Mathematical quantization

With a unique approach and presenting an array of new and intriguing topics, Mathematical Quantization offers a survey of operator algebras and related structures from the point of view that these objects are quantizations of classical mathematical structures. This approach makes possible, with minimal mathematical detail, a unified treatment of a variety of topics. Detailed here for the first time, the fundamental idea of mathematical quantization is that sets are replaced by Hilbert spaces. Building on this idea, and most importantly on the fact that scalar-valued functions on a set correspond to operators on a Hilbert space, one can determine quantum analogs of a variety of classical structures. In particular, because topologies and measure classes on a set can be treated in terms of scalar-valued functions, we can transfer these constructions to the quantum realm, giving rise to C*- and von Neumann algebras. In the first half of the book, the author quickly builds the operator algebra setting. He uses this as a unifying theme in the second half, in which he treats several active research topics, some for the first time in book form. These include the quantum plane and tori, operator spaces, Hilbert modules, Lipschitz algebras, and quantum groups. For graduate students, Mathematical Quantization offers an ideal introduction to a research area of great current interest. For professionals in operator algebras and functional analysis, it provides a readable tour of the current state of the field. Table of Contents c0015_c000 1 Mathematical Quantization 3 Preface 5 Contents 8 C0015_C001 11 Mathematical Quantization 11 Table of Contents -1 Chapter 1: Quantum Mechanics 11 1.1 Classical physics 11 1.2 States and events 12 1.3 Observables 16 1.4 Dynamics 19 1.5 Composite systems 23 1.6 Quantum computation 26 1.7 Notes 27 C0015_C002 28 Mathematical Quantization 28 Table of Contents -1 Chapter 2: Hilbert Spaces 28 2.1 Definitions and examples 28 2.2 Subspaces 32 2.3 Orthonormal bases 37 2.4 Duals and direct sums 40 2.5 Tensor products 44 2.6 Quantum logic 49 2.7 Notes 52 C0015_C003 53 Mathematical Quantization 53 Table of Contents -1 Chapter 3: Operators 53 3.1 Unitaries and projections 53 3.2 Continuous functional calculus 58 3.3 Borel functional calculus 62 3.4 Spectral measures 65 3.5 The bounded spectral theorem 69 3.6 Unbounded operators 71 3.7 The unbounded spectral theorem 74 3.8 Stone's theorem 76 3.9 Notes 80 C0015_C004 81 Mathematical Quantization 81 Table of Contents -1 Chapter 4: The Quantum Plane 81 4.1 Position and momentum 81 4.2 The tracial representation 85 4.3 Bargmann-Segal space 87 4.4 Quantum complex analysis 93 4.5 Notes 97 C0015_C005 99 Mathematical Quantization 99 Table of Contents -1 Chapter 5: C*-algebras 99 5.1 The algebras C(X) 99 5.2 Topologies from functions 103 5.3 Abelian C*-algebras 107 5.4 The quantum plane 109 5.5 Quantum tori 117 5.6 The GNS construction 124 5.7 Notes 131 C0015_C006 133 Mathematical Quantization 133 Table of Contents -1 Chapter 6: Von Neumann algebras 133 6.1 The algebras l1(X) 133 6.2 The algebras L1(X) 136 6.3 Trace class operators 139 6.4 The algebras B(H) 143 6.5 Von Neumann algebras 146 6.6 The quantum plane and tori 151 6.7 Notes 154 C0015_C007 155 Mathematical Quantization 155 Table of Contents -1 Chapter 7: Quantum Field Theory 155 7.1 Fock space 155 7.2 CCR algebras 158 7.3 Relativistic particles 163 7.4 Flat spacetime 167 7.5 Curved spacetime 169 7.6 Notes 172 C0015_C008 175 Mathematical Quantization 175 Table of Contents -1 Chapter 8: Operator Spaces 175 8.1 The spaces V (K) 175 8.2 Matrix norms and convexity 177 8.3 Duality 184 8.4 Matrix-valued functions 188 8.5 Operator systems 192 8.6 Notes 198 C0015_C009 199 Mathematical Quantization 199 Table of Contents -1 Chapter 9: Hilbert Modules 199 9.1 Continuous Hilbert bundles 199 9.2 Hilbert L1-modules 202 9.3 Hilbert C*-modules 205 9.4 Hilbert W*-modules 210 9.5 Crossed products 216 9.6 Hilbert *-bimodules 219 9.7 Notes 225 C0015_C010 227 Mathematical Quantization 227 Table of Contents -1 Chapter 10: Lipschitz Algebras 227 10.1 The algebras Lip0(X) 227 10.2 Measurable metrics 234 10.3 The derivation theorem 239 10.4 Examples 244 10.5 Quantum Markov semigroups 250 10.6 Notes 256 C0015_C011 257 Mathematical Quantization 257 Table of Contents -1 Chapter 11: Quantum Groups 257 11.1 Finite dimensional C*-algebras 257 11.2 Finite quantum groups 259 11.3 Compact quantum groups 264 11.4 Haar measure 268 11.5 Notes 271 C0015_References 273 Mathematical Quantization 273 Table of Contents -1 References 273 c0015_c000......Page 1 Mathematical Quantization......Page 3 Preface......Page 5 Contents......Page 8 1.1 Classical physics......Page 11 Table of Contents......Page 0 1.2 States and events......Page 12 1.3 Observables......Page 16 1.4 Dynamics......Page 19 1.5 Composite systems......Page 23 1.6 Quantum computation......Page 26 1.7 Notes......Page 27 2.1 Definitions and examples......Page 28 2.2 Subspaces......Page 32 2.3 Orthonormal bases......Page 37 2.4 Duals and direct sums......Page 40 2.5 Tensor products......Page 44 2.6 Quantum logic......Page 49 2.7 Notes......Page 52 3.1 Unitaries and projections......Page 53 3.2 Continuous functional calculus......Page 58 3.3 Borel functional calculus......Page 62 3.4 Spectral measures......Page 65 3.5 The bounded spectral theorem......Page 69 3.6 Unbounded operators......Page 71 3.7 The unbounded spectral theorem......Page 74 3.8 Stone's theorem......Page 76 3.9 Notes......Page 80 4.1 Position and momentum......Page 81 4.2 The tracial representation......Page 85 4.3 Bargmann-Segal space......Page 87 4.4 Quantum complex analysis......Page 93 4.5 Notes......Page 97 5.1 The algebras C(X)......Page 99 5.2 Topologies from functions......Page 103 5.3 Abelian C*-algebras......Page 107 5.4 The quantum plane......Page 109 5.5 Quantum tori......Page 117 5.6 The GNS construction......Page 124 5.7 Notes......Page 131 6.1 The algebras l1(X)......Page 133 6.2 The algebras L1(X)......Page 136 6.3 Trace class operators......Page 139 6.4 The algebras B(H)......Page 143 6.5 Von Neumann algebras......Page 146 6.6 The quantum plane and tori......Page 151 6.7 Notes......Page 154 7.1 Fock space......Page 155 7.2 CCR algebras......Page 158 7.3 Relativistic particles......Page 163 7.4 Flat spacetime......Page 167 7.5 Curved spacetime......Page 169 7.6 Notes......Page 172 8.1 The spaces V (K)......Page 175 8.2 Matrix norms and convexity......Page 177 8.3 Duality......Page 184 8.4 Matrix-valued functions......Page 188 8.5 Operator systems......Page 192 8.6 Notes......Page 198 9.1 Continuous Hilbert bundles......Page 199 9.2 Hilbert L1-modules......Page 202 9.3 Hilbert C*-modules......Page 205 9.4 Hilbert W*-modules......Page 210 9.5 Crossed products......Page 216 9.6 Hilbert *-bimodules......Page 219 9.7 Notes......Page 225 10.1 The algebras Lip0(X)......Page 227 10.2 Measurable metrics......Page 234 10.3 The derivation theorem......Page 239 10.4 Examples......Page 244 10.5 Quantum Markov semigroups......Page 250 10.6 Notes......Page 256 11.1 Finite dimensional C*-algebras......Page 257 11.2 Finite quantum groups......Page 259 11.3 Compact quantum groups......Page 264 11.4 Haar measure......Page 268 11.5 Notes......Page 271 References......Page 273 Presenting an array of new topics, "Mathematical Quantization" explores operator algebras. For graduate students, this text offers an introduction to a large area of active research, and for professionals in operating algebras and functional analysis, it provides a tour of the field No background in physics is really needed to understand the concept of mathematical quantization, which may be taken as nothing more than a formal analogy between sets and Hilbert spaces.