|
|
|
|
With tremendous relief, I have finished writing a book, and will return to putting out This Week's Finds on a roughly weekly basis. Let me briefly describe my book, which took so much more work than I had expected... and then let me start catching up on listing some of the stuff that's cluttering my desk!
1) Gauge Fields, Knots and Gravity, by John Baez and Javier de Muniain, World Scientific Press, to appear in summer 1994.
This book is based on a seminar I taught in 1992-93. We start out assuming the reader is familiar with basic stuff - Maxwell's equations, special relativity, linear algebra and calculus of several variables - and try to prepare the reader to understand recent work on quantum gravity and its relation to knot theory. It proved difficult to do this well in a mere 460 pages. Lots of tantalizing loose ends are left dangling. However, there are copious references so that the reader can pursue various subjects further.
Part 1. Electromagnetism Chapter 1. Maxwell's Equations Chapter 2. Manifolds Chapter 3. Vector Fields Chapter 4. Differential Forms Chapter 5. Rewriting Maxwell's Equations Chapter 6. DeRham Theory in Electromagnetism Part 2. Gauge Fields Chapter 1. Symmetry Chapter 2. Bundles and Connections Chapter 3. Curvature and the Yang-Mills Equations Chapter 4. Chern-Simons Theory Chapter 5. Link Invariants from Gauge Theory Part 3. Gravity Chapter 1. Semi-Riemannian Geometry Chapter 2. Einstein's Equations Chapter 3. Lagrangians for General Relativity Chapter 4. The ADM Formalism Chapter 5. The New Variables
2) Quantum Theory: Concepts and Methods, by Asher Peres, Kluwer Academic Publishers, 1994.
As Peres notes, there are many books that teach students how to solve quantum mechanics problems, but not many that tackle the conceptual puzzles that fascinate those interested in the foundations of the subject. His book aims to fill this gap. Of course, it's impossible not to annoy people when writing about something so controversial; for example, fans of Everett will be distressed that Peres' book contains only a brief section on "Everett's interpretation and other bizarre interpretations". However, the book is clear-headed and discusses a lot of interesting topics, so everyone should take a look at it.
Schroedinger's cat, Bell's inequality and Wigner's friend are old chestnuts that everyone puzzling over quantum theory has seen, but there are plenty of popular new chestnuts in this book too, like "quantum cryptography", "quantum teleportation", and the "quantum Zeno effect", all of which would send shivers up and down Einstein's spine. There are also a lot of gems that I hadn't seen, like the Wigner-Araki-Yanase theorem. Let me discuss this theorem a bit.
Roughly, the WAY theorem states that it is impossible to measure an operator that fails to commute with an additive conserved quantity. Let me give an example to clarify this and then give the proof. Say we have a particle with position q and momentum p, and a measuring apparatus with position Q and momentum P. Let's suppose that the total momentum p + P is conserved - which will typically be the case if we count as part of the "apparatus" everything that exerts a force on the particle. Then as a consequence of the WAY theorem we can see that (in a certain sense) it is impossible to measure the particle's position q; all we can measure is its position relative to the apparatus, q - Q.
Of course, whenever a "physics theorem" states that something is impossible one must peer into it and determine the exact assumptions and the exact result! Lots of people have gotten in trouble by citing theorems that seem to show something is impossible without reading the fine print. So let's see what the WAY theorem really says!
It assumes that the Hilbert space for the system is the tensor product of the Hilbert space for the thing being observed - for short, let's call it the "particle" - and the Hilbert space for the measuring apparatus. Assume also that A and B are two observables belonging to the observed system, while C is an observable belonging to the measuring apparatus; suppose that B + C is conserved, and let's try to show that we can only measure A if it commutes with B. (Our assumptions automatically imply that A commutes with C, by the way.)
So, what do we mean when we speak of "measuring A"? Well, there are various things one might mean. The simplest is that if we start the combined system in some tensor product state u(i) ⊗ v, where v is the "waiting and ready" state of the apparatus and u(i) is a state of the observed system that's an eigenvector of A:
Au(i) = a(i)u(i),
then the unitary operator U corresponding to time evolution does the following:
U(u(i) ⊗ v) = u(i) ⊗ v(i)
where the state v(i) of the apparatus is one in which it can be said to have measured the observable A to have value a(i). E.g., the apparatus might have a dial on it, and in the state v(i) the dial reads "a(i)". Of course, we are really only justified in saying a measurement has occured if the states v(i) are distinct for different values of i.
Note: here the WAY theorem seems to be restricting itself to nondestructive measurements, since the observed system is remaining in the state u(i). If you go through the proof you can see to what extent this is crucial, and how one might modify the theorem if this is not the case.
Okay, we have to show that we can only "measure A" in this sense if A commutes with B. We are assuming that B + C is conserved, i.e.,
U*(B + C)U = B + C.
First note that
<u(i), [A,B] u(j)> = (a(i) - a(j)) <u(i), Bu(j)>.
On the other hand, since A and B only act on the Hilbert space for the particle, we also have
<u(i), [A,B] u(j)> = <u(i) ⊗ v, [A,B] u(j) ⊗ v>
= <u(i) ⊗ v, [A,B+C] u(j) ⊗ v>
= (a(i) - a(j)) <u(i) ⊗ v, (B+C) u(j) ⊗ v>.
It follows that if a(i) - a(j) isn't zero,
<u(i), Bu(j)> = <u(i) ⊗ v, (B+C) u(j) ⊗ v>
= <u(i) ⊗ v, U*(B + C)U u(j) ⊗ v>
= <u(i) ⊗ v(i), (B + C) u(j) ⊗ v(j)>
= <u(i), Bu(j)> <v(i), v(j)> + <u(i), u(j)> <v(i), C v(j)>
but the second term vanishes since u(i) are a basis of eigenvectors and u(i) and u(j) correspond to different eigenvalues, so
<u(i), Bu(j)> = <u(i), Bu(j)> <v(i), v(j)>
which means that either <v(i), v(j)> = 1, hence v(i) = v(j) (since they are unit vectors), so that no measurement has really been done, OR that <u(i), B u(j)> = 0, which means (if true for all i,j) that A commutes with B.
So, we have proved the result, using one extra assumption that I didn't mention at the start, namely that the eigenvalues a(i) are distinct.
I can't say that I really understand the argument, although it's easy enough to follow the math. I will have to ponder it more, but it is rather interesting, because it makes more precise (and general) the familiar notion that one can't measure absolute positions, due to the translation-invariance of the laws of physics; this translation invariance is of course what makes momentum be conserved. (What I just wrote makes me wonder if someone has shown a classical analog of the WAY theorem.)
Anyway, here's the table of contents of the book:
Chapter 1: Introduction to Quantum Physics
1-1. The downfall of classical concepts 3
1-2. The rise of randomness 5
1-3. Polarized photons 7
1-4. Introducing the quantum language 9
1-5. What is a measurement? 14
1-6. Historical remarks 18
1-7. Bibliography 21
Chapter 2: Quantum Tests
2-1. What is a quantum system? 24
2-2. Repeatable tests 27
2-3. Maximal quantum tests 29
2-4. Consecutive tests 33
2-5. The principle of interference 36
2-6. Transition amplitudes 39
2-7. Appendix: Bayes's rule of statistical inference 45
2-8. Bibliography 47
Chapter 3: Complex Vector Space
3-1. The superposition principle 48
3-2. Metric properties 51
3-3. Quantum expectation rule 54
3-4. Physical implementation 57
3-5. Determination of a quantum state 58
3-6. Measurements and observables 62
3-7. Further algebraic properties 67
3-8. Quantum mixtures 72
3-9. Appendix: Dirac's notation 77
3-10. Bibliography 78
Chapter 4: Continuous Variables
4-1. Hilbert space 79
4-2. Linear operators 84
4-3. Commutators and uncertainty relations 89
4-4. Truncated Hilbert space 95
4-5. Spectral theory 99
4-6. Classification of spectra 103
4-7. Appendix: Generalized functions 106
4-8. Bibliography 112
Chapter 5: Composite Systems
5-1. Quantum correlations 115
5-2. Incomplete tests and partial traces 121
5-3. The Schmidt decomposition 123
5-4. Indistinguishable particles 126
5-5. Parastatistics 131
5-6. Fock space 137
5-7. Second quantization 142
5-8. Bibliography 147
Chapter 6: Bell's Theorem
6-1. The dilemma of Einstein, Podolsky, and Rosen 148
6-2. Cryptodeterminism 155
6-3. Bell's inequalities 160
6-4. Some fundamental issues 167
6-5. Other quantum inequalities 173
6-6. Higher spins 179
6-7. Bibliography 185
Chapter 7: Contextuality
7-1. Nonlocality versus contextuality 187
7-2. Gleason's theorem 190
7-3. The Kochen-Specker theorem 196
7-4. Experimental and logical aspects of inseparability 202
7-5. Appendix: Computer test for Kochen-Specker contradiction 209
7-6. Bibliography 211
Chapter 8: Spacetime Symmetries
8-1. What is a symmetry? 215
8-2. Wigner's theorem 217
8-3. Continuous transformations 220
8-4. The momentum operator 225
8-5. The Euclidean group 229
8-6. Quantum dynamics 237
8-7. Heisenberg and Dirac pictures 242
8-8. Galilean invariance 245
8-9. Relativistic invariance 249
8-10. Forms of relativistic dynamics 254
8-11. Space reflection and time reversal 257
8-12. Bibliography 259
Chapter 9: Information and Thermodynamics
9-1. Entropy 260
9-2. Thermodynamic equilibrium 266
9-3. Ideal quantum gas 270
9-4. Some impossible processes 275
9-5. Generalized quantum tests 279
9-6. Neumark's theorem 285
9-7. The limits of objectivity 289
9-8. Quantum cryptography and teleportation 293
9-9. Bibliography 296
Chapter 10: Semiclassical Methods
10-1. The correspondence principle 298
10-2. Motion and distortion of wave packets 302
10-3. Classical action 307
10-4. Quantum mechanics in phase space 312
10-5. Koopman's theorem 317
10-6. Compact spaces 319
10-7. Coherent states 323
10-8. Bibliography 330
Chapter 11: Chaos and Irreversibility
11-1. Discrete maps 332
11-2. Irreversibility in classical physics 341
11-3. Quantum aspects of classical chaos 347
11-4. Quantum maps 351
11-5. Chaotic quantum motion 353
11-6. Evolution of pure states into mixtures 369
11-7. Appendix: PostScript code for a map 370
11-8. Bibliography 371
Chapter 12: The Measuring Process
12-1. The ambivalent observer 373
12-2. Classical measurement theory 378
12-3. Estimation of a static parameter 385
12-4. Time-dependent signals 387
12-5. Quantum Zeno effect 392
12-6. Measurements of finite duration 400
12-7. The measurement of time 405
12-8. Time and energy complementarity 413
12-9. Incompatible observables 417
12-10. Approximate reality 423
12-11. Bibliography 428
3) Loop representations, by Bernd Bruegmann, Max Planck Institute preprint, available as gr-qc 9312001.
This is a nice review article on loop representations of gauge theories. Anyone wanting to jump into the loop representation game would be well advised to start here.
4) The fate of black hole singularities and the parameters of the standard models of particle physics and cosmology, by Lee Smolin, available in LaTeX format as gr-qc/9404011.
This is about Smolin's "evolutionary cosmology" scenario, which I already discussed in week31. Let me just quote the abstract:
A cosmological scenario which explains the values of the parameters of the standard models of elementary particle physics and cosmology is discussed. In this scenario these parameters are set by a process analogous to natural selection which follows naturally from the assumption that the singularities in black holes are removed by quantum effects leading to the creation of new expanding regions of the universe. The suggestion of J. A. Wheeler that the parameters change randomly at such events leads naturally to the conjecture that the parameters have been selected for values that extremize the production of black holes. This leads directly to a prediction, which is that small changes in any of the parameters should lead to a decrease in the number of black holes produced by the universe. On plausible astrophysical assumptions it is found that changes in many of the parameters do lead to a decrease in the number of black holes produced by spiral galaxies. These include the masses of the proton, neutron, electron and neutrino and the weak, strong and electromagnetic coupling constants. Finally,this scenario predicts a natural time scale for cosmology equal to the time over which spiral galaxies maintain appreciable rates of star formation, which is compatible with current observations that Ω = .1-.2
© 1994 John Baez
baez@math.removethis.ucr.andthis.edu
|
|
|
|