Talk:Gleason's theorem

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The term "Gleason's Theorem is pretty common among philosophers -- I put up a stub to see who would come out of the woodwork -- what I'd like to do is sketch a short proof, and give a decent bibliography. I put a pointer in the Bell article to see who might come this way. I've contributed before, but never started an article.

--Drewarrowood 08:08, 12 September 2006 (UTC)

I second "no deletion" and the need for improvement. The 1957 paper by A.M. Gleason ("Measures on the Closed Subspaces of a Hilbert Space", Journal of Mathematics and Mechanics 6: 885-93) is a classic paper on the foundations of quantum mechanics. It contains the first and (IMHO) still best derivation of the general quantum-mechanical rule for calculating the probabilities of measurement outcomes. (To understand its importance one has to bear in mind that probabilities of measurement outcomes are the only interference between quantum theory and experiments.)

--Ujm 08:27, 10 September 2006 (UTC)

What a foul trick, trying to rope a poor philosopher into doing this article. Well, I am such a person - although I didn't come here from Bell, I was just wondering what Wikipedia's coverage of the subject is like, and was disappointed to see it was merely a statement of the theorem.

I shall expand the article a bit, but I have no interest in sketching the proof...it is hideously complicated in its original form (i.e. Gleason's original paper) and even the elementary versions of it extend over several pages and are not easily summarised. Someone more used to identifying "key moves" in proofs and so forth is welcome to add a "proof section". The constructive proof can be found here, should anyone be interested. Maybe one day I'll do it, but not today.

But since, as Drewarrowood so slickly put it, the theorem is mostly used by philosophers, the focus of the article should probably be more on what the theorem actually says, why it is important, and what it is used for. So, I shall put in some blab about quantum logic, and how the theorem is a key ingredient in the derivation of the quantum formalism from logical structures (and how this works). Then, a brief bit about the philosophical implications. We really do not need to delete this article! It is of seminal importance to a serious field which is already not covered properly here: unfortunately questions of the interpretation of QM tend to be plagued with crankery, New Age flapdoodle, and positional soapboxing for various outlooks (many-worlds vs. Copenhagen, etc.).

Right. Now let me get cracking. Byrgenwulf 14:23, 23 September 2006 (UTC)

Ha! I just looked at who was commenting here...Herr Mohrhoff: you may remember my comment on your Koantum blog about Nietzsche...I never did get around to replying to you, since I have been caught up in the most awful fight here on Wikipedia. Anyway, feedback on my efforts here would be welcome: make sure I don't wander too far off into perspectivist diatribes! Byrgenwulf 14:30, 23 September 2006 (UTC)

"Gleason's theorem" has 10,400 hits on Google. So the article needs improvement at worst -- but certainly not deletion!!!

Yours truly, Ludvikus 15:10, 5 September 2006 (UTC)

However, there are more than on Gleason mathematician that have lived. And there does not appear to be a common reference to any "Gleason's theorem", or Gleason Theorem in my search of MacTutor and MathWorld. So the Author herein needs to justify his usage, or I shall be fored to agree with the Wikipedia Editor who recommended Deletion. So far, I'm Neutral on Deletion.

References:

MacTutor - http://www-history.mcs.st-and.ac.uk/history/Indexes/G.html

MathWorld - http://mathworld.wolfram.com/

Yours truly, Ludvikus 15:31, 5 September 2006 (UTC)

[edit] Great job!

I'd just like to thank the two editors concerned for turning, in five hours, a small stub into an article that I enjoyed reading (and will watch).

I'm more used to this taking a number of days, and intermediate steps, on Wikipedia, but this is a pretty motivating counterexample :)

RandomP 20:39, 23 September 2006 (UTC)

[edit] Uniqueness

For a Hilbert space of dimension 3 or greater, the only possible measure of the probability of the state associated with a particular linear subspace a of the Hilbert space will have the form Tr(P(a) W), where Tr is a trace class operator of the matrix product of the projection operator P(a) and the density matrix for the system W.

Is these only one such Tr?

If so, this should say the trace class operation
If not, it should say the only possible measures. Septentrionalis 22:35, 23 September 2006 (UTC)

I think I've corrected it now - the trace on a Hilbert space (more precisely, on the endomorphisms of a Hilbert space, as a partially-defined map) is unique, and usually referred to as "the trace" rather than "the trace class operation"; its domain is the set of trace class operators.

I've also replaced "matrix product" by "operator product", though "composition" might be more consistent with modern terminology; however, "density matrix" is traditional, and "matrix product" might be the right choice of terms if we want to keep this in a matrix mechanics-oriented view.

RandomP 23:19, 23 September 2006 (UTC)

Thanks: "trace" is correct, I think. I must confess I didn't check the statement of the theorem, I just left it as I found it...I think what it was trying before is that Tr is a (specific) operator that falls into the "trace class", as opposed to, say, the "inner product class"...

I'm also going to reword the theorem a tad, because it uses P in a different sense to how I used it later on (not having read the statement of the theorem given here, I didn't notice it). Nothing like a night's sleep to highlight all the slip-ups of the day before. Byrgenwulf 10:30, 24 September 2006 (UTC)

I also shifted the position of the W, since it could previously have been read to mean that the system is called W, when it is, in fact, the label for the density matrix. Byrgenwulf 10:36, 24 September 2006 (UTC)