Index

ABL rule, 375, 403, 405 birefringent retardation device, 401 absorbing boundary conditions, 172 Bloch sphere, 318 absorption of particles, 263 Bloch vector, 240 absorption times, 236 Bohm trajectory approach, 122 ac-conductance, 270 Bohm’s causal interpretation, 9, 122 acausality, 3 Bohm’s ontological interpretation, 122 acousto-optical modulation, 321 Bohmian approach, 227 action-angle variables, 221 Bohmian quantum mechanics, 192 additivity on disjoint regions, 157 Bohr–Wigner uncertainty relation, 80 Aharonov–Bohm energy measurement Boltzmann’s constant, 252 model, 73 Born’s probability interpretation, 8 anti-Zeno effect, 107, 115 Breit–Wigner resonance, 50 apparatus as part of the quantum Brownian motion, 188 system, 105 BSM approach, 20, 132, 226, 236 arrival states, 297 arrival time, 10, 156, 172, 175, 178, 179, calibration, 225, 226 223, 228–230, see also time of arrival canonical commutation relation, 221 arrival time measurement, 307 canonical time operator, 71 arrival-arrival correlation function, 301 carrier source, 265 arrival-time distribution, 125, see also Cauchy–Riemann conditions, 236 Kijowski’s distribution causality, 17, 249, 311 arrow of time, 404 causality conditions, 51 Arthurs–Kelly model, 93, 229 characteristic evolution timescale, 249 atomic clocks, 248 characteristic times, 59, 219, 220, 231 atomic orbit, 1 chiral molecules, 110 average passage instant, 44 classical Brownian motion, 163 clock, 183, 218, 219, 305 B¨uttiker–Landauer time, 21, 237, 315, clock coordinates, 220 316, 320 clock Hamiltonian, 223 back reaction, 228 clock precision, 218 back-action, 320 clock synchronization, 218 backflow effect, 14, 131 clock-particle system, 219 bandwidth, 104 clock-system complex, 219 barrier penetration, 99 clock-time operator, 222 beat envelope, 352 clocked dwell time, 225, 226 Bell inequality, 96 clocked Klein–Gordon equation, 210 Bell’s theorem, 17 clocked Schr¨odinger equation, 184, 186 Bell-type measurement, 373 coarse grained amplitude, 186 414 Index coarse graining, 184, 214 dissipative term, 252 coarse-grained projector, 157 duration, 2, 218, 227 collapse of the wave function, 6, 10, 18 dwell time, 20, 40, 42–44, 49, 70, 128, complex momentum, 39 129, 134, 192, 223, 225, 226, 234, complex potential, 12, 14, 51, 172, 263 235, 257, 273, 313 complex time, 21, 190, 191, 200, 236, dwell time measurement, 313 258, 320, 328, 338 condition of decoherence, 158 Einstein’s photon box, 92, 93 conditional measurement, 319 elements of reality, 404, 406 conditional time, 320 emissivity, 258, 265 conjugate variables, 2 emittance matrix, 270 consistency condition, 161 entangled state, 242 consistent histories, 18, 158 environment, 156, 159, 167, 168, 170, contextuality, 377 174, 178 continuous measurement, 17, 105, 173, EPR-type state, 373 174 Ersak equation, 109 continuous monitoring of decay, 105 evanescent electromagnetic waves, 218, correspondence-principle limit, 305 239 cost function, 242 evanescent photons, 218 counterfactuals, 401, 405, 406 evanescent wave, 59, 60, 256, 312, 327, cross-correlation, 309 333, 335, 354 crossing propagator, 161 Event Enhanced Quantum Theory crossing state, 15, 298 (EEQT), 18 crossing time, 156, 163, 174, 178 event time, 69, 85, 87, 88, 91, 92, 96, 220, 223 d’Alambert equation, 327 events time distribution, 8 De Broglie wavelength, 195, 198, 321 experimental tests of interaction times, decay, 100, 217, 218, 228 321 decoherence, 10, 106, 242, 248, 250, exponential decay, 53–55, 81, 83, 99, 251, 321 104, 110, 228 decoherence functional, 158 external time, 69, 70 decoherent histories approach, 153, extrapolated phase time, 46, 47, 232, 155–157, 159, 162, 172, 176–179 234 decohering fluctuations, 252 degree of fuzziness, 162 Faraday clock, 238 delay time, 48, 308, see also time delay Faraday constant, 239 density fluctuations, 274 Faraday polarization, 239 density of states, 50, 106, 257 Faraday rotation, 239 density of states matrix, 275 Feynman checkerboard model, 121 detector, 44, 156, 169, 170, 172 Feynman–Vernon influence functional detector model, 12, 47, 169 phase, 159 diffracted waves, 334, 354 fine-grained projector, 157 diffraction in time, 10, 94 Fleming decay bound, 109 Dirac electron, 122, 227 fluctuation-dissipation theorem, 252 Dirac equation, 148, 235, 365 Fokker–Planck equation, 160, 163 Dirac particles, 226 Fokker–Planck propagator, 163–165, Dirac velocity operator, 122 168 dispersion, 327, 350 forerunner, 61, 327, 331, 342 dispersion relation, 59, 327 fractal behavior, 193 Index 415 freely falling particle, 89 Josephson junction, 317, 326 frequency-band-limited detection, 61 jump time, 99, 101–104, 108, 116 frequency-band-limited wave, 61, 313 kick-clock, 245 gedanken experiment, 315, 390 Kijowski’s distribution, 15, 18, 19, 44, generalized Bardeen conductances, 267 82, 134, 172, 178, 229, 230, 244, 279, generalized two-state vector, 374, 375, 280, 293, 297, 299 378, 379 Klein–Gordon equation, 210, 330, 365 Golden rule, 99, 106 Kochen–Specker’s theorem, 17 Goos–H¨anchen effect, 318 Kramers–Kronig relations, 311 gravitational potential, 395 gravitational time dilation, 394, 395 Landau–Zener tunneling, 107 group delay, 308, 327, 336, 354 Langevin equation, 160 group delay time, 309 large time behaviour of survival group velocity, 59, 60, 327, 331, 332, probability, 57 349, 362, 366 Larmor clock, 21, 22, 196, 200, 214, guiding field, 122 230, 256, 257, 315 guiding wave, 125, 143 Larmor clock: optical, 218 Larmor frequency, 197, 256, 316 Hanbury–Brown–Twiss interferometry, Larmor precession, 22, 218, 226, 233, 309 258, 266, 318, 322 Hardy’s paradox, 391 Larmor time, 316, 322 Hartman effect, 22, 46, 312, 327 laser-cooled atoms, 321 Helmholtz equation, 327 Levinson’s theorem, 39, 51 heteronuclear coupling, 402 lifetime, 11, 48, 70, 79, 82, 95, 99, 103, heterostructures, 317 104, 115, 217, 223, 227 hidden variable theory, 9 lifetime-linewidth relation, 82, 84 hierarchy of density of states, 265 Lindblad form, 171 Hilgevoord–Uffink relation, 84, 92, 95, Lippmann–Schwinger equation, 33 96 local density of states, 256, 258, 264, homogeneous history, 157 265, 267, 269, 271 local dwell time, 258, 266 ideal quantum measurement, 375 local expectation value, 122 imaginary time, 328 local kinetic energy, 310 incident flux, 43, 234 local Larmor clock, 234, 256 incompleteness of quantum mechanics, local Larmor time, 276 3 locality, 131, 140, 143 inductive response, 272 lowering operator, 169 inelastic scattering, 268, 269 information transfer, 313 M¨oller operator, 32 inhomogenous history, 159 Mach-Zehnder interferometer, 401 initial error width, 249 magneto-optic trap, 321 injectivity, 258, 264 magnetorefractive layered structure, intensity-intensity interferometry, 309 239 intrinsic time, 69, 70 Mandelstam–Tamm relation, 7, 78, 79, inverse scattering, 14 82, 84, 85, 92, 93, 96 many-worlds interpretation, 10, 117, joint measurement of non commuting 410 observables, 22 Markovian approximation, 171, 250 416 Index massive shell, 397 path decomposition, 184 master equation, 18, 166, 171, 243, 250 path decomposition expansion, 161 matrix mechanics, 1, 2 path integral, 100, 183, 226 maximally symmetric operator, 284, path-integral approach, 21, 155, 184 285, 289, 291, 293 Pauli matrices, 110, 231 measurement of particle position, 124 Pauli’s “theorem”, 5, 12, 13, 71, 89, 90, measurement theory, 214 279, 290, 297 measuring device, 307, 381, 397 penetration time, 341, 352 metastable state, 104 period of applicability, 249 meter, 188 phase delay, 336, 354, 357 micromaser clock, 238, 240 phase shift, 224, 308 millisecond regime, 321 phase space, 229 monochromatic front, 61, 63 phase time, 46, 132, 237, 241 Moshinsky’s shutter, 77 phase velocity, 60, 357, 362 MOT, 321 photon detection, 19 Mott’s analysis of particle tracks, 177 photon number operator, 240 mover, 397 photonic bandgap, 312 photonic tunneling, 183, 240 Naimark’s theorem, 279, 287, 293 plasmons, 202 nano-technology, 183 Poincar´e recurrence, 116 negative delay, 51, 311 Poincar´e sphere, 318 negative energies, 13, 294 pointer, 189, 307 negative time, 22, 193, 258 pointer of a clock, 219, 228 nodal points, 235 pointer variables, 230 non-commuting observables, 20, 258 position-momentum uncertainty nonlinear autocorrelation, 311 relation, 4 nonlinear polarisability, 309 Positive Operator Valued Measure nonlocality, 132, 143, 321 (POVM), 6, 15, 16, 72, 74, 76, 86–91, nuclear spins, 402 221, 241, 279, 281, 283–285, 288, 293, 294 observable time, 71 post-selected system, 371 open system, 18 potentiality, 7, 8, 18 operational model, 22 pre-selected system, 371 optical clock, 239 preparation time and energy, 77 optical potential, 269 principle of indeterminacy, 4 oscillating potential, 19, 61, 258, 315 probability current density, 123, 135 pair creation, 124 probability functional, 249, 251 parametric down-conversion, 309 probability interpretation, 3 parametric time, 230, 248–251 product rule, 377 partial density of states, 256, 261, 263 projector approach of Brouard, Sala particle equation of motion, 122 and Muga, 132, see also BSM particle trajectories, 121 approach particle-antiparticle pair, 124, 213 protective measurements, 140, 407, 409 particle-meter coupling, 189 pulse-peak arrival time, 318 particle-meter Schr¨odinger equation, pump-probe paradigm, 306 214 passage time, 99–101, 109, 110, 116, quantile approach, 136 117 quantum beats, 96 Index 417 quantum Brownian motion, 159, 163, Schr¨odinger’s wave picture, 4 166, 167 Schr¨odinger equation, 327 quantum capacitance, 272 screen observable, 88 quantum clock, 10, 84, 85, 155, 218, 230 second quantization, 300 quantum clock approach, 227 sensitivity, 262, 263, 266 quantum cosmology, 153, 175 short time behaviour, 55 quantum decay, 100 signal velocity, 60, 328, 331, 341, 366 quantum gravitational time machine, simultaneous measurement, 229 396 slab, 239 quantum gravity, 153 small noise approximation, 250 quantum jump, 1, 3, 4, 7–9, 18, 99, 100, Smith’s lifetime matrix, 48 115, 116, 243, 244 sojourn time, 41, 42, 313, see also dwell quantum jump experiments, 100 time quantum measurement theory, 105, 111 source boundary condition, 62 quantum mechanics with sources, 13 source with a sharp onset, 61, 63 quantum noise, 250 sources in quantum mechanics, 12 quantum operating system, 397 space-time coarse grainings, 160 quantum trajectory, 18 space-time event, 218, 220 quasidistributions, 229 space-time probabilities, 155 space-time regions, 154 Rabi oscillation, 240 special states, 112 raising operator, 169 spectral theorem, 50, 287 Raman–Nath approximation, 241 spectrogram, 61 real clocks, 248 spin polarization, 256 real measurement, 305 spin precession, 256 reality of wave function, 144 spin rotation, 256 reduction of the wave function, 6 spin-dependent arrival-time distribu- reflection amplitude, 36, 225 tion, 137 reflection probability amplitude, 35 reflection time, 127, 227 spinor, 123, 137, 232, 260 reflectionprobability, 263 Split-Operator-Method, 247 refractive index, 239 spontaneous collapse, 18 relative error correlation time, 249 stationary-phase approximation, 48, relativistic causality, 23, 311 308, 311, 327 relativistic traversal time, 210 step potential, 31 reshaping of wave packets, 132 Stern-Gerlach experiment, 382, 392 resonance tunneling, 199 stimulated Raman transitions, 321 response time, 105, 106 stochastic model, 330, 352, 365 restricted Green function, 160 stochastic motion, 337 retrodiction, 373 stochastic processes, 249, 250 Stokes parameters, 318 Salecker–Wigner clock, 85, 220, 223, stopwatch, 11, 183, 184, 218, 223 226, 227, 230, 246 streak camera method, 308 scanning tunneling microscope (STM), strong measurement, 383 267 super-oscillations, 371 scattering matrix, 30, 34, 35, 37, 38, 40, superluminal arrival times, 310 44, 257, 259, 263 superluminal effects, 22, 60, 199, 205, scattering phase shift, 11, 311 206, 210, 212, 213, 218, 226, 227, scattering states, 9, 20, 259 311, 326, 331, 333, 337, 358, 362, 366 418 Index superluminal transmission time, 132 time-frequency representation of the superposition of time evolutions, 400 wave function, 61 survival probability, 53, 54, 82, 116 time-of-arrival distribution, 15, 178 systematic projector approach, 20 time-of-arrival operator, 15, 279, 281, 289, 294, 297 table of characteristic times, 101 time-to-amplitude convertors, 306 telegrapher’s equation, 328, 338, 352 total reflection, 335, 341, 360 temporal diffraction experiments, 96 toy model, 228 temporal interference, 93, 94 trajectory approach, 227 transition elements, 339 thermal bath, 248, 252 transition probability, 1, 9 thermodynamic arrow of time, 115 transition rate, 3, 106 thought experiments, 4 transition time between isomers, 110 time as a parameter, 6 transmission amplitude, 36, 225 time as an observable, 6, 16, 85, 86, 88, transmission probability, 203, 248, 263, 89, 91 276 time asymmetry, 403 transmission probability amplitude, 35 time covariance, 15, 17, 88, 230, 297 transmission speed, 226, 235 time delay, 11, 41, 44, 46, 48, 49, 51, transmission time, 22, 127, 227 70, 232, 306, 307, 309, 334 traversal time, 19, 61, 62, 183, 202, 314 time indeterminacy, 93 traversal time for tunneling, 276 time machine, 369, 393, 395 traversal-time operator, 214 time measurement, 84, 88, 305, 322 tunneling, 53, 103, 308, 326 time of arrival, 12, 13, 15, 18, 19, 93, tunneling electrons, 218, 239 99, 147, 223, 228–230, 279–281, 289, tunneling semiconductor, 218 294, 297, 298, 302, 327 tunneling time, 19, 99, 101, 104, 116, time of arrival for identical particles, 117, 128, 184, 218, 258, 326, 338, 300 360, 365 time of event, 3, 10, 17 two-level atom, 106, 240 time of first-photon detection, 244 two-level system detector, 169, 229 time of flight, 10, 50, 71, 218–220, 223, two-photon interferometer, 310 224, 313 two-slit diffraction experiment, 137, 186 time of passage, 79, 87, 93–95, 97, 220 two-state vector, 14, 369, 370, 372, 375, time of presence, 127 381, 408 time operator, 5, 12, 71, 88, 91, 155, 294, 305 ultra-cold atoms, 19, 321 time operator of Aharonovand Bohm, ultrafast laser, 306 12, 279, 289, 290, 292–294, 296, 297 uncertainty relation, 4 time resolution, 223 uniqueness of Bohm particle velocity, time resolution of a clock, 220, 225, 228 123 time reversal invariance, 35 uniqueness of probability current time reversibility, 319 density, 123 time symmetry, 369, 402, 404, 406 voltage probe, 268 time–energy entangled photons, 96 Von Neumann equation, 250 time–energy uncertainty relation, 69–96 Von Neumann measurement, 214, 375 time-dependent clock, 245 time-energy uncertainty relation, 4, 5, wave mechanics, 2 7, 11, 101, 104, 117, 155, 219, 279, wave-particle duality, 125, 326 299 waveguides, 308 Index 419 weak detector-environment coupling, Wigner function, 136, 167, 168, 176, 171 177 weak measurement, 22, 146, 190, 191, Wigner’s arrival-time distribution, 127 200, 226, 319, 328, 369, 379, 380, Wigner–Smith delay time matrix, 275 383, 385, 386, 390, 392, 402 Wilson cloud chamber, 177 weak value, 379, 381, 382 WKB approximation, 61, 234, 275, 315 weak vector, 392 weakly decoherent histories, 158 weakness condition, 402 Zeeman splitting, 240 weakness of the measurement, 385 Zeno effect, 16, 19, 101, 102, 105, 107, Wheeler–DeWitt equation, 153, 175, 110, 228 177 Zeno paradox, 188 white-light interference, 308 Zeno time, 102, 106, 110, 115, 116