On the use of Hilbert Spaces to represent states

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Hilbert spaces are defined by their linearity, inner product, and completeness, which are essential for representing quantum states. The discussion emphasizes that these properties facilitate the mathematical framework necessary for quantum mechanics, particularly through the Superposition Principle and continuous transformations between states. The use of complex numbers in Hilbert spaces is crucial for modeling interference effects and ensuring the evolution of physical systems. Various mathematical theorems, such as Gleason's Theorem, support the necessity of Hilbert spaces in quantum mechanics. The conversation also touches on the potential for alternative mathematical structures, like Rigged Hilbert Spaces, but concludes that Hilbert spaces remain fundamental for the representation of quantum states.
  • #31
Pulp, as you know Linearity is an axiom or postulate of QM, and as such it doesn't have any justification in the strict sense, and you won't find any, it doesn't need any. That's the thing with axioms

What you said is mathematically correct. But perhaps you can express those axioms as theorems of another equivalent set of axioms that, maybe, sound more familiar than the first ones.

For example, I have read in a lot of places that the conmutation relationship inherent in Heinsenberg Principle was an axiom of QM. This, being mathematically true, sounded to me that when "god created" QM he thought that, for no reason, the conmutator of p & x should be something different than 0. And I find that idea ridiculous.
Later, I've read in some other places (for example, in the Ballentine book mentioned in this post), that the form of P in x representation (and as a consequence, the conmutator between them an hence Heinsenberg principle) can be derived from the properties that is assumed that the space has (isotropy and such). And this sounds much more closer to the "real truth" than the axiom of the conmutator (However, perhaps we found that the son of QM that perhaps solves Planck length & GR problems take as more fundamental Heinsenberg principle than those properties of space, I am just saying that it does not sound likely -to me, just to me-)
And there are more examples. Bhobba told me here that Born rule can be derived through Gleasons theorems (I knew a similar derivation of Saunders). The use of complex numbers can be derived assuming that nature is continuous...
I mean, it seems (again, to me, just to me) that every abstract axiom in QM expressed as a mathematical relation can be expressed as a theorem assuming some previous axioms that sound (as always, to me, just to me) "more physical".
And here I arrive to Linearity. Is there any other axiom that you know that will sound more physical to me (perhaps you ask me how would you know what sounds more physical to me? I just think you know) and through which linearity can be demostrated?

Thanks to all for keep on trying to solve my doubts!
 
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  • #32
Gleason's theorem is a great theorem, but in what sense does it explain the Born rule? It doesn't explain why one would need probability.
 
  • #33
Gleason's theorem is a great theorem, but in what sense does it explain the Born rule? It doesn't explain why one would need probability

I don't know, Bhobba told me that and I believed and repeated it. I only read Saunders paper and I know that that paper indeed proves Born Rule from previous and reasonable axioms. Here is the post of Bhobba, perhaps I did not understand correctly:

If you are worried about the Born Rule - don't be. Its not a separate assumption - a very famous (but surprisingly not as well known as it should be) theorem called Gleasons theorem guarantees it:
http://kof.physto.se/theses/helena-master.pdf

Nevertheless, it is just an example. I don't want to miss the aim of this post which is (after a lot of very useful answers from all of you that cleared up a lot of mess in my mind) is to look for an argument to support the linearity hipothesis (if it is the case that there indeed is an argument out there).
 
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  • #34
TrickyDicky said:
This is simply not true. Perhaps they haven't given you reasons to consider alternatives, but I assure you that is not the case for many theorists and experimentalists that have the physical (and mathematical) maturity to realize that the current QM formulations are only really good approximations in the linear limit.
Theories try to accommodate the world we live in, not the other way around (as you seem to imply), and experiments tell us that the world is not linear, so any linear theory that tries to model our world will logically be superseded by a non-linear one.
Uh, what? I'm just saying that the assumption of linearity gives us a theory that hasn't been contradicted by experiments. It's ridiculous to accuse me of lacking physical and mathematical maturity because of this. And your claim about what I "seem to imply" is even sillier.
 
  • #35
atyy said:
Gleason's theorem is a great theorem, but in what sense does it explain the Born rule? It doesn't explain why one would need probability.

the_pulp said:
I don't know, Bhobba told me that and I believed and repeated it.
Once we have decided to look for a theory in which the states assign probabilities to the Hilbert subspaces of some Hilbert space, Gleason's theorem tells us we don't have the freedom to choose the probability assignments, because there's only one meaningful way to do them. (Without such a theorem, different probability assignments would have defined different theories).

What atyy is getting at is that Gleason's theorem doesn't tell us why we should make that kind of probability assignments in the first place.
 
  • #36
the_pulp said:
[...] an argument to support the linearity hypothesis (if it is the case that there indeed is an argument out there).
In this context, it is eye-opening to study Ballentine section 7.1 in which he derives the well-known half-integral angular momentum spectrum from nothing more than the SO(3) generators, their commutation relations, and the assumption that they are represented as Hermitian operators on an abstract Hilbert space.

AFAIK, there is no other method of deriving the angular momentum spectrum that does not involve use of such a linear Hilbert space.

Though this is not a conclusive "it-can't-be-anything-else" argument, it does set quite a high bar that alternative approaches must clear.
 
  • #37
strangerep said:
In this context, it is eye-opening to study Ballentine section 7.1 in which he derives the well-known half-integral angular momentum spectrum from nothing more than the SO(3) generators, their commutation relations, and the assumption that they are represented as Hermitian operators on an abstract Hilbert space.
Maybe I'm misinterpreting the intent of your statement, but it doesn't sound right.

In the proof idea as I know it, you don't presuppose a representation of SU(2) at all.

Instead, the argument proceeds (roughly) by looking at all representations, and invoking the fact that any element of the spectrum appears as an eigenvalue in some representation.

_____________________________________________On an unrelated note, while I mentioned I find the use of Hilbert spaces (and the linearity therein) as merely a matter of mathematical technique, I do find the fact the observable algebra is an algebra to be curious. Why do expressions like PX and P+X make sense??
 
  • #38
Fredrik said:
What atyy is getting at is that Gleason's theorem doesn't tell us why we should make that kind of probability assignments in the first place.

The way I see it is you have two choices - either a deterministic or statistical theory. The deterministic case however is contained in the statistical one - it simply means the probabilities are 0 or 1. What Gleasons Theorem shows is you can't assign 0 and 1 only to the subspaces of a vector space thus ruling out the deterministic case. The Kochen-Specker Theorem does as well but it really is a simple corollary to Gleasons Theroem.

The issue with Glesasons theorem is its hidden assumption of non contextuality - you assume the probably assignment does not depend on the way the rest of the vector space is partitioned off by a resolution of the identity. However this fits nicely in with my approach to QM which is based on invariance.

Thanks
Bill
 
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  • #39
bhobba said:
The way I see it is you have two choices - either a deterministic or statistical theory. The deterministic case however is contained in the statistical one - it simply means the probabilities are 0 or 1. What Gleasons Theorem shows is you can't assign 0 and 1 only to the subspaces of a vector space thus ruling out the deterministic case. The Kochen-Specker Theorem does as well but it really is a simple corollary to Gleasons Theroem.

The issue with Glesasons theorem is its hidden assumption of non contextuality - you assume the probably assignment does not depend on the way the rest of the vector space is partitioned off by a resolution of the identity. However this fits nicely in with my approach to QM which is based on invariance.

Thanks
Bill

Does one have to assume that measurements are projective operations?
 
  • #40
atyy said:
Does one have to assume that measurements are projective operations?

No. The idea is up to a multiplicative constant (the superposition principle implies this - superimposing a state with itself gives the same state) you know the outcome of any observation is an element of the vector space so you are assigning probabilities to the projection operators in a resolution of the identity (the space generated by multiplying any element by a constant is a subspace so automatically defines a projection operator). Gleasons theorem shows the usual trace formula for probabilities is the only one that can be defined. As mentioned there is a hidden assumption - namely the probability does not depend on what resolution of the identity a projection operator is part of - which is pretty much what a vector space is all about anyway (ie the elements do not depend on the whatever basis you chose) but is nonetheless an assumption.

You then associate each projection operator with a real number so a hermitian operator defines a resolution of the identity as the eigenvectors with the eigenvalues being the value associated with each projection operator.

Thanks
Bill
 
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  • #41
Hurkyl said:
Maybe I'm misinterpreting the intent of your statement, but it doesn't sound right.
Do you have a copy of Ballentine at hand? I believe what I said does indeed correspond to what he does.

In the proof idea as I know it, you don't presuppose a representation of SU(2) at all.

Instead, the argument proceeds (roughly) by looking at all representations, and invoking the fact that any element of the spectrum appears as an eigenvalue in some representation.
I haven't seen it done with that emphasis. Can you give me a (readable) reference?

On an unrelated note, while I mentioned I find the use of Hilbert spaces (and the linearity therein) as merely a matter of mathematical technique, I do find the fact the observable algebra is an algebra to be curious. Why do expressions like PX and P+X make sense??
Maybe because the symmetry/dynamical groups tend to be Lie groups, and physical representations are classified via Casimirs thereof? (Also, continuity and differentiability, etc, imply certain properties for the generators.)

BTW, if the same question is posed in the classical regime, we have things like ##J = X \times P##, and also ##H = P^2 + X^2##, etc. Is this curious or boring? :-)
 
  • #42
  • #43
strangerep said:
No. POVM's are a more general approach:

http://en.wikipedia.org/wiki/POVM

Indeed it is and the modern proof of Gleasons Theorem makes use of them instead of resolutions of the identity - but POVM's are derivable from projections of higher dimensional resolutions of the identity via Newmarks Theorem.

Thanks
Bill
 
  • #44
Helloooo! I opened this thread because I was asking about linearity!

And here I arrive to Linearity. Is there any other axiom that you know that will sound more physical to me (perhaps you ask me how would you know what sounds more physical to me? I just think you know) and through which linearity can be demostrated?

(Nevertheless, pretty interesting the talk about Gleasons Theorems)
 
  • #45
A question relating linearity and measurements:

Compare the classical and Schroedinger wave equations. Both have linear solution spaces. However, a solution of the classical equation is not considered a sate, because it contains only information about position, not velocity, and both can be measured. However, a solution of the Schroedinger equation is considered a state. Is this because the quantum notion of position or velocity is different from the classical one?
 
  • #46
atyy said:
A question relating linearity and measurements:

Compare the classical and Schroedinger wave equations. Both have linear solution spaces. However, a solution of the classical equation is not considered a sate, because it contains only information about position, not velocity, and both can be measured. However, a solution of the Schroedinger equation is considered a state. Is this because the quantum notion of position or velocity is different from the classical one?

I don't think so. The classical wave equation is second order w.r.t. time, the solution plus its time derivative is the state. The state should be something that determines the future evolution of the system and in the classical case the solution of the equation (at a given time) is not enough for initial conditions.
 
  • #47
strangerep said:
I haven't seen it done with that emphasis. Can you give me a (readable) reference?
Unfortunately, it's not something I've actually worked through. Here is wikipedia's page on the topic. The result they state near the bottom is what I remember -- the fact you derive is not a theorem about the behavior of SU(2) in a particular representation: instead it is a complete classification of all irreducible finite-dimensional representations.

Mulling it over, I think the statement you are referring to (I don't have the text) is effectively equivalent, just phrased differently.



BTW, if the same question is posed in the classical regime, we have things like ##J = X \times P##, and also ##H = P^2 + X^2##, etc. Is this curious or boring? :-)
Boring. On any particular state, P and X are definite numbers, and so P^2+ X^2 makes sense in the classical regime: "Measure position and momentum, square them, then add them".

I can dramatically point out the issue with spin states. If X and Y are spin about the X and Y axis, if we try to think of the spectrum of an operator as being the possible outcomes of a hypothetical measurement, and we try to interpret X+Y in the same way as I mentioned classically, we run into the problem that the interpretation says 1 + 1 = \sqrt{2}! (or more accurately, if we add either of \pm 1 and either of \pm 1, we get either of \pm \sqrt{2})
 
  • #48
Hurkyl said:
Mulling it over, I think the statement you are referring to (I don't have the text) is effectively equivalent, just phrased differently.
After a night's sleep, I now get what you were previously saying. A more careful revision of what I said should include some caveats about superselection sectors for different Casimir values.
I can dramatically point out the issue with spin states. If X and Y are spin about the X and Y axis, if we try to think of the spectrum of an operator as being the possible outcomes of a hypothetical measurement, and we try to interpret X+Y in the same way as I mentioned classically, we run into the problem that the interpretation says 1 + 1 = \sqrt{2}! (or more accurately, if we add either of \pm 1 and either of \pm 1, we get either of \pm \sqrt{2})
I wonder whether one hits the same issue if the quantum case were constructed using POVMs corresponding to spin-coherent states. It's been a while since I looked at these and it's too early in the morning right now... :-)
 
  • #49
bhobba said:
[...] POVM's are derivable from projections of higher dimensional resolutions of the identity via Newmarks Theorem.
I guess you meant "Neumark", or "Naimark"? For other readers, this is also mentioned in the Wiki page on POVMs previously linked. More specifically:
http://en.wikipedia.org/wiki/Neumark's_dilation_theorem

BTW, how does this work in the case of unbounded operators and continuous spectra? The construction of a "higher dimensional" Hilbert space seems like it might be a bit tricky in that context, and the Wiki page seems to deal only with bounded operators.
 
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  • #50
the_pulp said:
Helloooo! I opened this thread because I was asking about linearity!
Sorry if you've received the impression that your thread was being hijacked. The posts were indeed mostly relevant to your topic, but clearly that was not obvious. [Although... perhaps the stuff on Naimark's theorem should be moved to a different thread.]

Did you get (or study) the point I was trying to make in my earlier reference (post #36) to Ballentine's derivation of quantum angular momentum spectra?
 
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  • #51
Sorry if you've received the impression that your thread was being hijacked. The posts were indeed mostly relevant to your topic, but clearly that was not obvious. [Although... perhaps the stuff on Naimark's theorem should be moved to a different thread.]

Did you get (or study) the point I was trying to make in my earlier reference (post #36) to Ballentine's derivation of quantum angular momentum spectra?

No, I did not study it (I thought it was related to another thing, but it seems that it wasnt!). I am going to do it today at night. In the meantime if you can give me a short summary of what is the relation between that and my doubt, very welcome.
 
  • #52
the_pulp said:
No, I did not study [Ballentine] (I thought it was related to another thing, but it seems that it wasnt!). I am going to do it today at night. In the meantime if you can give me a short summary of what is the relation between that and my doubt, very welcome.
Ballentine ch1 explains how we have physically meaningful observable quantities, and that we wish to construct probability measures over them.

Doing this in a linear space rather than a nonlinear space is simply easier. There's no point using a technically less-convenient space if we don't need to. Similarly, a metric-topological space has lots more nice properties than more general topological spaces, so we don't use the latter unless the physics can't be modeled satisfactorily in the former. Rigged Hilbert space is a step towards a more general structure, but we don't need to use that for simple cases.

My point about the quantum angular momentum spectrum was about how a basic Hilbert space framework and Hermitian operators already gives an important physical result: the half-integral angular momentum spectrum.

Not sure what else I can usefully say here. Maybe after reading the earlier chapters of Ballentine, you should re-articulate whatever issues remain unresolved...
 
  • #53
the_pulp said:
Helloooo! I opened this thread because I was asking about linearity!

Another possible way I think one could motivate linearity is to note that elementary particles in quantum theory are considered point particles so they have no size (this also happens in the linear classical mechanics and electrodynamics theories and their idealized point particles) . Of course if that assumption was not physically exact I suppose it would introduce some non-linearity in the picture and QM as we know it would be just a linear approximation for a point.

We do know that we have to recur to QFT perturbative methods to have good approximations to certain experimental results (I'm thinking about the Lamb shift or the anomalous magnetic moment of the electron that are not so well approximated by either relativistic or NRQM).
 
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  • #54
the_pulp said:
Yes, that was sort of my route. I was not very sure about:
1) The linearity of the states (I think that when someone says to me that the state space should be linear because of the superposition principle it sounds like "the state space should be linear because the state space should be linear").
2) The use of complex numbers

And, related to the born rule, I was not using Gleasons theorems because I was thinking about Saunder's paper, but it is more or less the same (I will read your paper). Now, Ballentine and the link you sent me about complex numbers I think I've got the ideas a little bit more ordered.

Thanks!

there is nonlinear quantum theory.
 
  • #55
yoda jedi said:
there is nonlinear quantum theory.

Not in any of the various QM textbooks I've consulted, do you have some reference?
 

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