Quantum Geometry, Matrix Theory, and Gravity (1版)

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【簡介】 Building on mathematical structures familiar from quantum mechanics, this book provides an introduction to quantization in a broad context before developing a framework for quantum geometry in Matrix Theory and string theory. Taking a physics-oriented approach to quantum geometry, this framework helps explain the physics of Yang–Mills-type matrix models, leading to a quantum theory of space-time and matter. This novel framework is then applied to Matrix Theory, which is defined through distinguished maximally supersymmetric matrix models related to string theory. A mechanism for gravity is discussed in depth, which emerges as a quantum effect on quantum space-time within Matrix Theory. Using explicit examples and exercises, readers will develop a physical intuition for the mathematical concepts and mechanisms. It will benefit advanced students and researchers in theoretical and mathematical physics, and is a useful resource for physicists and mathematicians interested in the geometrical aspects of quantization in a broader context. 【目錄】 Preface The trouble with spacetime Quantum geometry and Matrix theory Part I. Mathematical Background: 1. Differentiable manifolds 2. Lie groups and coadjoint orbits Part II. Quantum Spaces and Geometry: 3. Quantization of symplectic manifolds 4. Quantum spaces and matrix geometry 5. Covariant quantum spaces Part III. Noncommutative field theory and matrix models: 6. Noncommutative field theory 7. Yang–Mills matrix models and quantum spaces 8. Fuzzy extra dimensions 9. Geometry and dynamics in Yang–Mills matrix models 10. Higher-spin gauge theory on quantum spacetime Part IV. Matrix Theory and Gravity: 11. Matrix theory: maximally supersymmetric matrix models 12. Gravity as a quantum effect on quantum spacetime 13. Matrix quantum mechanics and the BFSS model Appendixes References Index.

Introduction to Matrix Theory (2版)

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Linear algebra and matrix theory are among the most important and most frequently applied branches of mathematics. They are especially important in solving engineering and economic models, where either the model is assumed linear, or the nonlinear model is approximated by a linear model, and the resulting linear model is examined. This book is mainly a textbook, that covers a one semester upper division course or a two semester lower division course on the subject. The second edition will be an extended and modernized version of the first edition. We added some new theoretical topics and some new applications from fields other than economics. We also added more difficult exercises at the end of each chapter which require deep understanding of the theoretical issues. We also modernized some proofs in the theoretical discussions which give better overview of the study material. In preparing the manuscript we also corrected the typos and errors, so the second edition will be a corrected, extended and modernized new version of the first edition. Request Inspection Copy Sample Chapter(s) Preface Chapter 1: Vectors and Matrices Contents: Vectors and Matrices Vector Spaces and Inner-Product Spaces Systems of Linear Equations, and Inverses of Matrices Determinants Linear Mappings and Matrices Eigenvalues, Invariant Subspaces, Canonical Forms Special Matrices Elements of Matrix Analysis Readership: Written primarily for undergraduate and graduate students studying engineering, economics and business; can also be used in courses offered by the mathematics department, or by any kind of engineering and social sciences; can also be used by researchers in mathematics, economics, engineering; in addition, high school special courses, and teacher training courses can employ this textbook.

A Problem Based Journey from Elementary Number Theory to an Introduction to Matrix Theory

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Contents: Introduction Algebraic Structures The Natural Numbers The Integers The Real Numbers Introduction to Set Theory The Pigeonhole Principle and the Base 2 Number System Introduction to Group Theory Introduction to Matrix Theory Fibonacci Numbers, Determinants and Eigenvalues The Mathematics Behind Google's Page Rank and a Game of Numbers Bibliography Index Readership: High school students who love mathematics and mathematics teachers. Also good for professors of mathematics and maths education, students in the mathematical sciences and pre-service math teachers.