of a polynomial of the appropriate dimension of the external momenta(the UV degree of divergence of the Feynman loop integral)by suitable Moyal phases.Besides,if the noncommutative diagram has an ordinary counterpart(the diagram when space-time is commutative),the polynomial of the external momenta which carries the UV divergence is the same for both diagrams.This might lead us to think that noncommutativefield theories are always one-loop renormalizable,if their ordinary counterpart is;which would in turn render almost trivial the issue of the one-loop renormalizability of noncommutative field theories.One cannot be more mistaken.There are certain∗-deformations ofλφ4that are shown not to be renormalizable at the one-loop level:see ref.[18].It is common lore that U(N)Yang-Mills theories on noncommutative Minowski are one-loop renormalizable.Indeed,if one assumes that gauge,better,BRS invariance is preserved at the one-loop level,it is difficult to think otherwise.However,statements about the BRS invariance of afield theory are rigorous only if they are based either on explicit computations or on the Quantum Action Principle[19]plus BRS cohomology techniques [20].Since we lack a Quantum Action Principle for noncommutativefield theories,we should better carry out explicit computations,lest our statements will be erected on shaky ground.Even if we had a Quantum Action Principle at our disposal,it would always be advisable to check general results by performing explicit computations up to a few loops.In this paper we shall compute the complete UV divergent contribution to the1PI functional of4-dimensional noncommutative U(N)Yang-Millsfield theory for an arbitrary Lorentz gauge-fixing condition.We shall use dimensional regularization to carry out the computations.We shall thus show by explicit computation that this1PI functional is the sum of two integrated polynomials(with respect to the Moyal product)of thefield and its derivatives.Thefirst term is the noncommutative Yang-Mills action.This term is nontrivial in the cohomology of the noncommutative Slavnov-Taylor operator and gives rise to the renormalization of the coupling constant.The second term is exact in the cohomology of the noncommutative Slavnov-Taylor operator and gives rise to the wave function and gauge-fixing parameter renormalizations.This result constitutes a highly nontrivial check of the one-loop BRS invariance of the the theory;the high nontriviality of the check stemming from the fact that the UV divergence of each planar diagram contributing to the4-point function of the gaugefield has a structure which differs very much from that of the4-point tree-level contribution.The BRS invariance of the MS2(minimal subtraction)UV divergent part of the one-loop1PI functional leads,as we shall see,to a renormalized BRS invariant1PI functional up to order¯h.The layout of this paper is as follows.In Section2,we set the notation and display the Feynman rules for noncommutative U(N)Yang-Millsfield theory in an arbitrary Lorentz gauge.In Section3,we give the UV divergent divergent contribution to the one-loop1PI Green functions in the MS scheme of dimensional regularization and,from these data we obtain the MS UV divergent part of1PI functional written in an explicitly BRS invariant form.Section4is devoted to comments and conclusions.In the Appendices we give for the record the UV divergent contribution to each one-loop1PI diagram in the MS scheme.2.-Notation and Feynman rulesThe classical U(N)field theory on noncommutative Minkowski space-time is given by the Yang-Mills functionalS Y M=−1(2π)4d4p2θµνqµpνf(q)g(p).Here f(q)and q(p)are,respectively,the Fourier transforms of f(x)and g(x),the latter being two rapidly decreasing functions at infinity.θµνwill be taken either magnetic or light-like,thus unitarity holds[21].The reader is referred to ref.[22]for introductions to Noncommutative geometry.We shall denote1where S Y M has been given in eq.(1)andS gf=12B+∂µAµ)](x),S ext= d4x Tr Jµ⋆sAµ+H⋆sc (x),whereλis the gauge-fixing parameter.Taking into account thatN2−1a=0(T a)j1i1(T a)j2i2=T Rδj1i2δj2i1,one readily shows that the gaugefield propagator readsd4p p2T R g2δj2i1δj1i2 gµ1µ2+(λ′−1)pµ1pµ2δJµδΓδHδΓδ¯c =0.(3)For the one-loop,Γ1,contribution toΓ[Aµ,B,c,¯c;Jµ,H],the Slavnov-Taylor equation boils down toBΓ1=0,(4) whereB= d4x Tr δS clδAµ+δS clδJµ+δS clδc+δS clδH+Bδµ11j i 1µ22j i2G (0)µ1µ2AAj 1j 2i 1i 2(k )=T R g 2δj 2i 1δj 1i 2(−i )k 2µ11j i µ22i 233j i 3i S µ1µ2µ3AAAj 1j 2j 3i 1i 2i 3(k 1,k 2,k 3=−k 1−k 2)=ig 2T Rδj 4i 1δj 1i 2δj 2i 3δj3i 4e i [ω(k 1,k 2)+ω(k 3,k 4)](2g µ1µ3g µ2µ4−g µ1µ4g µ2µ3−g µ1µ2g µ3µ4)+(1243)+(1324)+(1342)+(1423)+(1432)1j i 12j i 2G (0)c ¯c j 1j 2i 1i 2(k )=T R δj 2i 1δj 1i2i T Re −iω(k 1,k 2)δj 2i 1δj i 2δj 1i −e iω(k 1,k 2)δj i 1δj 1i 2δj2iFigure 1:Feynman rules for noncommutative U (N )Yang-Mills theory:Propagators andvertices with no external fields.(i j k l )denotes a permutation of (1234).To regularize the Feynman integrals of our theory will shall use dimensional regu-larization.To define θµνin dimensional regularization we shall follow the philosophy in ref.[24]and say that θµνis an algebraic object which satisfies the following equationsθµν=−θνµ,ˆηµρθρν=0,p µθµρηρσθσνp ν≥0,∀p µ.Here ηρσand ˆηµρare,respectively,the “D-dimensional”and “D-4-dimensional”metrics asdefined in ref.[24].It is not difficult to convince oneself that,with the previous definition of θµν,the one-loop Feynman integrals do have a mathematically well-defined expressions and that the techniques used in ref.[24]to prove the Quantum Action Principle for dimen-sionally regularized ordinary field theories can also be employed here to conclude that at5µ22j i 21j i 11S (0)c ;sA j 1j 2i 1;i 2µ2(k 1)=i k 1µ2δj 2i 1δj1i 23µ3jµ22j i 21j i 1S (0)cA ;sA j 1j 2i 1i 2µ2;j3i 3µ3(k 1,k 2,k 3)=−iδj 2i 1δj 3i 2δj 1i 3e −iω(k 1,k 2)−δj 3i 1δj 1i 2δj2i 3e iω(k 1,k 2)g µ2µ31j i12j i 2j 3S (0)cc ;sc j 1j 2j3i 1i 2;i 3(k 1,k 2,k 3)=iδj 2i 1δj 3i 2δj 1i 3e −iω(k 1,k 2)−δj 3i 1δj 1i 2δj2i 3e iω(k 1,k 2)Figure 2:Feynman rules for noncommutative U (N )Yang-Mills theory:Vertices withinsertions of BRS variations.the one-loop level our dimensionally regularized noncommutative U (N )is BRS invariant.The so regularized Feynman diagrams are meromorphic functions of D ,with simple poles at D =4,if they are planar,and no poles,if they are nonplanar and P µθµρηρσθσνP ν>0for any linear combination,P ,of the external momenta.3.-The MS UV divergent part of the 1PI functionalThe fact that at the one-loop level only planar diagrams can be UV divergent and the fact that planar diagrams have the same UV degree as their ordinary field theory counterparts readily leads to the conclusion that the one-loop UV divergent 1PI functions are following:ΓAA ,ΓAAA ,ΓAAAA ,Γ¯c c ,Γ¯c Ac ,ΓJc ,ΓJAc and ΓHcc ,with obvious notation.We have computed the one-loop UV divergent contribution to all the divergent 1PI Green functions.Taking into account the results presented in the Appendices,one obtains the following values for these UV divergent part in the MS scheme:Γ(AA ),(pole )µ1µ2j 1j 2i 1i 2(p )=13−(λ′−1)N δj 2i 1δj 1i 2 p 2g µ1µ2−p µ1p µ2Γ(AAA ),(pole )µ1µ2µ3j 1j 2j 3i 1i 2i 3(p 1,p 2,p 3)=13−3Γ(AAAA ),(pole )µ1µ2µ3µ4j 1j 2j 3j 4i 1i 2i 3i 4(p 1,p 2,p 3,p 4)=13+2(λ′−1)N δj 4i 1δj 1i 2δj 2i 3δj 3i 4e i [ω(p 1,p 2)+ω(p 3,p 4)](2g µ1µ3g µ2µ4−g µ1µ4g µ2µ3−g µ1µ2g µ3µ4)+(1243)+(1324)+(1342)+(1423)+(1432)Γ(c ¯c ),(pole )j 2j 1i 2i 1(p 1)=− 12N δj 2i 1δj 1i 2p 21Γ(c ¯c A ),(pole )j 2j 1i 2i 1j 3i 3µ(p 2,p 1,p 3)=1(4π)2ε g 2T R 1−λ′−1(4π)2εg 2T R1+(λ′−1)Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−eiω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3 g µ2µ3Γ(ccH ),(pole )j 1j 2j 3i 1i 2i 3(p 1,p 2,p 3)=i1δΦi 11δΦi 22...δΦi n n,Γ(φ1φ2...φnK φ)i 1i 2...i ni =δsφi ·Γso that−2Z(1)g+2Z(1)A=13−1(4π)2ε2N 24(λ′−1) g2T R,−2Z(1)g+4Z(1)A=13−(λ′−1) g2T R.Hence,Z g=1−16g2T R,Z A=1−14(λ′−1)]g2T R.(7)Analogously,Z¯c Z c=1+12g2T R N,Z¯c Z A Z c=1−1(4π)2ελ′g2T R N,Z B=Z−1A,Zλ=Z2A and Z J=Z¯c.(8)That the Z s in eqs.(7)and(8)render UVfinite the one-loop1PI functional is a consequence of BRS invariance.Indeed,in view of eq.(6),it is not difficult to show that the singular contribution,Γ(pole),to the dimensionally regularized1PI functional can be cast into the formΓ(pole)=a(4π)2ε22(4π)2ε3+λ′(4π)2ελ′N T R g2,and B D is the linearized Slavnov-Taylor operator acting upon the space of formal∗-polynomials constructed with“D-dimensional”monomials of thefields and their deriva-tives.B D is defined as followsB D= d D x Tr δS clδAµ+δS clδJµ+δS clδc+δS clδH+Bδthe sum of two terms:thefirst term,the Yang-Mills term,is B D-closed and the second term is B D-exact(recall that B2D=0).The analogy with ordinary SU(N)Yang-Mills theory is apparent.And,indeed,as as in ordinary four-dimensional Yang-Mills theory we haveZ g=1−a∂µ+β∂∂λ− φγφ d4xφ(x)δdµ=−13N T R g4.9The other renormalization group coefficients read at the one-loop level:γA=+12 N T R g2,γc=+1δB=0,δΓ(pole)δJµ=0.Hence,up to order¯h,the MS renormalized1PI functional does satisfy both the gauge-fixing equation and the ghost equation:T R δΓMSrenδ¯c+∂µδΓMSrenThat the one-loop4-point function of the gaugefield has a MS UV divergent part which does not spoil the Slavnov-Taylor equation is,though,highly nontrivial.It demands that very delicate cancellations occur upon adding all the UV divergent4-point diagrams:unlike 2-and3-point diagrams,the sum of4-point diagrams of the same type is not proportional to tree-level4-point gauge vertex.That these delicate cancellations do happen is,of course, a consequence of the fact that BRS invariance holds indeed.In this regard,we invite the reader to go to Appendix C and eq.(6)and get acquainted with the momentum and colour structure of the UV divergent contributions reported there.Note that every4-point UV divergent contribution has an overall factor equal to N,for they come only from planar diagrams.As regards the actual value of the beta function,the anomalous dimensions of thefields and gauge-fixing parameter,we have found that they are those of SU(N)’s.This result is, of course,almost trivial[7,11],once it is shown that BRS holds.Indeed,all1PI planar diagrams but the4-point diagrams can be grouped in classes of planar diagrams of the same type,the sum of the diagrams in each class being proportional to the corresponding tree-level contributions.However,taking into account what it is at stake,computations which are both explicit and thorough are much welcomed.Finally,the computations presented in this paper werefinished more than a year ago. In the meantime two papers which overlap with it have appeared[25].Ourfindings are in agreement with theirs,but ours are more general.Appendix A.Gaugefield2-point functionThe diagrams which are UV divergent in dimensional regularization are the planar diagrams in Figure3.Note that the planar tadpole diagram is not singular at D=4in dimensional regularization.µ11ji1µ22ji2µ11ji1µ22ji2(i)(ii)Figure3:1PI UV divergent2-point Feynman diagrams for the gaugefield. The minimal UV divergent part of each diagram reads (i)=2i 112−(λ′−1)6−(λ′−1)(4π)2ε Nδj2i1δj1i216pµ1pµ2 .Appendix B.Gauge field 3-point functionThe UV divergent part of the 3-point function of the gauge field is obtained from the planar diagrams in Figure 4.µ11j µ22i 23j i 333p 2µ2i 2p 33µ3j i 3µ11j p 2µ2i 2p 33µ3j i 3µ11j (i )123(ii )123(iii )123Figure 4:1PI UV divergent 3-point Feynman diagrams for the gauge field.The UV divergent part of these diagrams read (i )123=i12+3(λ′−1)(4π)2ε134Ne −iω(p 1,p 2)δj 2i1δj 3i2δj 1i 3−e iω(p 1,p 2)δj 3i1δj 1i2δj 2i 3×(p 2−p 1)µ3g µ1µ2−(p 1+2p 2)µ1g µ2µ3+(p 2+2p 1)µ2g µ1µ3 ,(iii )123=i 112Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−e iω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3 ×(−p 1−2p 2)µ3g µ1µ2+(2p 1+p 2)µ1g µ2µ3+(p 2−p 1)µ2g µ1µ3 .Summing over permutations one obtains (I )=(i )123+(i )231+(i )312=i12+3(λ′−1)(4π)2ε134Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−e iω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3×(p 2−p 1)µ3g µ1µ2−(p 1+2p 2)µ1g µ2µ3+(p 2+2p 1)µ2g µ1µ3 ,(III )=(iii )123+(iii )213=i 112Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−eiω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3 × (p 1−p 2)µ3g µ1µ2+(p 1+2p 2)µ1g µ2µ3+(−2p 1−p 2)µ2g µ1µ3 .The sum of (I ),(II )and (III )yields the UV divergent contribution to the 1PI 3-point function of the gauge field.Note that (I),(II)and (III)are proportional to the 3-point vertex of the gauge field.Appendix C.Gauge field 4-point functionThe UV divergent contribution to the 4-point 1PI function of the gauge field is com-puted by summing over the UV divergent parts of the diagrams,and the appropriate permutations of these diagrams,in Figure 5.µ11j 444j i 4µ22i 23j i 3331p µ11j p 444j i 4p 2µ22i 2p 33µ3j i 31p µ11jp 444j i 4p 2µ2i 2p 33µ3j i 31p µ11jp 444j i 4p µ2i 2p 33µ3j i 3(i )1234(ii )1234(iii )1234(iv )1234Figure 5:1PI UV divergent 4-point Feynman diagrams for the gauge field.The UV divergent part of the diagrams in Figure 5read (i )1234=i124g µ1µ4g µ2µ3+5+5(λ′−1)24g µ1µ3g µ2µ4+−4−2(λ′−1)+(λ′−1)224 gµ1µ4gµ2µ3+ −4−2(λ′−1)+(λ′−1)22+(λ′−1)2(4π)2ε N×e i[ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj3i1δj1i2δj4i3δj2i4+e i[−ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj2i1δj4i2δj1i3δj3i4 2+17(λ′−1)24 gµ1µ4gµ2µ3+−58−(λ′−1)24−13(λ′−1)24 gµ1µ2gµ3µ4+e i[−ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj2i1δj3i2δj4i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj4i1δj1i2δj2i3δj3i4 −58−(λ′−1)28−(λ′−1)24−13(λ′−1)24 gµ1µ2gµ3µ4,(iii)1234=i 112+5(λ′−1)12 gµ1µ4gµ2µ3+172+(λ′−1)2472+7(λ′−1)224 1(4π)2ε N×e i[−ω(p1,p2)−ω(p1,p3)+ω(p2,p3)]δj3i1δj4i2δj2i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)−ω(p2,p3)]δj4i1δj3i2δj1i3δj2i4 7+7(λ′−1)12 gµ1µ4gµ2µ3+7+7(λ′−1)12 gµ1µ3gµ2µ4+−8−4(λ′−1)+(λ′−1)22+7(λ′−1)212 gµ1µ3gµ2µ4+7+7(λ′−1)12 gµ1µ2gµ3µ4 +e i[ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj3i1δj1i2δj4i3δj2i4+e i[−ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj2i1δj4i2δj1i3δj3i4 −8−4(λ′−1)+(λ′−1)22+7(λ′−1)22+7(λ′−1)2(4π)2ε N×e i[ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj3i1δj1i2δj4i3δj2i4+e i[−ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj2i1δj4i2δj1i3δj3i48+17(λ′−1)6 gµ1µ4gµ2µ3+−236 gµ1µ3gµ2µ4+−236 gµ1µ2gµ3µ4 +e i[−ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj2i1δj3i2δj4i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj4i1δj1i2δj2i3δj3i4−236 gµ1µ4gµ2µ3+8+17(λ′−1)6 gµ1µ3gµ2µ4+−236 gµ1µ2gµ3µ4 +e i[−ω(p1,p2)−ω(p1,p3)+ω(p2,p3)]δj3i1δj4i2δj2i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)−ω(p2,p3)]δj4i1δj3i2δj1i3δj2i4−236 gµ1µ4gµ2µ3+−236 gµ1µ3gµ2µ4+8+17(λ′−1)6 gµ1µ2gµ3µ4 ,(III)=(iii)1234+(iii)1324+(iii)1243==i 112+5(λ′−1)12 gµ1µ4gµ2µ3+172+(λ′−1)212+5(λ′−1)12 gµ1µ2gµ3µ4+e i[−ω(p1,p2)−ω(p1,p3)+ω(p2,p3)]δj3i1δj4i2δj2i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)−ω(p2,p3)]δj4i1δj3i2δj1i3δj2i4472+7(λ′−1)212+5(λ′−1)12 gµ1µ3gµ2µ4+172+(λ′−1)212−(λ′−1)12 gµ1µ4gµ2µ3+472+7(λ′−1)212+5(λ′−1)12 gµ1µ2gµ3µ4,(IV)=(iv)1234+(iv)1324+(iv)1243+(iv)1432+(iv)1342+(iv)1423==−i(4π)2ε N×e i[−ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj2i1δj3i2δj4i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj4i1δj1i2δj2i3δj3i4+e i[−ω(p1,p2)−ω(p1,p3)+ω(p2,p3)]δj3i1δj4i2δj2i3δj1i4+e i[ω(p1,p2)+ω(p1,p3)−ω(p2,p3)]δj4i1δj3i2δj1i3δj2i4+e i[ω(p1,p2)−ω(p1,p3)−ω(p2,p3)]δj3i1δj1i2δj4i3δj2i4+e i[−ω(p1,p2)+ω(p1,p3)+ω(p2,p3)]δj2i1δj4i2δj1i3δj3i4(gµ1µ4gµ2µ3+gµ1µ3gµ2µ4+gµ1µ2gµ3µ4).The sum(I)+(II)+(III)+(IV)is the UV divergent part of the1PI4-point gauge function in the MS scheme.Note that (I),(II),(III)and(IV)are not proportional to the tree-level4-point vertex.1j i 1122j i 21j i 1122j i 23µj i 331j i 1122j i 23µj i 33(i )(ii )(iii )Figure 6:1PI UV divergent Feynman digrams for the ghost functions.Appendix D.Ghost functionsThe diagrams needed to compute the ghost self-energy and the ghost-boson vertex function are shown in Figure 6.From them,we have obtained the following results(i )=−i12N δj 2i1δj 1i 2p 21,(ii )=3i(4π)2εg21+(λ′−1)Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−e iω(p 1,p 2),δj 3i 1δj 1i 2δj 2i 3 p 1µ3,(10)(iii )=i (4π)2εg 2 1+(λ′−1)Ne −iω(p 1,p 2)δj 2i 1δj 3i 2δj 1i 3−eiω(p 1,p 2)δj 3i 1δj 1i 2δj 2i 3 p 1µ3.Appendix E.Functions with external fieldsThe diagrams which contribute at the one-loop level are shown in Figure 7.The results are:(i )=−i12N δj 2i 1δj 1i 2p 1µ2,(ii 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introduction to psychology书
introduction to psychology书
《心理学导论：思想与行为的认识之路》（第11版）是一本非常经典的心理学入门书，其特色是以读者为本，利于学习。
作者都是教学经验丰富的心理学家，他们在书中对有关心理学的基本问题做了详尽的解答，系统地讲述了心理学各分支领域的基本知识和运态。
在撰写方式上，作者将一门严肃的学科赋予生动活泼的表达，同时指引人们自觉运用心理学原理解决实际问题，培养良好的心理素质。
读者可以将本书中抽象的文字变成鲜活的体验，从而在愉悦的精神享受中学习心理学知识。
通过阅读这本心理学导论，读者会真切地感受到心理学和日常生活息息相关，所学知识终身受用，从而引导自己生活得更加幸福和美好。
这部心理学导论几乎每三四年修订一次，以确保内容的时效性和准确性。
耶鲁大学心理学导论中英文字幕16
在以后的两节课里我们将讨论有关社会心理学的内容This is going to begin a two-lecture sequence on social psychology on有关我们如何看待自己如何看待看待他人how we think about ourselves, how we think about other people,如何看待其他群体的成员how we think about other groups of people.我们已经讨论过很多人类心理能力We've talked a lot about the capacities其中一些能力包括of the human mind and some of these capacities适应和处理物质世界的involve adapting and dealing with the material world.因此我们必须选择食物必须在世界中漫游So, we have to choose foods, we have to navigate around the world,必须识别物体we have to recognize objects,必须能够理解自然界的相互作用we have to be able to understand physical interactions.但可能我们进化成的心理But probably the most interesting aspect of最有趣的方面是我们能够理解他人与他人打交道our evolved minds is our capacity to understand and deal with other people.我们对他人如何工作非常感兴趣We are intensely interested in how other people work.二零零五年有个大新闻The story that was a dominant news story in 2005 was this.你们中的一些人那些看不到屏幕的人And some of you--this--for those of you who aren't seeing the screen,这是有关詹妮弗·安妮丝顿和布拉德·皮特的离婚案is the separation of Jennifer Aniston and Brad Pitt.我还记得我第一次听说这事是在什么地方I remember where I was when I first heard about this.这非常有趣就是记得作为心理学家And it's an interesting sight. Just remember--stepping back. As psychologists我们必须对肯定的事情提出疑问我们要质疑所谓的常识we have to question the natural. We have to take things并研究它们刚刚发生的事情是that are commonsense and explore them. And one thing which just happens is,我们为这些事情所着迷we're fascinated by this stuff.我们对名人的生活感兴趣We're fascinated by the lives of celebrities.我们对他人的社会生活非常感兴趣We're fascinated by the social lives of other people.为什么会这样呢这是个有趣的问题And it's an interesting question to ask why.这是我将在之后两节课中And this is one of the questions会讲到的问题之一which I'm going to deal with in the next couple of lectures但在我开始讲社会心理学理论之前but before I get to the theory of social psychology我想先讨论一项个体差异I want to talk about an individual difference.几周前我们用了一节课的时间So, we devoted a lecture early on--of a couple of weeks ago,讲述人与人之间智力和人格上的个体差异to individual differences across people in intelligence and personality.我想讲有关人在社会属性方面的差异I want to talk a little bit about an individual difference in our social natures然后给大家做个测验and then I want people to do a test that will explore看看你们处于什么位置where you stand on a continuum.你面前放着的那张纸就是测验That test is the piece of paper you have in front of you.没有拿到的同学请举手Anybody who doesn't have it please raise your hand助教会发给你and one of the teaching fellows will bring it to you.不要着急你现在还不知道怎么做You don't know what to do yet with it so don't worry.这个测试是马尔科姆·格拉德威尔The test was developed actually by Malcolm Gladwell他是一位科普作家在他著作《引爆流行》中开发的who is a science writer-- in his wonderful book The Tipping Point.格拉德威尔在测验介绍中And as he introduces the test,他详述了斯坦利·米尔格拉姆的另一个实验Gladwell recounts another experiment done by Stanley Milgram,当然米尔格拉姆因服从实验闻名但他也做了很多有趣的事of course famous for his obedience work but he did a lot of interesting things.在一项经典实验中他将包裹给从内布拉斯加州的奥马哈人中And one classic study he did was he gave a package to 160 people randomly chosen随机挑选的一百六十人然后让这些人无论如何将包裹送给in Omaha, Nebraska and he asked these people to get the package somehow这在很多年前还没有因特网and this was many years ago before the internet,没有电邮要将包裹送到一个在波士顿工作before e-mail?To get the package to a stockbroker who worked in Boston却住在马萨诸塞州雪伦市的股票经纪人but lived in Sharon, Massachusetts.他发现大多数人竟然做到了What he found was that most people were able to do it.当然没有人认识这个人Nobody, of course, knew this man但他们认识那些可能认识这个人的人but they knew people who might know people who would know this man.所以大多数人成功了大多数人都能将包裹送至这个人So, most people succeeded. Most people were able to get the packages to this man这就是六度分隔理论and it took at maximum six degrees of separation,著名的短语的出处which is where the famous phrase comes about我们与他人的间隔只有六个人而已that we're all separated from another person by six degrees of separation.这不是普遍正确的This is not true in general.只在美国内做过实验This was a very--a single experiment done within the United States,但这个想法非常吸引人but the idea is appealing,人们是通过一系列人脉与其他人联系的that people are connected to one another via chains of people.米尔格拉姆发现特别有趣的是But what Milgram found that was particularly interesting was半数的被试仅通过两个人就将包裹送到了that in about half of the cases these packages went through two people.也就是说如果标出人与人间的关系That is, if you plot the relationships between people--我们以这个房间里的人为例We can take each person in this room,将彼此认识的人连线find everybody you know and who knows you and draw a line,如果我们真这么做的话我们看到的不会是一张网but if we were to do this you wouldn't find an even mesh of wires.相反你会发现有些人会成为节点Rather, you'd find that some people are clusters.这些人就是格拉德威尔所谓的联系员这有点像空中交通Some people are what Gladwell calls "Connectors." it's like air traffic.以前的空中交通是与各个目的地的连线Air traffic used to be everything flew to places local但现在更像一个枢纽系统例如芝加哥的奥黑尔或纽瓦克to it but now there's a system of hubs, chicago O'Hare for instance or Newark都是许多飞机经过的地方而有些人就是枢纽where planes fly through. Some people are hubs. 这些人就是那种认识很多人的人Some people are the sort of people who know a lot of people. 这房间里的一些人可能是枢纽这不难发现Some people in this room might be hubs, and it is not impossible to find out.这张纸上有二百五十个从曼哈顿电话簿上The piece of paper you have here is 250 names chosen randomly随机抽取的名字from a Manhattan phone book.其中包含了不同种族They capture a range of ethnicities,世界不同地区不同原国籍的人different parts of the world, different national origins.下面就是我想让你们做的我给大家五分钟时间Here's what I'd like you to do. And I'll give about five minutes for this.浏览这些名字并圈出那些你认识的人的名字Go through these names and circle how many people you know.圈名字的规则是Now, the rules of this are,圈出那些你们彼此认识的人的名字to know somebody you have to--they have to know you back.如果是个名人这里一个名字是约翰逊So, if it's a celebrity--Well, here--one of the names here is Johnson.我听说过埃尔文·约翰逊但埃尔文·约翰逊从没听说过我Now, I've heard of Magic Johnson but Magic Johnson has never heard of me,所以不能圈这个名字另外我们系主任是玛西亚·约翰逊so I cannot circle it.On the other hand, our department chair is Marcia Johnson.她认识我所以我就可以圈她开始浏览并圈出你认识的人She has heard of me, so I could circle it. Go through and circle it.圈出所有你认识同时也认识你的人Circle all the people you know who know you.那些就是和你有联系的人Those are the people you're connected to.如果你认识的人里有重名的圈两次If you know more than one person with the same last name, circle it twice.如果你没有拿到这张纸而你又想参与进来If you don't have this piece of paper and you want to participate,请举手助教会发给你please raise your hand and one of the teaching fellows will bring it to you. 在大家做这个的时候我想多说两句I'm going to talk a little bit more about this while people go through this.人际关系的问题很多情况下都很有趣The issue of connections between people is intellectually interesting还可以让我们对人们如何互动for many reasons and might allow us to develop some generalizations有一个归纳当然六度空间理论about how people interact. The game of Six Degrees of Separation has, of course,也成了演员凯文·贝肯根的著名电影轶事turned into a famous movie trivia thing revolving around the actor Kevin Bacon,我想之所以选他是因为他名字压韵吧I think chosen just because it rhymes with "Separation." 凯文·贝肯的六度空间游戏是这么玩的And the game of "Six Degrees of Kevin Bacon" is played by任何一名演员计算接近凯文·贝肯taking any actor and computing how many steps要经过多少步it would take to get to Kevin Bacon.有些计算机科学家已经将这个开发出来了And some computer scientists have developed this. 他们检查国际电影数据库中They've gone through each of the quarter million actors二十五万名男女演员and actresses on the international movie database然后计算出他们的贝肯数贝肯数就是指他们联系到贝肯and computed their "Bacon number." and the Bacon number is the number of steps所需要的步骤数it takes for them to get to Kevin Bacon.比如爱德华·阿斯纳在电影《JFK》中与凯文·贝肯合作So for instance, Ed Asner was in the movie Change of--; "JFK" With Kevin Bacon.那么爱德华·阿斯纳的贝肯数为一So, Ed Asner has a Bacon number of one.埃尔维斯·普雷斯利在《修女变身》中与爱德华·阿斯纳合作Elvis Presley was in the movie "Change of Habit" with Ed Asner而这是他与凯文·贝肯最紧密的联系了and that's his closest connection to Kevin Bacon.所以埃尔维斯·普里斯利的贝肯数为二So, Elvis Presley has a Bacon number of two.结果发现如果你观察二点五抱歉It turns out that if you look at the 2.5--sorry,电影数据库中的二十五万人并计算他们的贝肯数the quarter million people on the movie database and compute their Bacon number,平均贝肯数为二点八这就是联系到凯文·贝肯的the average Bacon number is 2.8. That's how many steps your average person所需平均人数is away from Kevin Bacon.然后你可以在演员中计算出联系最多人的一个You could then, for any actor or actress, compute the most connected one.联系最多的那位演员So, the most connected one would be the one for将是这二十五万人中平均来说最多人联系的whom the quarter million are, on average, the most connected to.联系最多演员的答案相当出人意料And the answer of the most connected actor or actress is reasonably surprising.大家想猜猜吗Does anybody want to guess?我先给大家一个错误答案I'll start you off with the wrong answer and this,顺便说一句可以在该网站上找到答案不是约翰·韦恩by the way, can be found on this web site. It's not John Wayne.约翰·韦恩确实拍了很多电影一百八十部电影John Wayne has been in many movies, 180 movies,在六十多年里但他不是最多联络人in fact, over sixty years, but he isn't well connected at all because mostly他几乎只出现在西部电影中所以我们不停地看到同一些人he was in westerns so we saw the same people over and over again.梅丽尔·斯特里普也不是Meryl Streep also isn't it因为梅丽尔·斯特里普很不幸只拍好电影所以because Meryl Streep has the misfortune of playing only in good movies. So,她与像亚当·桑德勒和约翰·克劳德·范·戴姆这些演员没联系she has no connection with people like Adam Sandler and John Claude Van Damme.猜一下有其他猜测吗克里斯托弗·沃肯有点靠谱了Guess. Any guesses? Christopher Walken is a good one.我们来查查看我只认识少数几个人We could look it up. I only know a few names here.克里斯托弗·沃肯不是最多的尼古拉斯·凯奇是个有趣例子Christopher Walken is not a finalist. Nicolas Cage is an interesting case.尼古拉斯·凯奇拍过好电影吗Has Nicolas Cage been in good movies?我不是想我这是自找麻烦了I don't want to get-- i'm going to get more controversial than I want to.一个类似临时演员的人他最多算个二流演员请再说一遍 A guy who is one step above an extra.He's like a B-list actor at best.Pardon me?最佳联络人The most connected guy?最佳联络人我想说你是对的是罗德·斯泰格尔The most connected guy, and I think this shows that you're right,is Rod Steiger.他是演艺史上的最佳联络人He's the most connected actor in the history of acting不是因为他比其他人参演了更多电影because it isn't that he's been in more movies than everybody else.迈克尔·凯恩可能是世界上出演电影最多的人Michael Caine has probably been in the most movies of any person on earth,但罗德出演了各种类型的电影他出演了《码头风云》but he's been in all sorts of movies. He was in "On the Waterfront",《炎热的夏夜》以及像《无妄之灾》这类烂片"In the Heat of the Night," and really bad movies like "Carpool".他参演戏剧和犯罪连续剧惊险片西部片恐怖电影He's been in dramas and crime serials, thrillers, westerns, horror movies,科幻片和音乐剧science fiction, musicals.有些人就像罗德·斯泰格尔有些人在他们日常生活里Now, some people are like Rod Steiger. So, some people in their day-to-day lives与人有很多互动我想从互动中可以了解的是have many interactions and I think one of the things we know from interacting我们能从人群中识别出这些人with people is we can distinguish them from other people.有多少人已经做完了How many people have finished their things right now?好我认识学院里的一个人Okay. I know one person in the department是我知道的认识最多人的人who is one of the most connected people I know on earth.如果我想如果我真的想联系拉扎斯菲尔德我会去找他说If I wanted--If I really had to talk to Rumsfeld, i'd go to this person and say,你能帮我联系上菲尔德吗如果我想整某个人"Can you get me in touch with Rumsfeld?" if I wanted to get somebody whacked,我会去问他I'd ask this guy.在学院里我还认识其他的人据我所知Then I know someone else in the department and, as best I know,我是她唯一认识的人I'm the only person she knows.有多少人的得分低于十分So, how many people scores below ten on this?有多少人得分在十分到二十分之间How many between ten and twenty?二十分到三十分之间呢Between twenty and thirty?三十分到四十分呢Thirty and forty?四十分到五十分之间呢Between forty and fifty?五十分到六十分呢Fifty and sixty?有多少人得分超过六十分How many people scored above sixty?有人得分超过六十分吗Anybody above sixty?格拉德维尔在很多地方做过这个实验Gladwell has done this in a lot of places.在大学生群体里平均分是二十一分The average is twenty-one among a college crowd.有些人得分超过一百Some people score as high as over 100.年龄越大得分越高可能很明显The older you are, the more-- the higher you tend to score, maybe obviously,不是在这个国家呆的时间越长得分越高not--the longer you've been in the country the higher you tend to score.记者的分数理所当然会高些学者的分数不那么高Journalists tend to score reasonably high, academics not so high,格拉德维尔指出有些人就是有天赋and--but what Gladwell points out is some people have the gift.有些人比其他人更有社交天赋Some people are more social than others并与很多人以各种有趣的形式保持联系and this connects in all sorts of interesting ways.人脉问题涉及社会因素The issue of connection has social factors这是社会学家为来耶鲁念书给出的一个很好理由and it's one answer that sociologists give for why it's good to go to Yale.一个答案是因为这里有丰富的学术资源So, one answer is, well, because of the great intellectual benefits.先不说这个很讽刺的Put that aside. Let's be more cynical here.另一个答案是你能交到很牛的朋友Another answer is that you develop powerful friends.这很接近了但社会学家给出的答案And that's closer, but the interesting answer sociologists不是你交到多少厉害的朋友come to is it's not so much you develop powerful friends;而是你认识了多少很牛的人在耶鲁你认识了很多人rather,you develop powerful acquaintances. Through Yale you know a lot of people他们不一定是好朋友但他们是相识的人and they don't have to be close friends but they are acquaintances.社会学家指出你生活中的很多方面And sociologists point out that for a lot of aspects of your life,例如找工作熟人很重要人脉很重要like getting a job, acquaintances matter, connections matter, 你通过进入像耶鲁这样的地方and the connections you establish by going to而建立起来的人脉将为你未来做很好的铺垫a place like Yale hold you in good stead for the rest of your life,除了学术资源这个地方能给你带来above and beyond any intellectual qualities that this place may offer.这就是我在下节课以及以后的两节半课所要说的内容Here's what we're going to do for thenext lecture and a half,two lectures.我们首先讨论自我We're first going to talk about the self.然后我们讨论自我和他人Then we're going to talk about the self and other;基本上我们对自己的看法basically, differences between how we think of ourselves和我们对他人的看法是存在差异的我们将专门讨论and how we think about other people. Then we're going to talk exclusively about我们如何看待他人然后讨论how we think about other people and then we'll talk about我们如何看待像哈佛学生或同性恋或黑人这样的群体how we think about groups like Harvard students or gay people or black people.我将从我最喜欢的研究结果开始这是有关自我的内容I'll start with my favorite finding of all time and this is about the self.有关焦点效应我每天早晨都很匆忙And this is about the spotlight effect. So, my mornings are often rushed因为我有两个孩子有时我没有定闹钟because I have two kids. So, I get up and sometimes I don't set the alarm就会起晚我挣扎着下床把孩子们叫起来and I get up late; I stagger out of bed; I wake the kids; 问候佣人做好准备做早餐I greet the servants; I get ready; I make breakfast.我冲出房间通常在大概下午三点时有人会说I run out of the house and then usually around 3 o'clock somebody points out,某次是一个流浪汉说我耳朵上有一大块剃须膏in one case a homeless man, that I have a big glob of shaving cream in my ear or因为我剃须时都没心思看镜子because I neglected to actually look in the mirror while I shaved. 又一次我参加一个派对我发现我的衬衫上没穿正Or I have once been to a party and I found my shirt was misaligned,严重地没穿正不是一个扣子seriously misaligned, not one button but--Anyway,总之那时候我的感觉很不成熟so--and so I feel when this happens I'm very immature.我甚至感觉这是世界末日And I basically feel this is the end of the world,这实在太丢脸而且每个人都注意到了问题是this is humiliating and everybody notices. And so the question is,当这发生时有多少人真的注意到了呢焦点效应就是how many people notice when something happens? And the spotlight effect--Well,在讨论我最喜欢的实验之前before talking about my favorite experiment ever,我先播一段《辛普森一家》there is an episode of "The Simpsons"它充分说明了焦点效应that provides a beautiful illustration of the spotlight effect.这段短片还对心理测验给出了个漂亮的例证And then it has a beautiful illustration of psychological testing,我让大家一个一个看so I'll give you them quickly one after the other.因此汤姆·季洛维奇一位社会心理学家So, Tom Gilovich, a social psychologist,对焦点效应问题很感兴趣was interested in the question of the spotlight effect,当我们穿粉红衬衫上班或耳朵上粘了剃须膏或其他什么which is when we wear a pink shirt to work,shaving cream in our ear or whatever,我们真会过高地估计别人对此的注意吗do we systematically overestimate how much other people notice?他就此做了一系列实验其中一个实验是这样做的He did a series of experiments. And in oneexperiment what he did was他从心理学导论课上找到一些被试he got in the subjects ...Standard Intro Psych drill.跟他们说我希望你们明天穿件T恤And said,"I want you to wear a T-shirt for the next day我希望T恤上有图and I want it to have a picture on it,"然后他让他们穿上他们认为and he got them to wear T-shirts that had pictures on it最尴尬图片的T恤that were the most embarrassing pictures that they could have on it.结果发现如果你问人们It turns out that if you ask peopleT恤上印什么画最难以接受what's the worst picture to have on the T-shirt位居第一的答案是希特勒和巴瑞·曼尼洛that you are wearing, the number one answer is Hitler tied with Barry Manilow.画在T恤上最好的图画是The best pictures to have on your T-shirt马丁·路德·金和杰瑞·宋飞are Martin Luther King Jr. And Jerry Seinfeld.结果发现人们It turns out that people--他让他们穿着T恤到处走动一天然后问他们And then he had them go about their day and asked them,有多少人注意到了你的T恤"How many people noticed your T-shirt?"然后心理学家又去问被试周围的人And then the psychologists went around and they asked the people,你们有多少人注意到了这个人的T恤"How many of you noticed this person's T-shirt?"结果发现他们大概错误估计了两倍And it turned out they got it wrong by a factor of about two. 他们认为有一百个人注意到但其实只有五十个人注意到They thought, say, 100 noticed, but fifty people noticed.通过多次研究季洛维奇和And across study after study after study Gilovich他的同事找到了支持焦点效应的证据and his colleagues have found support for the spotlight effect,也就是你认为人们注意到你which is that you believe that people are noticing但其实并没有他们正忙着注意他们自己you all the time but they aren't. They're busy noticing themselves.知道这点非常有用季洛维奇之所以对此感兴趣And this is actually a useful thing to know. Gilovich got interested in this是因为他对后悔心理学感兴趣because he's interested in the psychology of regret.结果发现如果你真的去问那些临终的人And it turns out that if you actually ask dying people, 或是年纪很大的人你这辈子最后悔的事是什么or really old people basically, "What do you regret from your life?"他们不约而同地都对他们没有尝试过的事情感到后悔they regret the things as a rule that they didn't try.但当你问他们为什么不尝试时But when you asked them why they didn't try it他们的回答是这样我看起来会很愚蠢the answers tended to be "I would look silly."很有趣的是结果发现其实人们并不像你想的And it turns out, interesting to know, that people just don't care那么关心你as much as other people think you are.你可以将这看成好事或坏事You could take that as good news or bad news但聚光灯并不像我们想的那样聚焦在我们身上but the spotlight is not on us as much as we think it is.季洛维奇发现的第二个效应是透明度效应There's a second effect Gilovich discovers called "The transparency effect."透明度效应非常有趣And the transparency effect is quite interesting.透明度效应是指The transparency effect is that we believe我们高估自己的透明度that we're more transparent than we are.我需要一个自认为不会撒谎的人I need somebody up here who thinks that he or she is a bad liar.我只要你说三个句子Just--I just need you to say three sentences.我甚至会提前告诉你I'll even tell you what it is ahead of time.我将问你三个问题I'm going to ask you three questions:你去过伦敦吗你有弟弟妹妹吗"Have you been in London? Do you have a younger sibling?"你喜欢寿司吗我希望你回答这三个问题and "Do you like sushi?" I want you to answer with one of those answers there.但对其中一个问题说谎I want you to lie about one of them.在座各位的任务就是识别出The task will be for everybody else to recognize猜一下你哪个问题说谎了and guess which one you're lying about.你愿意做吗Do you want to go up?我会写下哪个问题你要说谎Yeah. And I will even write down which one you should lie on.我想你就这个问题说谎行吗So, I want you to lie as to that number. Okay?你去过伦敦吗不我没去过伦敦Have you ever been in London? No, I have not been in London. 你有弟弟妹妹吗有的我弟弟妹妹Do you have a younger sibling? Yes, I have a younger sibling. 你喜欢寿司吗不我不喜欢寿司Do you like sushi? No, I do not like sushi.好我们来投票她其中一个回答在说谎Ok, let's have a vote. She was lying about one of them.谁觉得第一个是谎话Who votes for one?谁认为是第二个Who votes for two?谁认为是第三个Who votes for three?大多数人觉得第二个或第三个Pretty much of a tie between two and three.你自己说你哪个说谎了第三个You could say which one you were lying. Three.这个效应这个效应有两方面The effect--there are two aspects of the effect.一方面是人们其实很擅长说谎One aspect is people are actually quite good at lying.很少人站在那里It is a rare person who couldn't stand up there而所有人都知道他们在说什么谎and everybody would figure out what they're lying about,但透明度效应是我们并不这么认为but the transparency effect is we don't feel that way.我们通常认为我们露馅了We often feel like things bleed out of us人们常常高估其他人and so people will systematically overestimate the extent to看穿自己秘密的程度一般来说which other people notice their secrets. And this is actually, in general,为什么有时说谎很难why it's sometimes difficult to teach or to tell stories是因为我们通常高估别人识谎的程度because we constantly overestimate how much other people know.我们认为自己很透明We think of ourselves as more transparent than we are.第二个社会心理学现象是人们总是自我感觉很好A second social psychological phenomena is you think you're terrific.如果我去问别人你认为在这学期的心理学导论课上表现如何If I asked people, "How well areyou doing in intro Psych this semester?"然后让你相对班里其他同学的表现and I asked you to give yourself a percentage rating relative 给自己定个百分等级如果每个人都打分准确的话to the rest of the class, then if everybody was accurate,或至少没有系统误差分数应该等于百分之五十or at least not systematically biased, the number should add up to 50%.也就是你们中有一半高于平均水平Roughly half of you are doing better than average有一半低于平均水平and roughly half of you are doing worse than average.结果发现人们往往It turns out though that people will systematically戏剧性地都认为自己高于平均水平and dramatically view themselves as better than average.他们都认为自己高于平均水平They will view themselves as better than average无论是问及他们是否是好学生when asked how good they are as a student,好老师好爱人尤其是好司机时答案都一样as a teacher, as a lover, and particularly, as a driver. 每个开车的人都认为自己是个好司机Everybody who drives thinks that he or she is a wonderful driver.这被称为沃博艮湖效应This has been called the "Lake Wobegon effect"这个词来源于盖瑞森·凯勒的小说based on Garrison Keillor's story about a place关于一个儿童都超常的地方where all the children are above average.沃博艮湖效应在心理学中包括了And the Lake Wobegon effect in psychology involves自认为高于平均水平的系统误差a systematic bias to see ourselves as better than average.但心理学家不明白的是为什么会出现沃博艮湖效应What psychologists don't really know is why the Lake Wobegon effect exists,并就此提出了很多设想其中一个我们获得的自然反馈and there are a couple of proposals. One is the nature of the feedback we get.在我们生活中很多时候只有在表现不错的时候得到反馈So, for a lot of aspects of your life you only get feedback when you're good,在我们成功做到某些事时得到反馈在一个正常的多产的when you do something good. In a normal, productive,健康的欢乐的环境中人们不会因为你做错事而尖叫healthy,happy environment, people don't scream at you about how bad you're doing而在你做得好时称赞你but they compliment how good you are and这可能会导致人们在某些方面产生that could lead to an inflated self-esteem膨胀性的自我评价on the part of people in certain domains.另一个可能是好的标准不一样Another possibility is there's different criteria for goodness.例如好司机的标准当我问你们评价自己开车怎样时For a driver, for instance, when I ask you to rank how good you are as a driver,人们通常会认为他们会说我比一般人好些what people often do is they think--they say, "I'm better than average,"但人们关注的往往只是开车的一个方面but what they do is they focus on one aspect of their driving.有人可能说So, some of you might say,我停车停得很好所以我是个好司机"Hey, I'm just a great parallel parker so I'm a great driver."其他人可能说我非常小心所以我是好司机Others might say, "I'm very careful, great driver."还有人可能说我插队插得比别人好所以我是好司机Others might say, "I take chances no one。
介绍理论英文作文范文
介绍理论英文作文范文英文：Introduction to Theory。
Theory is a fundamental aspect of many fields,including science, mathematics, and philosophy. It is essentially a set of ideas or principles that explain a particular phenomenon or concept. The purpose of theory is to provide a framework for understanding and explaining the world around us.In science, theory is used to explain observed phenomena and make predictions about future observations. For example, the theory of evolution explains how species change over time and how new species arise. In mathematics, theory is used to prove theorems and develop new mathematical concepts. In philosophy, theory is used to explore concepts such as ethics, metaphysics, and epistemology.One of the key features of theory is that it must be testable and falsifiable. This means that it must be possible to conduct experiments or make observations that could potentially disprove the theory. If a theory cannot be tested or is not falsifiable, it is not considered a valid scientific theory.Another important aspect of theory is that it must be supported by empirical evidence. This means that there must be observations or data that support the theory. The more evidence there is to support a theory, the more likely itis to be accepted as true.In summary, theory is a crucial element in many fields, providing a framework for understanding and explaining the world around us. It must be testable and supported by empirical evidence to be considered a valid scientific theory.中文：理论介绍。
Chapter 6 Introduction to Information Theory 信息理论知识
Chapter 6 Introduction to InformationTheory（PP330~399）This chapter will introduceinformation sources codingandchannel codingfrom the information theory angle of viewThe first part of this chapter: Source Coding Theorem{}1,216.1 Math models for information sources a . Discrete sources (DMS & non-DMS but stationary):,....,, is oneof the possible letter and (), 1b . Analog sources ( B L kLk k k k x x x x p P X x p ====å（）（）and-limited sources):sin[2(/2)() (6.11)22(/2)n W t n W X t X W W t n W p p ¥-¥-æö=-ç÷-èøå6.2 A Logarithm ic m easure of inform ationInform ation of event 1()log log (),()The unit is nats (base e) peroutp bits (base 2)o ut of the source r .i i i i i X x I x P x P x x ===-Self -information M utual informati （）(|)(;)log (;)()i j i j j i i P x y I x y I y x P x ==onconditional self-information Average mutual information1111F o r a l l i (t o t a l n )a n d j (t o t a l m ),(;)(,)(;)(,)(,)l o g ()()n m i j i j i j n m i j i j i j i j I X Y P x y I x y P x y P x y P x P y ======åååå1(|)log log (|) (|)i j i j i j I x y P x y P x y ==-\i j i i j I(x ;y )=I(x )-I(x |y )x R x T Loss()log ()i j i P x y P x =The average self-information (source entropy )11()()()()log ()Then we can get an important relationshi (;)()(|) p:n ni i i i i i I X Y H X H X Y H X P x I x P x P x ====-=-ååx T Loss xR example()()log () Information measures for continuous random variabl(|)()log (|)((es ;)H X p x P x dx H X Y p x p x y dxdyI X Y H ¥-¥¥¥-¥-¥=-=-=òòòdifferential entropy :average conditional entropy :average mutual information：)(|)() (|).X H X Y H Y H Y X -=-For Y continuous and X d *, see(32-20,21) in the fourth e iscr di ete tioninstead of åò6.3 Lossless coding of informationsourcesSource coding can be classified into(1) Lossless codingThe goal is that minimize the number of bitsbut the source can be perfectly reconstructed.(2)Lossy coding (Coding for analog sources)The goal is that the data are compressedsubject to a maximum tolerable distortion.6.3-1 The Lossless Source Coding TheoremShannon’s first theorem, 1948It gave limit on lossless source coding and known as the lossless source coding theorem: Let X denote DMS with entropy H(X). There exists a lossless source code for this source at any rate R if R>H(X). There exists no lossless source code for this source at rates R<H(X).Note: the unit of R is coded bits per source output, it is termed the code rate .The Fundamental Rate of Discrete Stationary Sources Coding that Lossless Recovery is Possible : Entropy Rate or Source Entropy12121112For a , the entropy of a block of random variables is ()() (6.3-5)T he entropy per letter is1()()T he defined aslim discrete stationary source entropy rate ()kk ii i k k k H X X X H XX X X H H X X X kX H X ¥-====åL L L 121()lim ()T he existance of this lim it can be verified(PP337~339).k k k H X H X X X k®¥®¥=L6.3-2 Lossless Coding AlgorithmTwo typical approaches for lossless coding of discrete source:(1)Huffman coding algorithman example of variable-length coding algorithm.(2)Lempel-Ziv algorithman example of fixed-length coding algorithm.1uniquely decodable2instantaneously decodableprefix condition can satisfy the two requirements The requirements of encoding are .(3) higher en :any codeword is not the prefix of any other codeword （）（）coding efficiency with the lower number of bits per output symbol or letter When the output letters or symbols are equally probable, we can use fixed-length encoding.（）Variable-length source codingWhen the output letters or symbols are not equally probable, we use variable-length code words.In order to increase the efficiency of coding：the symbols that occur more frequentlyare assigned shorter codewords (entropy coding).1k The definition of the encoding efficiency: ()/(),is one of the possible symbols n is the length of codeword of Lkk k k kH X R R nP a a a ==åSource coding theorem gives bounds of the entropy coding:For noiseless, prefix condition satisfied VLC: ()()1, ()/()1/j symbol by symbol H x R H x J symbols at a time H x R J H x J£<+£<+1 2 3 4 010 110 111aa a aprefix conditionis not satisfied Prefix condition:As the example we exam two code trees123401011111aaaaAn example of VLC of lossless coding: Huffman coding algorithm•Huffman coding algorithm is an optimalentropy coding algorithm. It is avariable-length coding algorithm andcan be used for both DMS and discretestationary sources.Huffman source coding algorithm(1952)* Based on the known source letter probability.* Optimum in the sense that the codewords satisfy the prefixcondition and the average number of bits required per symbol is minimum .It is not necessarily unique (Fig 6.3-5).0.350.300.200.100.0050.0050.04111100000011.65.35.15.05.016.3 1 and Fig 6.3-4Example -An example of FLC of lossless coding: Lempel-Ziv source encoding For Discrete stationary sources (the output letters are statistically dependent)Lempel-Zip source coding algorithm* Widely used in the compression of computer files (unix, ms-dos)* Designed to be independent of the source statistics. 6.3-4F ig If the sequence is much longer, the efficiency will increase a lot.From the source 1010110100100111010..44/80h =*Coding for analog sources: sampling, quantization, coding.*Quantization will introduce distortion. Optimumquantization means minimum coding 6-4 Lossy data compression (Coding for analog sources)rate R (bits/sample) with certain distortion D, or minimum distortion with certain coding rate.*What is the relationship between D and R?Rate-distortion function R(D), andDistortion-rate function D(R)Shannon Third Theorem(1959):Source Coding with a Fidelity CriterionA memoryless source X can be encoded at the rate of R for a distortion not exceeding D if R>R(D). Conversely, for any code with rateR<R(D) the distortion exceeds D.Note: Shannon gave lower bound of R(D) for the case of any discrete-time, continuousamplitude, memoryless source with zeromean and finite variance (MSE measure).22222 1log (/)(0)() (6.4-16)20() is the variance of gaussian source output and R iscoding rate (bit per sample). Actually it x x g x x D D R D D s s s s 죣ï=íï>îFor gaussian source,the rate distortion function is 2221010 is the upper bound of R(D)()2, (6.4-17) or,10log ()610log (6.4-18)Actually it is the upper bound of D(R)R g x g x D R D R R s s -==-+The distortion rate function isPDF of the amplitude is Gaussian* Bounds on R(D) and D(R)for any (any gaussian and non-gaussian amplitude distribution ) discrete-time, continuous amplitude, memoryless sourc e with zero mean and finite variance (MSE measure):[]2*222()*22*2211()()log 2log ()221, ()22()2, 6()()6()()1, ()log 22x g R H X R x g g g g x R D H X eD R D Dor D R D R ethus H X H X dB R D R D dB where H X e s p s p p s ---º-££ºº££ºéùéù=-=-ëûëû=g 10*R(D)D(R)D (R)10log D (R)Shannon, 1948Berger,19712x s up bound **lower bound (R ,D )( )upbound of differential entropy H(X)The second part of this chapter: Channel Coding TheoremChannel coding is the first choice of techniques for reliable transmission1.Orthogonal modulation :Increasing M, BER will decrease, but BW will increase too(grows exponentially with k : )k is the bits carried by a symbol.2.Coding : We choose M useful code words from total. Increasing Nwill reduce BER. But BW will increase (grows linearly with k ）. It can be used in the case of both BW limited and power limited. k is the bits carried by a code word.M N q )(1(2)2k Tc c /n is fixed, reducing k for im prove BER,r increased and w increased linearlyc c b W r r n k »=6-5Cannel models and channel capacity 6.5-1 Channel models*When we discuss coding and decoding, we often use the concept of composite channel. It includes modulator, waveform channel (here AWGN ch. assumed) and demodulator and detector)(Fig.6.5-1). We use BSC, DMC, D I C O C, Waveform Channels etc. channel models frequently.TX: Channel encoding: binary code modulation: BPSKRX: Hard decision decoding1.Binary Symmetric Channel (BSC ):Fig.6.5-2. It is memoryless and it can be described by Conditional Probabilities that relate possible outputs to the possible inputs .2.Discrete Memoryless Channels (DMC ):Fig.6.5-3, discrete q-ary input to modulator and Q-ary discrete output from detection. It can be described by probability transition matrix and conditional probabilities.[], where ()ij ij i j p p P y x =ºPTX: Channel encoding: binary code; modulation: M-arymemoryless modulator RX: soft decision decodingq possible symbols input (q=M) Q possible symbols output3.Discrete Input Continuous Output Channel (DICOC)(Q=infinite),It means the input to modulator is discrete and selected from a finite symbol set and the output of thedemodulator is unquantized. This channel model can be described by X(discrete), Y (analog) and their conditional probability density functions. In AWGN DICOC case, when a symbol inputs to the channel,12-1{,,, } q X x x x =L22011()/2for input alphabet {,,...),1(), k=0,1,...,q-1 (6.5-6)2For any given input , 1,2,...,the corresponding output sequencek q y x k i i i X x x x p y X x e i n Y X G s ps---=====+i sequence X 1211221(,,The ...,,...)() condition that the channel is memoryless ma y be ex (6.5-8) pres e s d asn n n n i i i i p y y y X u X u X u p y X u =======Õ4.AWGN Waveform Channels(ideal physical channels assumed)BW=W, C(f)=1. y(t)=x(t)+n(t),Alternatively, we have an equivalent discrete-time channel described by the conditional pdf. 20()/012121, ,,and are the sets of .1() (6.5-18)For the input sequence of length N without memory,(,.....,,,.....)() (6samples .5i i i i i i i i y x N i i NN N i i i y x n y x n p y x e N p y y y x x x p y x p --==+=ºÕ-19)Shannon Second Theorem(1948): The Noisy Channel Coding Theorem Reliable communication over a discrete memoryless channel is possible if the communication rate R satisfies R<C, where C is the channel capacity. At the rates higher than capacity, reliable communication is impossible.Note: The limit of communication rate is the capacity.6.5-2 Channel capacity1.DMC( with quantization after demodulation)The mutual information (received)I(x j ;y i )=I(x j )-I(x j /y i ), j: 0~q-1, i: 0~Q-1 The average mutual informationThe Capacity :1100(;)(,)(;)q Q j i j i j i I X Y P x y I x y --===åå11()()00(/)max (;)max ()(/)log ()j j q Q i j j i j P x P x j i i P y x C I X Y P x P y x P y --====ååThe unit of C are received bits per input symbol into the channel (bits/symbol) or bits/channel use .Example 6.5-1: BSC AWGN channelC=p log2p+(1-p)log2(1-p)=1-H(p) (6.5-29)Fig.6.5-4 is the relationship of C~p for AWGN BSC.Fig.6.5-5 is the relationship of C~SNR of AWGN BSC(with binary modulator and binary demodulator, when SNR is increased , p is decreased and C is increased monotonically. Fig.6.5-6 is the relationship of C~SNR for binary symmetric input, continuously output channel (see 6.5-30;6.5-31;6.5-32).H(p): binary entropy functionBSCBinary equal-prob. Input，continuous output (binary input AWGN ch.)2. Discrete in, continuous out ( AWGNmemoryless ) channel1()1(/)m ax (/)()log()w here ()(/)()j q j j j P x j q k k k p y x C p y x P x dy p y p y p y x P x -¥-¥=-===åòå3. Band-limited waveform AWGN channelwith an input power constraint (Shannon, 1948):,2/~22s A C ()Continuous input, continuous output channel model.1lim max (;) bits/sec(/)()(;)(/)()log ()()T p x C I X Y Tp y x p x I X Y p y x p x dx dy orp x p y ®¥¥¥-¥-¥==òò21Alternatively using the sampling vectors and , the minium sampling rate 2W(samples/sec),so in T s 2econds the number of samples are .(;)(/)(/)()logT=N )W (N N N N Ni i i i i i i i i I p y x p y x p x dy dx p y ¥¥-¥-¥===åòòX Y x y 2220()/2()/0{} are independent zero-mean Gaussian RV.1(), and with AWGN channel21(/)i xi i i x i xy x N i i x p x ep y x eNs p s p ---==22()10220222()02021m ax (;)log 122 log 1,,2m ax (;)log 1log 1 ( 6.5-43)NxN N p x i x xav avxav N N p x av I x y N N W T N N P P TWP I x y W T W N P C W W N s s s s=æö\=+ç÷èøæö=+ç÷èø==æö\=+ç÷èøæö=+ç÷èøåQ N=2WTThe capacity of a band-limited AWGNwaveform channel with a band-limited and average power-limited input.(Shannon, 1948 )20log 1 ( 6.5-43)av P C W WN æö=+ç÷èø0double effects:transm ission rate can be incre :(1) P , the C (2)W , C 1.44, w ith 1. increasing W , the ; 2. but increasing W , the N and w ill be i ased ncreased S and N R PN W ¥®¥®¥®¥®D iscu ssion s 2020b 0.(3) W hen w e express the C as m axim um Rlog 1 (6.5-4bandw and 5)R log 1 (6.5-46)W 21 idth efficien C w i cy ll be (6.5-49) .N decr a ede s rP R W N W P r N W r x æö<+ç÷èøæö=<+ç÷èø->Crelation ~ is given in Figure 4.6-1W on page 229RW hen r==1, 0;W r=0, 1.6;, exp(ln 2ln ),T his C is the upper lim its of data rate for reliable com m unications over a noisy .b bbb db N db N r r r N N x x x x =®-®¥»- channel.Noisy channel coding theorem (Shannon 194If the coding is used for reliable communications , we have There exist channel codes that makes it possible to achieve reliable communication, 8a): with as small an error probability as desired, if the transmission rate R<C, where C is the channel capacity. If R>C, it is not possible to make the probability of error tend toward zero with any code.6.6Achieving channel capacity withorthogonal signals21/220T he exact form for sym bol error rate of M -ary orthogonal signals is1112221 exp 2If it is upper bounded b ( refer to 4.4-10)M y x M sP e dx y dy N p p x -¥--¥-¥éùæö=-·êúç÷èøêúëûéùæöêú·--ç÷ç÷êúèøëûòò0/201y the of M -1events that (,) (,) for i 1, union bound further upper bounding (/by and , w e canobtains N i s C r s C r Q N s e x x -<¹0(/2ln 2)/20M (2ln 2)/2,when /2ln 2 1.39 (1.42dB),M , then P 0. It is not a very tight upper bound. Two different upper bounds for Q( ) can lead another bound of .12(ln )42b b k N M b M k M P eN P e M P e x g x g -----=>=®¥®£<2ln 2)0 (6.6-11)1 (ln )4, From /,ln 2 1.6dB, M , the the second bound n Pe 0.k s bb M where N k g g g g x g g ìïïíïî==>=-®¥®。
INTRODUCTION TO NUMBER THEORY
MARTIN KLAZAR (lecture notes)
These lecture notes cover the one-semester course Introduction to Number ´ Theory (Uvod do teorie ˇ c´ ısel, MAI040) that I have been teaching on the Faculty of Mathematics and Physics of Charles University in Prague since 1996. Needless to say, I do not claim any originality of the material presented here (in most cases I attempted to give references to the corresponding secondary sources that I used). July 2006 Martin Klazar
Another notation is ⌈α⌉ = min{m ∈ Z : m ≥ α} and α = min({α}, 1 − {α}) (distance to the nearest integer). The following fundamental theorem in diophantine approximation is due to Peter Dirichlet (1805–1859). Theorem (Dirichlet, 1842). 1. For every α ∈ R and Q ∈ N, Q ≥ 2, there exist numbers p, q ∈ Z such that 1 ≤ q < Q and 1 p . ≤ α− q qQ 2. For every irrational α ∈ R the inequality α− p 1 < 2 q q
算法 Lecture01概要
Copyright Li Zimao @ 2010-2011-1 SCUEC
First Example: Compute an
Write a program to compute an, given a real number a and a positive integer n, suppose n is a power of 2?
Question: How many pairs will there be in one year?
(image of Leonardo Fibonacci from http://www.math.ethz.ch/fibonacci)
1. In the beginning: (1 pair) 2. End of month 1: (1 pair) Rabbits are ready to mate. 3. End of month 2: (2 pairs) A new pair of rabbits are born. 4. End of month 3: (3 pairs) A new pair and two old pairs. 5. End of month 4: (5 pairs) ... 6. End of month 5: (8 pairs) ... 7. after 12 months, there will be 233 rabits
about adding, multiplying, dividing numbers, and extracting square roots and computing π.
– /wiki/Jaguar_(computer)
Copyright Li Zimao @ 2010-2011-1 SCUEC
关于半正定矩阵Hadamard积的矩阵不等式
证明: 因 A, B 0, M 为酉矩阵, 则 A = A* , B
= B* , MM* = M* M = I , 并且在 ( 1) 中取 a = 1, b = 0, 即可得到 ( 7) 。
定理 5: 设 A, B, M Cn n , 且 A, B, M 均为酉矩 阵, 则
I + A- 1B oB-1A 2 A - 1M oB- 1M M - 1A oM - 1B ( 8)
第 4卷 第 1期 2009年 3月
贵阳学院学报 (自然科学版 ) (季刊 ) JOURNAL OF GU IYANG COLLEGE N atural Sc iences ( Quarterly)
Vo.l 4 No. 1 M ar. 2009
*
关于半正定矩阵 H adam ard积的矩阵不等式
曾 诚1, 汤凤香2, 何淦瞳 3
A* A oB* B A* oB* A oB ( 3)
证明: 在 ( 1) 中取 a = 1, b = 0, M = I , 即可得
到 ( 3)。
推论 3[ 10] : 设 A, B Cn n , 且 A, B 0, 则
A 2 oB2
A oB 2
( 4)
证明: 因 A, B 0, 则 A = A* , B = B* , 并且
On the M atrix Inequalitiy for the Hadam ard P roduct of Positive Sem idefinite M atrices
ZENG Cheng1, TANG Feng-x iang2, H e G an- tong3
( 1. M aths D epartm ent, G uiyang U n ive rs ity, Gu iy ang Gu izhou 550005; 2. M aths D epa rtm en t, J iam usiU n iversity, J iam usiH e ilong jiang 154007; 3. Co llege o f N atura l Sc ience, G uizhou Un iv ers ity, G uiyang Gu izhou 550025, Ch ina) Abstrac t: T here ex ist great differences between positive sem idefin ite m a trix and positive defin ite m atr ix in the inequa lity research. M oore P enrose inverse is used to rep lace the general reverse w hen positiv e defin ite m atrix is extended to pos-i tive sem idefin ite m atr ix. Based on the m ethod o f b lo ck m atr ix and Schur com plem ent, w e obta in severa lH adam ard product inequalities abou t positive sem idefin ite ma tr ices. K ey word s: Schur comp lement; positive sem idefin ite m atrix; M oo re- Penrose inverses; H adam ard product; m atr ix inequa lities
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arXiv:hep-th/0012145v3 29 Jul 2001
Department of Mathematics, University of California Davis Davis, CA 95616 USA schwarz@ February 1, 2008
∗ Research supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation grant PHY-95-14797. † Research supported in part by NSF grant DMS-9801009
7 Morita equivalence 7.1 Morita equivalence of associative algebras . . . . . . . . . . . . . . . . . . . . 7.2 Gauge Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Invariance of BPS spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Noncommutative instantons 8.1 Instantons on Tθ4 . Definition and a simple example. . . . . . . . . . . . . . . 8.2 Instanton action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Noncommutative orbifolds 9.1 Noncommutative toroidal orbifolds . . . . . . . . . . . . . . . . . . . . . . . 9.2 K-theory of orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 K-theory of noncommutative Z2 orbifolds . . . . . . . . . . . . . . . . . . . . 10 Literature
8 10 11 11 13 14 15 15 17 17 18 19 22 22 24 26 28 32 36 37 37 39 40 45 47 48 49 51 51 53 57 58 58 60 62 62 64 66 67
4 Compactifications 4.1 Compactification on a circle. Relation between IKKT and BFSS models 4.2 Compactification on a regular T 2 . . . . . . . . . . . . . . . . . . . . . 4.3 Compactification on a noncommutative T 2 . . . . . . . . . . . . . . . . 4.4 Compactifications on T d and Tθd . . . . . . . . . . . . . . . . . . . . . . 4.5 Noncommutative geometry from a constant curvature background . . . 5 Noncommutative geometry 5.1 Algebras of functions and vector bundles . . . . . . . . . . . 5.2 Noncommutative Rd spaces . . . . . . . . . . . . . . . . . . 5.3 Endomorphisms and connections . . . . . . . . . . . . . . . 5.4 Involutive algebras . . . . . . . . . . . . . . . . . . . . . . . 5.5 Noncommutative tori . . . . . . . . . . . . . . . . . . . . . . 5.6 Projective modules over noncommutative tori . . . . . . . . 5.7 Connections on noncommutative tori . . . . . . . . . . . . . 5.8 K-theory, Chern character . . . . . . . . . . . . . . . . . . . 5.9 Modules with nondegenerate constant curvature connection 5.10 Heisenberg modules as deformations of vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction to M(atrix) theory and noncommutative geometry
Anatoly Konechny 1 ∗ and Albert Schwarz 2 † 1 Department of Physics, University of California Berkeley and Theoretical Physics Group, Mail Stop 50A-5101 LBNL, Berkeley, CA 94720 USA konechny@
Abstract Noncommutative geometry is based on an idea that an associative algebra can be regarded as ”an algebra of functions on a noncommutative space”. The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang-Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang-Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics. In this paper we give a mostly self-contained review of some aspects of M(atrix) theory, of Connes’ noncommutative geometry and of applications of noncommutative geometry to M(atrix) theory. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and SO(d, d|Z)-duality, an elementary discussion of instantons and noncommutative orbifolds. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and to mathematicians who would like to learn about some new problems arising in theoretical physics.
Contents
1 Introduction 2 Yang-Mills theory reduced to a point 3 Matrix models 3.1 IKKT matrix model . . . .ห้องสมุดไป่ตู้. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 8 8