Consistent Quantum TheoryThis book presents what is known as the consistent quantum theory approach to quantum mechanics at a level accessible to university students in physics, chemistry, mathematics, and computer science, and can be used as a supplement to standard textbooks. It provides a clear explanation of points not yet adequately treated in traditional texts and which students find confusing, as do their teachers. The book will also be of interest to physicists and philosophers working on the foundations of quantum mechanics. |
Contents
Introduction | 1 |
12 Quantum states and variables | 2 |
13 Quantum dynamics | 3 |
14 Mathematics I Linear algebra | 4 |
15 Mathematics II Calculus probability theory | 5 |
16 Quantum reasoning | 6 |
17 Quantum measurements | 8 |
18 Quantum paradoxes | 9 |
135 Weak detectors in internal arms | 188 |
Dependent contextual events | 192 |
142 Classical analogy | 193 |
143 Contextual properties and conditional probabilities | 195 |
144 Dependent events in histories | 196 |
Density matrices | 202 |
152 Density matrix as a preprobability | 203 |
153 Reduced density matrix for subsystem | 204 |
Wave functions | 11 |
22 Physical interpretation of the wave function | 13 |
23 Wave functions and position | 17 |
24 Wave functions and momentum | 20 |
25 Toy model | 23 |
Linear algebra in Dirac notation | 27 |
32 Linear functionals and the dual space | 29 |
33 Operators dyads | 30 |
34 Projectors and subspaces | 34 |
35 Orthogonal projectors and orthonormal bases | 36 |
36 Column vectors row vectors and matrices | 38 |
37 Diagonalization of Hermitian operators | 40 |
38 Trace | 42 |
39 Positive operators and density matrices | 43 |
310 Functions of operators | 45 |
Physical properties | 47 |
42 Toy model and spin half | 48 |
43 Continuous quantum systems | 51 |
44 Negation of properties NOT | 54 |
45 Conjunction and disjunction AND OR | 57 |
46 Incompatible properties | 60 |
Probabilities and physical variables 51 Classical sample space and event algebra | 65 |
52 Quantum sample space and event algebra | 68 |
53 Refinement coarsening and compatibility | 71 |
54 Probabilities and ensembles | 73 |
55 Random variables and physical variables | 76 |
56 Averages | 79 |
Composite systems and tensor products | 81 |
62 Definition of tensor products | 82 |
63 Examples of composite quantum systems | 85 |
64 Product operators | 87 |
65 General operators matrix elements partial traces | 89 |
66 Product properties and product of sample spaces | 92 |
Unitary dynamics | 94 |
72 Unitary operators | 99 |
73 Time development operators | 100 |
74 Toy models | 102 |
Stochastic histories | 108 |
82 Classical histories | 109 |
83 Quantum histories | 111 |
84 Extensions and logical operations on histories | 112 |
85 Sample spaces and families of histories | 116 |
86 Refinements of histories | 118 |
87 Unitary histories | 119 |
The Born rule | 121 |
92 Singletime probabilities | 124 |
93 The Born rule | 126 |
94 Wave function as a preprobability | 129 |
Alpha decay | 131 |
96 Schrodingers cat | 134 |
Consistent histories | 137 |
102 Consistency conditions and consistent families | 140 |
103 Examples of consistent and inconsistent families | 143 |
104 Refinement and compatibility | 146 |
Checking consistency | 148 |
113 Initial and final projectors | 149 |
114 Heisenberg representation | 151 |
115 Fixed initial state | 152 |
116 Initial pure state Chain kets | 154 |
118 Intrinsically inconsistent histories | 157 |
Examples of consistent families 121 Toy beam splitter | 159 |
122 Beam splitter with detector | 165 |
123 Timeelapse detector | 169 |
124 Toy alpha decay | 171 |
Quantum interference | 174 |
132 Toy MachZehnder interferometer | 178 |
133 Detector in output of interferometer | 183 |
134 Detector in internal arm of interferometer | 186 |
154 Time dependence of reduced density matrix | 207 |
155 Reduced density matrix as initial condition | 209 |
156 Density matrix for isolated system | 211 |
157 Conditional density matrices | 213 |
Quantum reasoning | 216 |
Toy beam splitter | 219 |
163 Internal consistency of quantum reasoning | 222 |
164 Interpretation of multiple frameworks | 224 |
Measurements I | 228 |
172 Microscopic measurement | 230 |
173 Macroscopic measurement first version | 233 |
174 Macroscopic measurement second version | 236 |
175 General destructive measurements | 240 |
Measurements II | 243 |
182 Wave function collapse | 246 |
183 Nondestructive SternGerlach measurements | 249 |
184 Measurements and incompatible families | 252 |
185 General nondestructive measurements | 257 |
Coins and counterfactuals | 261 |
192 Quantum coins | 262 |
193 Stochastic counterfactuals | 265 |
194 Quantum counterfactuals | 268 |
Delayed choice paradox | 273 |
202 Unitary dynamics | 275 |
203 Some consistent families | 276 |
204 Quantum coin toss and counterfactual paradox | 279 |
205 Conclusion | 282 |
Indirect measurement paradox | 284 |
212 Unitary dynamics | 286 |
213 Comparing Min and Mout | 287 |
214 Delayed choice version | 290 |
215 Interactionfree measurement? | 293 |
216 Conclusion | 295 |
Incompatibility paradoxes | 296 |
222 Value functionals | 298 |
223 Paradox of two spins | 299 |
224 Truth functionals | 301 |
225 Paradox of three boxes | 304 |
226 Truth functionals for histories | 308 |
Singlet state correlations | 310 |
232 Spin correlations | 311 |
233 Histories for three times | 313 |
234 Measurements of one spin | 315 |
235 Measurements of two spins | 319 |
EPR paradox and Bell inequalities | 323 |
242 Counterfactuals and the EPR paradox | 326 |
243 EPR and hidden variables | 329 |
244 Bell inequalities | 332 |
25 Hardys paradox | 336 |
252 The first paradox | 338 |
253 Analysis of the first paradox | 341 |
254 The second paradox | 343 |
255 Analysis of the second paradox | 344 |
Decoherence and the classical limit | 349 |
262 Particle in an interferometer | 350 |
263 Density matrix | 352 |
264 Random environment | 354 |
265 Consistency of histories | 356 |
Quantum theory and reality | 360 |
272 Quantum vs classical reality | 361 |
273 Multiple incompatible descriptions | 362 |
274 The macroscopic world | 365 |
275 Conclusion | 368 |
Bibliography | 371 |
| 377 | |
| 383 | |
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Common terms and phrases
analogous apparatus assign beam splitter Boolean algebra Born rule chain operators channel classical mechanics classical physics commute complex numbers conditional probabilities consider consistency conditions consistent family corresponding counterfactual decoherence decomposition defined density matrix depend detected detector discussed in Sec dynamics earlier eigenvalues equation event algebra example family of histories framework given Hilbert space identity incompatible initial data inner product interaction interferometer interval kets linear macroscopic measurement outcome momentum mutually orthogonal notation orthonormal basis paradox particular phase space photon pointer possible pre-probability projectors quantum coin toss quantum description quantum mechanics quantum particle quantum reasoning quantum system quantum theory refinement result S₂ sample space Schrödinger's sense spin spin-half particle stochastic subspace superposition suppose t₁ t₂ tensor product toy model truth functional unitary time development unitary time evolution vector wave function collapse wave packet zero