Also available at http://math.ucr.edu/home/baez/week33.html
May 10, 1994
This Week's Finds in Mathematical Physics - Week 33
John Baez
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, ISBN 0-7923-2549-4.
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) x 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) x v) = u(i) x 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
__ = (a(i) - a(j)) ____.
On the other hand, since A and B only act on the Hilbert space for the
particle, we also have
____ = ____
= ____
= (a(i) - a(j)) ____.
It follows that if a(i) - a(j) isn't zero,
____ = ____
= ____
= ____
= ____ + ____
but the second term vanishes since u(i) are a basis of eigenvectors
and u(i) and u(j) correspond to different eigenvalues, so
____ = ____
which means that either = 1, hence v(i) = v(j)
(since they are unit vectors), so that no measurement has
really been done, OR that ____ = 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 Omega = .1-.2
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