|WikiProject Physics||(Rated Start-class, Mid-importance)|
- 1 Early and orphan comments
- 2 Great job!
- 3 Uniqueness
- 4 (Mildly) Off-topic: wikitex error
- 5 Bohm's theory
- 6 The other Gleason's Theorem
- 7 possible missing qualification in definition of state.
- 8 Theorem was not expressed meaningfully
- 9 Difficulties in "Application" section
- 10 Link to atom
- 11 Confusing wording in the introduction
Early and orphan comments
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.
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)
- 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)
(Mildly) Off-topic: wikitex error
In the Application paragraph, we find the following wiki text:
- We let A represent an observable with finitely many potential outcomes: the eigenvalues of the Hermitian operator A, i.e. . An "event", then...
At least with my settings, there's a spurious "-" inserted after in the HTML. Does this happen to anyone else?
RandomP 21:13, 24 September 2006 (UTC)
I like this article very much, but I would suggest adding one sentence. Currently, the article says:
"The theorem is often taken to rule out the possibility of hidden variables in quantum mechanics."
If you say this, I think that you need to add a caveat. Gleason's theorem doesn't apply to Bohm's theory, which is the only popular hidden variable theory nowadays. Gleason's theorem assumes that you begin by describing a particle by a state in Hilbert space, but Bohm doesn't do that. (For him, a particle has a definite position at all times.) This is a really big loophole, because other hidden-variable approaches could just dispense with Hilbert space altogether.
Personally, I don't really like Bohm's theory, but it does provide a nice counterexample to most "general" statements about hidden-variable theories (as evidenced here)! Sthinks (talk) 07:39, 9 December 2007 (UTC)
The other Gleason's Theorem
There is another candidate for this title, on the weight enumerators of binary self-dual codes (Pless, Vera (1982). Introduction to the theory of error-correcting codes. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons. p. 116. ISBN 0-471-08684-3.). I presume they are not the same? Richard Pinch (talk) 08:06, 11 July 2008 (UTC)
- It is of course the same Gleason, but it is not the same theorem. However, a quick look on Google gives < 700 results for the theorem on weight enumerators and > 45,000 for the theorem on probability measures. Thus while the self-dual codes one seems to be important in its field, and hence worth including, it should probably go on another, disambiguated, page -- point being that if a Bayesian asked a random person about Gleason's theorem, and wasn't met with a blank stare, he would in all likelihood expect his subject to start talking about Hilbert space. Unless information theory people are less likely to put up webpages on their subject than physicists, but that hypothesis is doubtful.--184.108.40.206 (talk) 16:05, 22 July 2008 (UTC)
possible missing qualification in definition of state.
Shouldn't there be some sort of maximality constraint on the x1...xn in clause 2 in the definition of state? Otherwise the sum of probabilities for x1, x2 would bave to be 1 (by clasue 2), and also the sum of probabilities for x3, ..., xn would have to be 1 (by clause 2), giving the sum over x1...xn as 2, (contrary to clause 2).
Theorem was not expressed meaningfully
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.
This is not expressed right.
(1) The system W and the density matrix for the system W are represented by the same symbol W. The matrix W is not defined or quantified over.
(2) The set on which the probability measure is to be defined is not specified. From context, presumably it is the set of "states", which is identified with the lattice of (closed?) linear subspaces, which in turn is identified with the lattice of orthogonal projections (onto closed subspaces).
This should be clarified, since an elementary definition of state is a unit vector in H (called a state vector), or the 1-dimensional linear subspace it spans.
(3) There must be some additional hypothesis or qualification on the probability measure that relates it to the Hilbert space.
Maybe it's compatible with the lattice of subspaces in some sense? For example, monotone with respect to the lattice and sums to 1 with the complement?
Then this is not, strictly speaking, a measure (meaning something defined on a sigma-algebra of subsets of something), but rather a function on the orthocomplemented lattice of closed subspaces of H, that slightly generalizes a measure. I would have to think this through.
(4) Presumably P(a) means the orthogonal projection onto the subspace a, but this should be stated.
(5) As pointed out by other commenters, Tr is not an operator. It is the (unique) trace defined on the set of trace-class operators on the Hilbert space.
P(a) will be a bounded operator, never trace-class unless it has finite-dimensional image. So W has to be trace class to assure that P(a) W is trace-class and the trace is allowable.
Since W is a hanging (unquantified) variable, we have to quantify it. I might guess that we should say "every measure (of a certain type) on the space of states can be represented in the form a → Tr(P(a)W) for some trace-class self-adjoint operator W on H".
Would we then interpret W as an observable?
Requiring that observables are trace-class is pretty strong. Many of the most important observables are not even bounded operators. But this difficulty is generally prevalent in quantum mechanics, so maybe it's not the point.
There's a better statement of Gleason's theorem at quantum logic. But this version's a mess.
- Simple solution -- I just copied the theorem from there. It gets everything right. Note that the previous version was so unclear that I even mixed up observables and states.
Difficulties in "Application" section
I also did some fixing-up of the Application section. Here Gleason's theorem is stated somewhat more correctly, but serious problems remain.
The most conspicuous one is that an observable A is introduced and induces a finite sublattice. But then suddenly it is forgotten, and P is defined on the full lattice. Yet similar notation is used. Atoms are defined only through A and the operator has to have distinct eigenvalues by assumption. So the Hilbert space is finite dimensional.
Very confusing. The paragraph on A is irrelevant and should be removed.
Also, the identity involving P(y) is treated incorrectly, being introduced as an observation instead of a definition or requirement.
Finally, it is not obvious how this statement relates to the one in the introduction, mostly because the notation has not been brought in line. (Nor was it in line before I fixed the introduction.)
- Upshot: all the needed definitions are at quantum logic. There, all the questions in this section and the previous one are answered.220.127.116.11 (talk) 18:20, 23 April 2015 (UTC)
Link to atom
From the article, as I found it:
Confusing wording in the introduction
The last sentences of the introduction were:
This implies that the Standard Quantum Logic can be viewed as a manifold of interlocking perspectives that cannot be embedded into a single perspective. Hence, the perspectives cannot be viewed as perspectives on one real world. So, even considering one world as a methodological principle breaks down in the quantum micro-domain.
This was confusing to me. Is 'perspectives' a technical word? Does 'manifold' have the usual meaning in math/physics? If so, the corresponding wikipedia articles should be linked. But looking at the body of the article, there is no further mention of manifolds or perspectives. It seems like this section is not meaningful (or at best very unclear) so I have removed it. If I am incorrect in doing so, I would be interested in what the intended meaning was. — Preceding unsigned comment added by Doublefelix921 (talk • contribs) 11:10, 7 May 2017 (UTC)