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.