The world according to quantum mechanics : why the laws of physics make perfect sense after all

The book summarises contemporary knowledge about the theory of atomic and molecular clusters. New results are discussed on a high theoretical level. Access to this field of research is given by an explanation of the various subjects in introductory chapters An invaluable supplement to standard textbooks on quantum mechanics, this introduction to the general theoretical framework of contemporary physics focuses on conceptual, epistemological, and ontological issues. Probability : basic concepts and theorems -- The principle of indifference -- Subjective probabilities versus objective probabilities -- Relative frequencies -- Adding and multiplying probabilities -- Conditional probabilities and correlations -- Expectation value and standard deviation -- A (very) brief history of the "old" theory -- Planck -- Rutherford -- Bohr -- de Broglie -- Mathematical interlude -- Vectors -- Definite integrals -- Derivatives -- Taylor series -- Exponential function -- Sine and cosine -- Integrals -- Complex numbers -- A (very) brief history of the "new" theory -- Schrodinger -- Born -- Heisenberg and "uncertainty" -- Why energy is quantized -- The Feynman route to Schrodinger (stage 1) -- The rules of the game -- Two slits -- Interference -- The propagator as a path integral -- The time-dependent propagator -- A free particle -- A free and stable particle -- Special relativity in a nutshell -- The principle of relativity -- Lorentz transformations : general form -- Composition of velocities -- The case against positive K -- An invariant speed -- Proper time -- The meaning of mass -- The case against K = 0 -- Lorentz transformations : some implications -- 4-vectors -- The Feynman route to Schrodinger (stage 2) -- Action -- How to influence a stable particle? -- Enter the wave function -- The Schrodinger equation -- Why quantum mechanics? -- The classical probability calculus -- Why nontrivial probabilities? -- Upgrading from classical to quantum -- Vector spaces -- Compatible an incompatible elementary tests -- Noncontextuality -- The core postulates -- The trace rule -- Self-adjoint operators and the spectral theorem -- Pure states and mixed states -- How probabilities depend on measurement outcomes -- How probabilities depend on the times of measurements -- The rules of the game derived at last -- The classical forces : effects -- The principle of "least" action -- Geodesic equations for flat spacetime -- Energy and momentum -- Vector analysis : some basic concepts -- Curl and Stokes's theorem -- Divergence and Gauss's theorem -- The Lorentz force -- Curved spacetime -- Gravity -- The classical forces : causes -- Gauge invariance -- Fuzzy potentials -- Maxwell's equations -- A fuzzy metric -- Einstein's equation -- Aharonov-Bohm effect -- Fact and fiction in the world of classical physics -- Quantum mechanics resumed -- The experiment of Elitzur and Vaidman -- Observables -- The continuous case -- Commutators -- The Heisenberg equation -- Operators for energy and momentum -- Angular momentum -- The hydrogen atom in brief -- Spin -- Spin 1/2 -- A Stern-Gerlach relay -- Why spin? -- Beyond hydrogen -- Spin precession -- The quantum Zeno effect -- Composite systems -- Bell's theorem : the simplest version -- "Entangled" spins -- Reduced density operator -- Contextuality -- The experiment of Greenberger, Horne, and Zeilinger -- Uses and abuses of counterfactual reasoning -- The experiment of Englert, Scully, and Walther -- Time-symmetric probability assignments -- Quantum statistics -- Scattering billiard balls -- Scattering particles -- Symmetrization -- Bosons are gregarious -- Fermions are solitary -- Quatum coins and quantum dice -- Measuring Sirius -- Relativistic particles -- The Klein-Gordon equation -- Antiparticles -- The Dirac equation -- The Euler-Lagrange equation -- Noether's theorem -- Scattering amplitudes -- QED -- A few words about renormalization -- Beyond QED -- QCD -- Electroweak interactions -- Higgs mechanism -- Pitfalls -- Standard axioms : a critique -- The principle of evolution -- The eigenstate-eigenvalue link -- Interpretational strategy -- Spatial aspects of the quantum world -- The two-slit experiment revisited -- The importance of unperformed measurements -- Spatial distinctions : relative and contingent -- The importance of detectors -- Spatiotemporal distinctions : not all the way down -- The shape of things -- Space -- The macroworld -- Questions of substance -- Particles -- Scattering experiment revisited -- How many constituents? -- An ancient conundrum -- A fundamental particle by itself -- Manifestation -- "Creation" in a nutshell -- The coming into being of form -- Bottom-up or top-down? -- Whence the quantum-mechanical correlation laws? -- How are "spooky actions at a distance" possible? -- Why the laws of physics are just so -- The stability of matter -- Why quantum mechanics (summary) -- Why special relativity (summary) -- Why quantum mechanics (summary continued) -- The classical or long-range forces -- The nuclear or short-range forces -- Fine tuning -- Quanta and Vedanta -- The central affirmation -- The poises of creative consciousness Contents......Page 12 Preface......Page 6 Overview......Page 20 1.1 The principle of indifference......Page 22 1.3 Relative frequencies......Page 23 1.4 Adding and multiplying probabilities......Page 24 1.5 Conditional probabilities and correlations......Page 26 1.6 Expectation value and standard deviation......Page 27 2.2 Rutherford......Page 28 2.3 Bohr......Page 29 2.4 de Broglie......Page 31 3.1 Vectors......Page 34 3.2 Definite integrals......Page 36 3.3 Derivatives......Page 38 3.5 Exponential function......Page 42 3.6 Sine and cosine......Page 43 3.7 Integrals......Page 44 3.8 Complex numbers......Page 46 4.1 Schrodinger......Page 50 4.2 Born......Page 52 4.3 Heisenberg and "uncertainty"......Page 54 4.4 Why energy is quantized......Page 57 5.2 Two slits......Page 60 5.2.1 Why product?......Page 61 5.2.3 Why proportional to BA?......Page 62 5.3 Interference......Page 63 5.3.1 Limits to the visibility of interference fringes......Page 64 5.4 The propagator as a path integral......Page 66 5.5 The time-dependent propagator......Page 67 5.7 A free and stable particle......Page 69 6.1 The principle of relativity......Page 72 6.2 Lorentz transformations: General form......Page 73 6.3 Composition of velocities......Page 77 6.4 The case against positive K......Page 78 6.5 An invariant speed......Page 80 6.6 Proper time......Page 81 6.7 The meaning of mass......Page 82 6.8 The case against K = 0......Page 83 6.9 Lorentz transformations: Some implications......Page 84 6.10 4-vectors......Page 87 7.1 Action......Page 88 7.2 How to inuence a stable particle?......Page 89 7.4 The Schrodinger equation......Page 90 A Closer Look......Page 94 8.1 The classical probability calculus......Page 96 8.2 Why nontrivial probabilities?......Page 98 8.4 Vector spaces......Page 99 8.4.2 Subspaces and projectors......Page 101 8.4.3 Commuting and non-commuting projectors......Page 103 8.5 Compatible and incompatible elementary tests......Page 105 8.6 Noncontextuality......Page 107 8.8 The trace rule......Page 109 8.9 Self-adjoint operators and the spectral theorem......Page 111 8.10 Pure states and mixed states......Page 112 8.11 How probabilities depend on measurement outcomes......Page 113 8.12 How probabilities depend on the times of measurements......Page 114 8.12.1 Unitary operators......Page 115 8.12.2 Continuous variables......Page 118 8.13 The rules of the game derived at last......Page 119 9.1 The principle of "least" action......Page 120 9.2 Geodesic equations for at spacetime......Page 123 9.3 Energy and momentum......Page 124 9.4 Vector analysis: Some basic concepts......Page 126 9.4.1 Curl and Stokes's theorem......Page 127 9.4.2 Divergence and Gauss's theorem......Page 129 9.5 The Lorentz force......Page 130 9.5.1 How the electromagnetic field bends geodesics......Page 132 9.6 Curved spacetime......Page 134 9.6.2 Raising and lowering indices......Page 135 9.6.3 Curvature......Page 136 9.6.4 Parallel transport......Page 137 9.7 Gravity......Page 139 10.1 Gauge invariance......Page 142 10.2 Fuzzy potentials......Page 143 10.2.1 Lagrange function and Lagrange density......Page 144 10.3 Maxwell's equations......Page 145 10.3.1 Charge conservation......Page 147 10.4 A fuzzy metric......Page 148 10.4.1 Meaning of the curvature tensor......Page 149 10.5 Einstein's equation......Page 150 10.6 Aharonov–Bohm effect......Page 151 10.7 Fact and fiction in the world of classical physics......Page 153 10.7.1 Retardation of effects and the invariant speed......Page 155 11.1 The experiment of Elitzur and Vaidman......Page 158 11.2 Observables......Page 160 11.3 The continuous case......Page 161 11.4 Commutators......Page 162 11.6 Operators for energy and momentum......Page 163 11.7 Angular momentum......Page 164 11.8 The hydrogen atom in brief......Page 166 12.1 Spin 1/2......Page 172 12.1.1 Other bases......Page 174 12.1.2 Rotations as 2 2 matrices......Page 175 12.1.3 Pauli spin matrices......Page 178 12.2 A Stern–Gerlach relay......Page 179 12.3 Why spin?......Page 181 12.4 Beyond hydrogen......Page 182 12.5 Spin precession......Page 185 12.6 The quantum Zeno effect......Page 186 13.1 Bell's theorem: The simplest version......Page 188 13.2 "Entangled" spins......Page 190 13.2.1 The singlet state......Page 191 13.3 Reduced density operator......Page 192 13.4 Contextuality......Page 193 13.5.1 A game......Page 196 13.5.2 A fail-safe strategy......Page 197 13.6 Uses and abuses of counterfactual reasoning......Page 198 13.7 The experiment of Englert, Scully, and Walther......Page 203 13.7.1 The experiment with shutters closed......Page 204 13.7.2 The experiment with shutters opened......Page 205 13.7.3 Inuencing the past......Page 206 13.8 Time-symmetric probability assignments......Page 209 13.8.1 A three-hole experiment......Page 211 14.2 Scattering particles......Page 214 14.2.1 Indistinguishable macroscopic objects?......Page 216 14.4 Bosons are gregarious......Page 217 14.5 Fermions are solitary......Page 218 14.6 Quantum coins and quantum dice......Page 219 14.7 Measuring Sirius......Page 220 15.1 The Klein–Gordon equation......Page 224 15.2 Antiparticles......Page 225 15.3 The Dirac equation......Page 226 15.4 The Euler–Lagrange equation......Page 227 15.5 Noether's theorem......Page 229 15.6 Scattering amplitudes......Page 230 15.8 A few words about renormalization......Page 231 15.8.1 . . . and about Feynman diagrams......Page 234 15.9 Beyond QED......Page 235 15.9.2 Groups......Page 236 15.9.3 Generalizing QED......Page 237 15.9.4 QCD......Page 238 15.9.5 Electroweak interactions......Page 239 15.9.6 Higgs mechanism......Page 240 Making Sense......Page 242 16.1 Standard axioms: A critique......Page 244 16.2 The principle of evolution......Page 246 16.3 The eigenstate{eigenvalue link......Page 248 17. Interpretational strategy......Page 250 18.1 The two-slit experiment revisited......Page 252 18.1.1 Bohmian mechanics......Page 253 18.2 The importance of unperformed measurements......Page 254 18.4 The importance of detectors......Page 256 18.5 Spatiotemporal distinctions: Not all the way down......Page 257 18.7 Space......Page 259 19. The macroworld......Page 262 20.2 Scattering experiment revisited......Page 266 20.3 How many constituents?......Page 267 20.4 An ancient conundrum......Page 268 20.5 A fundamental particle by itself......Page 269 21.2 The coming into being of form......Page 270 21.3 Bottom-up or top-down?......Page 271 21.4 Whence the quantum-mechanical correlation laws?......Page 272 21.5 How are "spooky actions at a distance" possible?......Page 273 22.1 The stability of matter......Page 276 22.2 Why quantum mechanics (summary)......Page 277 22.4 Why quantum mechanics (summary continued)......Page 279 22.5 The classical or long-range forces......Page 280 22.6 The nuclear or short-range forces......Page 281 22.7 Fine tuning......Page 283 23. Quanta and Vedanta......Page 286 23.1 The central a rmation......Page 287 23.2 The poises of creative consciousness......Page 288 Appendix A. Solutions to selected problems......Page 290 Bibliography......Page 296 Index......Page 302 An Invaluable Supplement To Standard Textbooks On Quantum Mechanics, This Introduction To The General Theoretical Framework Of Contemporary Physics Focuses On Conceptual, Epistemological, And Ontological Issues. 1. Probability : Basic Concepts And Theorems. 1.1. The Principle Of Indifference. 1.2. Subjective Probabilities Versus Objective Probabilities. 1.3. Relative Frequencies. 1.4. Adding And Multiplying Probabilities. 1.5. Conditional Probabilities And Correlations. 1.6. Expectation Value And Standard Deviation -- 2. A (very) Brief History Of The Old Theory. 2.1. Planck. 2.2. Rutherford. 2.3. Bohr. 2.4. De Broglie -- 3. Mathematical Interlude. 3.1. Vectors. 3.2. Definite Integrals. 3.3. Derivatives. 3.4. Taylor Series. 3.5. Exponential Function. 3.6. Sine And Cosine. 3.7. Integrals. 3.8. Complex Numbers -- 4. A (very) Brief History Of The New Theory. 4.1. Schrodinger. 4.2. Born. 4.3. Heisenberg And Uncertainty. 4.4. Why Energy Is Quantized --^ 5. The Feynman Route To Schrodinger (stage 1). 5.1. The Rules Of The Game. 5.2. Two Slits. 5.3. Interference. 5.4. The Propagator As A Path Integral. 5.5. The Time-dependent Propagator. 5.6. A Free Particle. 5.7. A Free And Stable Particle -- 6. Special Relativity In A Nutshell. 6.1. The Principle Of Relativity. 6.2. Lorentz Transformations : General Form. 6.3. Composition Of Velocities. 6.4. The Case Against Positive K. 6.5. An Invariant Speed. 6.6. Proper Time. 6.7. The Meaning Of Mass. 6.8. The Case Against K = 0. 6.9. Lorentz Transformations : Some Implications. 6.10. 4-vectors -- 7. The Feynman Route To Schrodinger (stage 2). 7.1. Action. 7.2. How To Influence A Stable Particle? 7.3. Enter The Wave Function. 7.4. The Schrodinger Equation --^ 8. Why Quantum Mechanics? 8.1. The Classical Probability Calculus. 8.2. Why Nontrivial Probabilities? 8.3. Upgrading From Classical To Quantum. 8.4. Vector Spaces. 8.5. Compatible And Incompatible Elementary Tests. 8.6. Noncontextuality. 8.7. The Core Postulates. 8.8. The Trace Rule. 8.9. Self-adjoint Operators And The Spectral Theorem. 8.10. Pure States And Mixed States. 8.11. How Probabilities Depend On Measurement Outcomes. 8.12. How Probabilities Depend On The Times Of Measurements. 8.13. The Rules Of The Game Derived At Last -- 9. The Classical Forces : Effects. 9.1. The Principle Of Least Action. 9.2. Geodesic Equations For Flat Spacetime. 9.3. Energy And Momentum. 9.4. Vector Analysis: Some Basic Concepts. 9.5. The Lorentz Force. 9.6. Curved Spacetime. 9.7. Gravity -- 10. The Classical Forces : Causes. 10.1. Gauge Invariance. 10.2. Fuzzy Potentials. 10.3. Maxwell's Equations. 10.4. A Fuzzy Metric. 10.5. Einstein's Equation. 10.6. Aharonov-bohm Effect. 10.7 Fact And Fiction In The World Of Classical Physics -- 11. Quantum Mechanics Resumed. 11.1. The Experiment Of Elitzur And Vaidman. 11.2. Observables. 11.3. The Continuous Case. 11.4. Commutators. 11.5. The Heisenberg Equation. 11.6. Operators For Energy And Momentum. 11.7. Angular Momentum. 11.8. The Hydrogen Atom In Brief -- 12. Spin. 12.1. Spin 1/2. 12.2. A Stern-gerlach Relay. 12.3. Why Spin? 12.4. Beyond Hydrogen. 12.5. Spin Precession. 12.6. The Quantum Zeno Effect --^ 13. Composite Systems. 13.1. Bell's Theorem : The Simplest Version. 13.2. Entangled Spins. 13.3. Reduced Density Operator. 13.4. Contextuality. 13.5. The Experiment Of Greenberger, Horne, And Zeilinger. 13.6. Uses And Abuses Of Counterfactual Reasoning. 13.7. The Experiment Of Englert, Scully, And Walther. 13.8. Time-symmetric Probability Assignments -- 14. Quantum Statistics. 14.1. Scattering Billiard Balls. 14.2. Scattering Particles. 14.3. Symmetrization. 14.4. Bosons Are Gregarious. 14.5. Fermions Are Solitary. 14.6. Quantum Coins And Quantum Dice. 14.7. Measuring Sirius -- 15. Relativistic Particles. 15.1. The Klein-gordon Equation. 15.2. Antiparticles. 15.3. The Dirac Equation. 15.4. The Euler-lagrange Equation. 15.5. Noether's Theorem. 15.6. Scattering Amplitudes. 15.7. Qed. 15.8. A Few Words About Renormalization. 15.9. Beyond Qed -- 16. Pitfalls. 16.1. Standard Axioms : A Critique. 16.2. The Principle Of Evolution. 16.3. The Eigenstate-eigenvalue Link --^ 17. Interpretational Strategy -- 18. Spatial Aspects Of The Quantum World. 18.1. The Two-slit Experiment Revisited. 18.2. The Importance Of Unperformed Measurements. 18.3. Spatial Distinctions : Relative And Contingent. 18.4. The Importance Of Detectors. 18.5. Spatiotemporal Distinctions : Not All The Way Down. 18.6. The Shapes Of Things. 18.7. Space. 19. The Macroworld -- 20. Questions Of Substance. 20.1. Particles. 20.2. Scattering Experiment Revisited. 20.3. How Many Constituents? 20.4. An Ancient Conundrum. 20.5. A Fundamental Particle By Itself -- 21. Manifestation. 21.1. Creation In A Nutshell. 21.2. The Coming Into Being Of Form. 21.3. Bottom-up Or Top-down? 21.4. Whence The Quantum-mechanical Correlation Laws? 21.5. How Are Spooky Actions At A Distance Possible? -- 22. Why The Laws Of Physics Are Just So. 22.1. The Stability Of Matter. 22.2. Why Quantum Mechanics (summary). 22.3. Why Special Relativity (summary). 22.4 Why Quantum Mechanics (summary Continued). 22.5. The Classical Or Long-range Forces. 22.6. The Nuclear Or Short-range Forces. 22.7. Fine Tuning -- 23. Quanta And Vedanta. 23.1. The Central Affirmation. 23.2. The Poises Of Creative Consciousness. Ulrich Mohrhoff. Includes Bibliographical References (p. 277-281) And Index. An invaluable supplement to standard textbooks on quantum mechanics, this unique introduction to the general theoretical framework of contemporary physics focuses on conceptual, epistemological, and ontological issues. The theory is developed by pursuing the question : what does it take to have material objects that neither collapse nor explode as soon as they are formed? The stability of matter thus emerges as the chief reason why the laws of physics have the particular form that they do. The first of the book's three parts familiarizes the reader with the basics through a brief historical survey and by following Feynman's route to the Schrodinger equation. The necessary mathematics, including the special theory of relativity, is introduced along the way, to the point that all relevant theoretical concepts can be adequately grasped. Part II takes a closer look at this. As the theory takes shape, it is applied to various experimental arrangements. Several of these are central to the discussion in the final part, which aims at making epistemological and ontological sense of the theory. Pivotal to this task is an understanding of the special status that quantum mechanics attributes to measurements - without dragging in "the consciousness of the observer". Key to this understanding is a rigorous definition of "macroscopic" which, while rarely even attempted, is provided in this book An invaluable supplement to standard textbooks on quantum mechanics, this unique introduction to the general theoretical framework of contemporary physics focuses on conceptual, epistemological, and ontological issues. The theory is developed by pursuing the question: what does it take to have material objects that neither collapse nor explode as soon as they are formed? The stability of matter thus emerges as the chief reason why the laws of physics have the particular form that they do.The first of the book's three parts familiarizes the reader with the basics through a brief historical survey and by following Feynman's route to the Schrödinger equation. The necessary mathematics, including the special theory of relativity, is introduced along the way, to the point that all relevant theoretical concepts can be adequately grasped. Part II takes a closer look. As the theory takes shape, it is applied to various experimental arrangements. Several of these are central to the discussion in the final part, which aims at making epistemological and ontological sense of the theory. Pivotal to this task is an understanding of the special status that quantum mechanics attributes to measurements -- without dragging in "the consciousness of the observer." Key to this understanding is a rigorous definition of "macroscopic" which, while rarely even attempted, is provided in this book.