Quantum Geometry, Matrix Theory, and Gravity

This book describes quantum geometry in the framework of Matrix Theory, which offers a quantum theory of space-time and matter.

Harold C. Steinacker (Author)

9781009440783, Cambridge University Press

Hardback, published 11 April 2024

420 pages
26.1 x 18.5 x 2.8 cm, 0.99 kg

'This text provides perhaps the first bridge for the uninitiated into this interesting field of research. The introductory chapters of this text are perhaps its finest, which beautifully define and explain the essential concepts needed for the proceeding discussions. Moreover, this book provides more than just a broad survey of historical results in the field, in the latter quadrants of the text, recent research is outlined in full, giving the reader a current outline of the field.' Physics Book Reviewer

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.

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.

Subject Areas: Statistical physics [PHS]