Quantum geometry, matrix theory, and gravity:
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 hel...
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| Format: | Buch |
| Sprache: | English |
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Cambridge ; New York ; Port Melbourne ; New Delhi ; Singapore
Cambridge University Press
2024
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| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | 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. |
| Beschreibung: | xvi, 402 Seiten |
| ISBN: | 9781009440783 |
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| 520 | 3 | |a 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. | |
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Datensatz im Suchindex
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| adam_text |
Л^^ИИЕкЯН^Н page xii Preface Part IMathematical background 1 Differentiable manifolds 1.1 1.2 2 Symplectic manifolds and Poisson structures 1.1.1 Hamiltonian vector fields The relation with Hamiltonian mechanics Lie groups and coadjoint orbits 2.1 2.2 CPn-1 as symplectic reduction of C" (Co)adjoint orbits as symplectic manifolds Part II Quantum spaces and geometry 3 Quantization of symplectic manifolds 3.1 3.2 3.3 3.4 4 Quantum spaces and matrix geometry 4.1 4.2 vii The Moyal-Weyl quantum plane R^ 3.1.1 Canonical Moyal-Weyl quantum plane or quantum mechanical phase space 3.1.2 General Moyal-Weyl quantum plane R|" 3.1.3 Weyl quantization 3.1.4 Derivatives and inner derivations 3.1.5 Coherent states on the Moyal-Weyl quantum plane Star products and deformation quantization Compact quantum spaces from coadjoint orbits 3.3.1 The fuzzy sphere 5^ 3.3.2 Fuzzy CPX“1 3.3.3 *Quantized coadjoint orbits The fuzzy torus 7^ Quasi-coherent states The abstract quantum space A4 4.2.1 t/(l) connection, would-be symplectic form, and quantum metric on Μ 81 4.2.2 Differential structure of quasi-coherent states 1 3 6 9 11 13 17 19 29 31 37 37 39 40 46 47 54 57 57 66 69 71 76 78 79 87
Contents viii 4.3 4.4 4.5 4.6 4.7 4.8 5 5.2 5.3 5.4 5.5 102 102 106 110 112 113 114 115 118 118 124 128 133 140 143 147 150 153 155 159 161 165 167 169 169 170 171 Part 111 Noncommutative field theory and matrix models 175 Noncommutative field theory 6.1 6.2 89 89 90 93 95 99 The fuzzy four-sphere 5^ 5.1.1 Semiclassical limit and bundle structure The fuzzy hyperboloid H^ 5.2.1 Semiclassical limit and Poisson structure 5.2.2 Irreducible tensor fields on 77 4 5.2.3 50(4, 1) intertwiners 5.2.4 Divergence-free vector fields on 774 and reconstruction *Relation to twistor space Cosmological к = — 1 quantum spacetime and higher spin 5.4.1 Semiclassical structure 5.4.2 Effective FLRW metric on Λί3,1 5.4.3 Algebra of functions and higher spin 5.4.4 Spacelike 50(3, 1) substructure and associated identities 5.4.5 Divergence-free vector fields on M3.1 5.4.6 Reconstruction of divergence-free vector fields 5.4.7 *Local bundle geometry 5.4.8 General к = — 1 cosmologies Cosmological к = 0 quantum spacetime Covariant quantum spaces 5.1 6 The semiclassical structure of At 4.3.1 Almost-local quantum spaces and Poisson tensor 4.3.2 Relating the algebraic and geometric structures 4.3.3 Quantum tangent space Quantization map, symbol, and Semiclassical regime *Complex tangent space and quantum Kähler manifolds 4.5.1 Coherent states and quantization map for quantum Kähler manifolds 101 Examples of quantum (matrix) geometries 4.6.1 The fuzzy sphere ^ revisited 4.6.2 The Moyal-Weyl quantum plane revisited 4.6.3 *Quantized coadjoint orbits revisited Further structures and properties 4.7.1 Dimension,
oxidation, and reduction 4.7.2 Band-diagonal matrices Landau levels and quantum spaces Noncommutative scalar field theory String modes and trace formulas 6.2.1 The off-diagonal string symbol and regularity 6.2.2 The string representation of the propagator 177 178 190 197 198
Contents ix 6.3 6.4 6.5 6.6 6.7 6.2.3 Trace formulas on classical and quantum spaces 6.2.4 String modes on fuzzy S# 6.2.5 String modes on covariant quantum spaces 6.2.6 Factorized string modes for product spaces String modes at one loop in NC field theory Higher loops and pathological UV/IR mixing Diffeomorphisms and symplectomorphisms on quantum spaces Noncom mutative Yang-Mills theory I *Relation to string theory 200 206 207 208 208 213 215 216 220 7 Yang-Mills matrix models and quantum spaces 7.1 Almost-commutative matrix configurations and branes 7.2 Quantization: Matrix integral and is regularization 7.2.1 * Single-matrix models 7.3 Nontrivial vacua and emergent quantum spaces 7.3.1 Commutative backgrounds 7.4 Yang-Mills gauge theory on quantum spaces from SSB 7.4.1 Fluctuations and gauge fields 7.4.2 *The matrix energy-momentum tensor 7.4.3 Branes, transversal fluctuations, and scalar fields 7.4.4 Tangential fluctuations and gauge fields 7.4.5 Stacks of branes and nonabelian Yang-Mills theory II 7.5 *Gauge fixing and perturbative quantization 7.6 *Effective matrix action 7.6.1 Reducible versus irreducible backgrounds 7.6.2 *Stabilization of covariant backgrounds 223 226 228 230 232 235 236 236 237 239 240 241 243 246 248 250 8 Fuzzy extra dimensions 8.1 Fuzzy extra dimensions in the Yang-Mills gauge theory 8.2 Fluctuation modes for fuzzy extra dimensions 8.3 Fuzzy extra dimensions in matrix models 8.4 *Self-intersecting fuzzy extra dimensions and chiral fermions 252 252 253 255 257 9 Geometry and dynamics in Yang-Mills matrix models 9.1 Frame and tensor fields on
generic branes 9.1.1 Effective metric and frame 9.1.2 Frame and tensor fields 9.1.3 Gauge transformations and (symplectic) diffeomorphisms 9.1.4 Noncommutative gauge fields and field strength 9.1.5 Geometric YM action on symplectic branes 9.1.6 Nonabelian gauge theory on curved symplectic branes 9.2 Weitzenböck connection, torsion, and Laplacian 9.2.1 Divergence constraint 9.2.2 Effective frame 260 260 261 263 264 265 266 269 271 274 275
Contents 9.3 10 9.2.3 Example: Fuzzy S? 9.2.4 Divergence-free frames and metric 9.2.5 *Contraction identities 9.2.6 Effective Laplacian Prc-gravity: Covariant equations of motion for frame and torsion 9.3.1 Riemann tensor and torsion 9.3.2 Ricci scalar, torsion, and Einstein-Hilbert action 9.3.3 Οπ-shell Ricci tensor and effective energy-momentum tensor 9.3.4 Example: Spherically symmetric solutions Higher-spin gauge theory on quantum spacetime 10.1 10.2 10.3 10.4 Cosmic FLRW spacetime M3.1 Deformed spacetime 3.1 and higher spin 10.2.1 Frame and metric 10.2.2 Divergence constraint on covariant quantum spaces 10.2.3 Higher-spin gauge trafos and volume-preserving diffeos 10.2.4 Geometric backgrounds and frame reconstruction 10.2.5 Local normal coordinates Linearized ho gauge theory and fluctuation modes 10.3.1 Diagonalization of D2 and fluctuation spectrum 10.3.2 Inner product, Hilbert space, and no ghosts 10.3.3 Linearized frame, metric, and Schwarzschild solution Prc-gravity and matter Part IV Matrix theory and gravity 11 Matrix theory: Maximally supersymmetric matrix models 11.1 11.2 12 The IKKT or 1IB matrix model at one loop ILLI Fermions 11.1.2 ^ = 4 noncommutative super-Yang-Mills 11.1.3 One-loop effective action in Minkowski signature 11.1.4 The one-loop effective action for the Euclidean IKKT model 11.1.5 Selection and stabilization of branes at one loop 11.1.6 Flux terms and other soft SUSY breaking terms 11.1.7 UV/IR mixing, supergravity, string theory, andAf = 4 SYM 11.1.8 Compactification of target space Numerical and nonperturbative aspects 11.2.1 Euclidean
model 11.2.2 Lorentzian model Gravity as a quantum effect on quantum spacetime 12.1 12.2 12.3 Heat kernel and induced gravity on commutativespaces Higher-derivative local action on simple branes Einstein-Hilbert action from fuzzy extra dimensions 275 276 277 279 280 284 285 286 287 291 292 293 293 294 295 296 301 305 306 308 312 314 317 319 319 321 323 323 327 328 329 332 338 340 340 341 342 342 344 346
Contents 12.4 13 Gravity on covariant quantum spacetime 12.4.1 Gravitational action on A43·1 without extra dimensions 351 352 12.4.2 Einstein-Hilbert action from fuzzy extra dimensions Λ2 354 12.4.3 Induced gravity in Euclidean signature 358 12.4.4 One-loop gravitational action and vacuum stability 359 Matrix quantum mechanics and the BFSS model 363 13.1 Matrix quantum mechanics 363 13.2 The BFSS model and its relation with M-theory 366 13.3 Relation between the BFFS and IKKT models 368 13.4 The BMN model 369 13.5 Holographic aspects and deep quantum regime 370 Appendix A Gaussian integrals over matrix spaces 372 Appendix В Some SO(D) group theory 373 AppendixC T orsion identities Appendix D Some integrals 381 383 Appendix E Functions on coadjoint orbits 384 Appendix F Glossary and notations 386 References and Further Reading 389 Index 400 |
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| spelling | Steinacker, Harold C. Verfasser (DE-588)1334229414 aut Quantum geometry, matrix theory, and gravity Harold C. Steinacker, University of Vienna, Austria Cambridge ; New York ; Port Melbourne ; New Delhi ; Singapore Cambridge University Press 2024 xvi, 402 Seiten txt rdacontent n rdamedia nc rdacarrier 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. Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Matrix-Modell (DE-588)4518453-7 gnd rswk-swf Geometrische Quantisierung (DE-588)4156720-1 gnd rswk-swf Stringtheorie (DE-588)4224278-2 gnd rswk-swf Gravitation (DE-588)4021908-2 gnd rswk-swf Geometric quantization Gravity / Mathematical models String models Matrices Geometrische Quantisierung (DE-588)4156720-1 s Quantenfeldtheorie (DE-588)4047984-5 s Matrix-Modell (DE-588)4518453-7 s Gravitation (DE-588)4021908-2 s Stringtheorie (DE-588)4224278-2 s DE-604 Erscheint auch als Online-Ausgabe 978-1-009-44077-6 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035219638&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Steinacker, Harold C. Quantum geometry, matrix theory, and gravity Quantenfeldtheorie (DE-588)4047984-5 gnd Matrix-Modell (DE-588)4518453-7 gnd Geometrische Quantisierung (DE-588)4156720-1 gnd Stringtheorie (DE-588)4224278-2 gnd Gravitation (DE-588)4021908-2 gnd |
| subject_GND | (DE-588)4047984-5 (DE-588)4518453-7 (DE-588)4156720-1 (DE-588)4224278-2 (DE-588)4021908-2 |
| title | Quantum geometry, matrix theory, and gravity |
| title_auth | Quantum geometry, matrix theory, and gravity |
| title_exact_search | Quantum geometry, matrix theory, and gravity |
| title_full | Quantum geometry, matrix theory, and gravity Harold C. Steinacker, University of Vienna, Austria |
| title_fullStr | Quantum geometry, matrix theory, and gravity Harold C. Steinacker, University of Vienna, Austria |
| title_full_unstemmed | Quantum geometry, matrix theory, and gravity Harold C. Steinacker, University of Vienna, Austria |
| title_short | Quantum geometry, matrix theory, and gravity |
| title_sort | quantum geometry matrix theory and gravity |
| topic | Quantenfeldtheorie (DE-588)4047984-5 gnd Matrix-Modell (DE-588)4518453-7 gnd Geometrische Quantisierung (DE-588)4156720-1 gnd Stringtheorie (DE-588)4224278-2 gnd Gravitation (DE-588)4021908-2 gnd |
| topic_facet | Quantenfeldtheorie Matrix-Modell Geometrische Quantisierung Stringtheorie Gravitation |
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