Kowalski – Page 56 – E. Kowalski's blog

The Kochen-Specker argument, and the spectral theory script

Somewhat later than I had hoped, I have updated the script of my spectral theory course. The version currently found online is complete as far as the material I intended to put in is concerned, but there are a few places where I haven’t written down all details (in particular for the proof of the Weyl law for the Dirichlet Laplace operator in an open subset of Euclidean space). I am also aware of quite a few small problems in the last chapter on Quantum Mechanics, due partly to notation problems (for the Fourier transforms, and for “physical” versus mathematical normalizations). I will need to re-read the whole text carefully to correct this; on the other hand, thanks to lists of corrections that I have already received from a few students, the number of typos is much less than before… I will however continue updating the PDF file as I continue checking parts of the text.

What delayed this version for a long time was the write-up of the last section on “The interpretation of Quantum Mechanics”; of course it’s in some sense an extraneous part of the script, since spectral theory barely enters in it, but I found it important to at least try to connect the mathematical framework with the actual physics. (This partly explains all the reading I’ve done recently about these issues). It is equally obvious that I am not the most knowledgeable person for such a discussion, but after all, there are good authorities that claim that no one really understands this question anyway…

What I end up discussing contains however one little mathematical result, which is cute and interesting independently of its use in Quantum Mechanics; it is a theorem of S. Kochen and E.P. Specker which states the following:

There does not exist any map
$latex f\,:\, \mathbf{S}^2\rightarrow \{0,1\}$
where S² is the sphere in R³ with the property that, whenever
$latex x,y,z$
are pairwise orthogonal unit vectors, we have
$latex f(x)+f(y)+f(z)=2$
or in other words, two of the three values are equal to 1, and the other is equal to 0.

How this result enters into discussions of the interpretation of Quantum Mechanics is described by M. Jammer in his book on the subject (not the same as his book on the development of Quantum Mechanis, but another one, equally evanescent as far as the internet is concerned); more recently, J. Conway and S. Kochen have combined it with the Einstein-Podolsky-Rosen argument (or paradox) to derive what they call the “Free Will Theorem”, which is an even stronger version of the unpredictability of properties of Spin 1 particles (those to which the Kochen-Specker argument applies). Conway has given lectures in Princeton on this result and its history and consequences, which are available as videos online.

Coming back to the result above, considered purely from the mathematical point of view, it is interesting to notice that both the original proof and the version used by Conway-Kochen (which is due to A. Peres) show that the hypothetical map does not exist even for some finite sets of points on the sphere. It is of some interest to get a smallest possible set of such points. The proof I gave in the script, however, which is taken from Jammer’s book (who attributes it to R. Friedberg) is maybe theoretically slightly more complicated, but it is also somewhat more conceptual in that one doesn’t have to be puzzled so much at the reason why one finite set of vectors or another is really fundamental.

“Würfeln” is German for playing dice

Since finishing Max Jammer’s book on the history of Quantum Mechanics, I’ve read a few more (and more popular) books, articles or reviews about the same general subject. One very striking thing — very obvious because of the outstanding level of the earlier book — was that none of the other texts gave any kind of feeling for the fact that the foundational work (until the middle 30’s at least) was very much a German-speaking affair. A few other languages are represented (de Broglie and the Curies in French, Dirac in English, Bohr at least partly in Danish), but their numbers are dwarfed by those of the German-speaking masters (Planck, Sommerfeld, Born, Einstein, Pauli, Schrödinger, Jordan, Heisenberg, etc). One anecdote emphasizes this clearly: the Indian physicist S. N. Bose sent a crucial paper in the form of a letter to Einstein (presumably in English), asking him if he could arrange for a German translation to be made and for its publication (Einstein did the translation himself).

Jammer gives most of the important quotations (and the crucial words in others) in the original language, with a translation. The other texts I’ve read, even if they briefly mention the original language, give only English translations of older quotes, with rarely a word of German appearing. (Of course, there is a lot of later literature which was first written in English). For most of the quotations, it seems there is no “official” translation, so it’s hard to judge their correctness.

For instance, it seems every source gives a slightly different version of the so-called “God doesn’t play dice” citation. The German original (in a letter from Einstein to Max Born in 1926; Born had been the first to give the standard interpretation of the modulus square of the “wave function” as giving the probability density of finding a quantum particle at a given point) is the following:

Die Quantenmechanik ist sehr Achtung gebietend. Aber eine innere Stimme sagt mir, dass das noch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, dass der Alte nicht würfelt.

which translates fairly literally (the best I can do…) as

Quantum mechanics is very imposing. But an inner voice tells me, that this is not yet the real McCoy. The theory provides a lot, but it brings us little closer to the secrets of the Old Man. At least I am certain, that the Old Man doesn’t play dice.

What is mostly missing from most of the translations I’ve seen is the informality and playfulness of the language. There’s wahre Jakob, which seems really equivalent to the real McCoy. And of course there is der Alte — I have no idea what would be a colloquially equivalent word in English; I can’t say at all whether it really refers to a deity or not (and if yes, at what level of formality). And I also wonder if there isn’t some slight difference of emphasis or subtlety of meaning in the verb würfeln, which contains in a single word the meaning to to play dice (jouer au dé). [Interestingly, it seems that würfeln also means to dice in the cooking-sense of cutting in dices.]

More animals

And now for something completely different: the amazing African Jacana (or Actophilornis africana, for the cognoscenti), another denizen of the Zürich rain forest, which is distinguished by having (relative to size) the longest toes, and the longest claw on the rear toe (if I believe, as I have no reason not to, the official Masoala rainforest guide). Here is a first picture:

Notice the baby Jacana on the left (the adult is likely to be the father, since the male takes care of the eggs in this species).

Here is a closer view of the baby:

It gets its big toes pretty young…

Kronecker-Weber by deformation, or: another bad reference

I have already mentioned two instances of pretty bad references in which I am involved (here and there). Here’s a third one: in Remark I.5.4 in my introductory notes on automorphic forms, L-functions and number theory (published in the proceedings of a school held at the Hebrew University in Jerusalem in March 2001), I state

Remark 1.5.4. The Kronecker-Weber Theorem, as stated here, bears a striking resemblance
to the L-function form of the modularity conjecture for elliptic curves (explained
in de Shalit’s lectures). One can prove Theorem 1.5.2 by following the general principles of
Wiles’s argument [Tu] (deformation of Galois representations, and computation of numerical
invariants in a commutative algebra criterion for isomorphism between two rings).

where the helpful-looking [Tu] leads rather disappointingly to:

[Tu] Tunnell, J.: Rutgers University graduate course (1995–96).

About a year and a half ago, R. Rhoades asked me if there was any more information available about this. The answer was that I had my own handwritten lecture notes of the original course taught by J. Tunnell at Rutgers (of course, maybe other people who had participated had their own). I said that I’d try to get those notes scanned, but it’s only in the last two days that I’ve finally started doing so — thanks to the recent installation in the ETH Library of a pretty fancy scanning machine, which makes the process essentially painless.

I’ve only scanned the first notebook and part of the second for the moment (enough to contain what Tunnell did about the GL(1) analogue of the modularity theorem of Wiles and its application to the Kronecker-Weber theorem):

First notebook

Second notebook

Unfortunately, it’s not clear how useful these will be to anyone, except future historians of the teaching of the proof of Fermat’s Great Theorem. The lectures are not entirely linear (there are notes about a parallel seminar on Serre’s Conjecture and of a few other lectures in the middle), they are in French, and the quality of the scan is not perfect (the second notebook was particularly cheap, and the ink on one side of a page is partly visible on the other side).

Qui dit mieux?

Analytic number theorists often work with multiple sums and integrals. In fact, sums are sometimes so congenial that the more there are, the merrier, and it may be quite a deep step to split a single sum into two. A famous examples is found, for instance, in Iwaniec’s celebrated bilinear form of the remainder term in the linear sieve, where one goes from something like

$latex \sum_{d<D}{|r_d|}$

to an expression involving two variables (say n and m)

$latex \sum_{m<M}{\sum_{n<N}{\alpha_m\beta_n r_{mn}}}$

with

$latex MN=D,$

and more or less unknown (but essentially bounded) coefficients α and β. (For a very clear discussion of why this is of crucial importance in some important problems of analytic number theory, and why the second form is more useful than the first, see for instance the Section entitled “The remainder term” in this survey paper of J. Friedlander).

In this spirit of increasing sums unboundedly, here is the record-holder I’ve seen so far (if memory serves): it is equation (29) in a paper of M. Young on non-vanishing of central values of L-functions of elliptic curves. Here is a screenshot:

where one counts no less than 11 summation signs.

Does any reader have a better example at hand? Examples involving the composition of more than 11 derived functors are also welcome for this friendly competition.