Contents 11.3 Time-dependent fluctuations 279 11.3.1 Fluctuation-dissipation theorem 280 11.3.2 Kubo formula 11.4 Heisenberg uncertainty relation 282 11.4.1 Density matrix of a damped oscillator 12 Transitions and dissipatio 12.1 Complicating the damped oscillator:Towards a qubit 12.1.1 Delocalization criterion 12 2 Spin-boson model 12 3 Shifted oscillators 12 4 Shake e-up and P(E 2.5 Orthogonality catastrophe 12.6 Workout of P(E) 12.7 Transition rates and delocalization 38 12.8 Classification of environments 12.8.1 Subohmic 12.820hmic 12.8.3 Superohmic 12.9 Vacuum as an environment Exercises 310 Solutions 312 PARTV RELATIVISTIC QUANTUM MECHANICS 315 13 Relativisticquantum mechanics 13.1 Principles of the theory of relativity 13.1.1 Lorentz transformation 13.1.2 Minkowski spacetime 13.1.3 The Minkowski metric 323 13.1 Four-vectors 13.2 Dirac equation 13.2.1 Solutions of the Dirac equation 13.2.2 Second quantization 13.2.3 Interaction with the electromagnetic field 3.3.1 13.3.2 Perturbation theory and divergences 13.4 Renormalization Exercises 348 Solutions 351 Index 352 ix Contents 11.3 Time-dependent fluctuations 279 11.3.1 Fluctuation–dissipation theorem 280 11.3.2 Kubo formula 281 11.4 Heisenberg uncertainty relation 282 11.4.1 Density matrix of a damped oscillator 283 Exercises 286 Solutions 288 12 Transitions and dissipation 290 12.1 Complicating the damped oscillator: Towards a qubit 290 12.1.1 Delocalization criterion 292 12.2 Spin–boson model 292 12.3 Shifted oscillators 294 12.4 Shake-up and P(E) 296 12.5 Orthogonality catastrophe 297 12.6 Workout of P(E) 298 12.7 Transition rates and delocalization 301 12.8 Classification of environments 302 12.8.1 Subohmic 304 12.8.2 Ohmic 305 12.8.3 Superohmic 306 12.9 Vacuum as an environment 307 Exercises 310 Solutions 312 PART V RELATIVISTIC QUANTUM MECHANICS 315 13 Relativistic quantum mechanics 317 13.1 Principles of the theory of relativity 317 13.1.1 Lorentz transformation 318 13.1.2 Minkowski spacetime 321 13.1.3 The Minkowski metric 323 13.1.4 Four-vectors 324 13.2 Dirac equation 326 13.2.1 Solutions of the Dirac equation 330 13.2.2 Second quantization 333 13.2.3 Interaction with the electromagnetic field 336 13.3 Quantum electrodynamics 337 13.3.1 Hamiltonian 338 13.3.2 Perturbation theory and divergences 339 13.4 Renormalization 343 Exercises 348 Solutions 351 Index 352