Bohmian Mechanics (eBook)

Detlef Dürr studied physics in Münster, Germany, where he obtained his PhD in physics in 1978. After four post-doc years at Rutgers in the group of Joel Lebowitz working with Sheldon Goldstein, he was awarded a Heisenberg fellowship (1985-1989), during which he joined forces with Sheldon Goldstein and Nino Zanghì to develop the statistical analysis of Bohmian mechanics - a cooperation which continues to this day. In 1989 he became professor of mathematics at the University of Munich. His research interests are non-equilibrium statistical mechanics, foundations of statistical mechanics, Bohmian mechanics and the foundations of quantum theory.Stefan Teufel studied physics in Munich, Germany, where he was awarded his PhD in mathematics in 1998. His PhD advisor was Detlef Dürr. After one year as a post doc at Rutgers with Sheldon Goldstein he joined the group of Herbert Spohn at the Technical University of Munich. In 2004 he became lecturer in mathematics at Warwick University, UK. Since 2005 he has been full professor of mathematics at the University of Tübingen. His research interests include adiabatic and semiclassical problems in quantum dynamics, exponential asymptotics and Bohmian mechanics.

Preface 5
Contents 9
Introduction 13
Ontology: What There Is 13
Extracts 13
In Brief: The Problem of Quantum Mechanics 16
In Brief: Bohmian Mechanics 18
Determinism and Realism 21
References 22
Classical Physics 23
Newtonian Mechanics 24
Hamiltonian Mechanics 25
Hamilton--Jacobi Formulation 36
Fields and Particles: Electromagnetism 38
No fields, Only Particles: Electromagnetism 46
On the Symplectic Structure of the Phase Space 50
References 54
Symmetry 55
Chance 60
Typicality 62
Typical Behavior. The Law of Large Numbers 65
Statistical Hypothesis and Its Justification 74
Typicality in Subsystems:Microcanonical and Canonical Ensembles 77
Irreversibility 91
Typicality Within Atypicality 92
Our Atypical Universe 100
Ergodicity and Mixing 101
Probability Theory 107
Lebesgue Measure and Coarse-Graining 107
The Law of Large Numbers 113
References 118
Brownian motion 119
Einstein's Argument 120
On Smoluchowski's Microscopic Derivation 124
Path Integration 128
References 129
The Beginning of Quantum Theory 130
References 136
Schrödinger's Equation 137
The Equation 137
What Physics Must Not Be 143
Interpretation, Incompleteness, and =||2 147
References 151
Bohmian Mechanics 152
Derivation of Bohmian Mechanics 154
Bohmian Mechanics and Typicality 158
Electron Trajectories 160
Spin 165
A Topological View of Indistinguishable Particles 173
References 178
The Macroscopic World 179
Pointer Positions 179
Effective Collapse 185
Centered Wave packets 189
The Classical Limit of Bohmian Mechanics 192
Some Further Observations 197
Dirac Formalism, Density Matrix,Reduced Density Matrix, and Decoherence 197
Poincaré Recurrence 204
References 206
Nonlocality 207
Singlet State and Probabilities for Anti-Correlations 211
Faster Than Light Signals? 214
References 215
The Wave Function and Quantum Equilibrium 216
Measure of Typicality 216
Conditional Wave Function 218
Effective Wave function 221
Typical Empirical Distributions 223
Misunderstandings 228
Quantum Nonequilibrium 229
References 230
From Physics to Mathematics 231
Observables. An Unhelpful Notion 231
Who Is Afraid of PVMs and POVMs? 237
The Theory Decides What Is Measurable 245
Joint Probabilities 246
Naive Realism about Operators 248
Schrödinger's Equation Revisited 249
What Comes Next? 252
References 253
Hilbert Space 254
The Hilbert Space L2 256
The Coordinate Space 2 258
Fourier Transformation on L2 261
Bilinear Forms and Bounded Linear Operators 271
Tensor Product Spaces 274
References 281
The Schrödinger Operator 282
Unitary Groups and Their Generators 282
Self-Adjoint Operators 287
The Atomistic Schrödinger Operator 297
References 301
Measures and Operators 302
Examples of PVMs and Their Operators 306
Heisenberg Operators 308
Asymptotic Velocity and the Momentum Operator 309
The Spectral Theorem 314
The Dirac Formalism 314
Mathematics of the Spectral Theorem 316
Spectral Representations 325
Unbounded Operators 327
Unitary Groups 335
H0=-/2 336
The Spectrum 344
References 347
Bohmian Mechanics on Scattering Theory 348
Exit Statistics 349
Asymptotic Exits 356
Scattering Theory and Exit Distribution 359
More on Abstract Scattering Theory 361
Generalized Eigenfunctions 364
Towards the Scattering Cross-Section 371
The Scattering Cross-Section 372
Born's Formula 373
Time-Dependent Scattering 375
References 381
Epilogue 382
References 383
Bibliography 384
Index 389