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.

Geometry of Quantum States

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Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.

Quantum Stochastic Processes and Noncommutative Geometry

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The classical theory of stochastic processes has important applications arising from the need to describe irreversible evolutions in classical mechanics; analogously quantum stochastic processes can be used to model the dynamics of irreversible quantum systems. Noncommutative, i.e. quantum, geometry provides a framework in which quantum stochastic structures can be explored. This book is the first to describe how these two mathematical constructions are related. In particular, key ideas of semigroups and complete positivity are combined to yield quantum dynamical semigroups (QDS). Sinha and Goswami also develop a general theory of Evans-Hudson dilation for both bounded and unbounded coefficients. The unique features of the book, including the interaction of QDS and quantum stochastic calculus with noncommutative geometry and a thorough discussion of this calculus with unbounded coefficients, will make it of interest to graduate students and researchers in functional analysis, probability and mathematical physics.