# On the use of Hilbert Spaces to represent states

1. May 11, 2012

### the_pulp

There are a lot of coherent definitions of Hilbert spaces. Lets take the wikipedia one and let me ask you some cuestions:
A Hilbert space:

1) Should be linear
2) Should have an inner product (the bra - ket rule to get an amplitude)
3) Should be complete (every cauchy sequence should be convergent)

Why states should have these properties?

Thanks!!!

Pd: I would like to receive answers that are not dependent of the results of QM. For example, we can make an argument that starts by accepting the Heisenberg principle and from this, perhaps, we can derive that the state should be represented by an element of a Hilbert space. But I want to go all the way around. I would like to prove or explain by really basic hypothesis why 1), 2) and 3) should hapend and then, for example, prove Heisenberg principle (adding first, all the other stuff such as the Born rule, evolution equation and such).

2. May 11, 2012

### mathman

There is an item in the general physics forum which seems to be asking the same question.

3. May 11, 2012

### the_pulp

I saw that one, but it doesnt. That one tries to represent classical mechanics by the "states/operators" language".
Here what I want to know is why the elements of a Hilbert space are suitable to represent states and why we are not using, for example, some topological space (which are more general and dont have a notion of distance) or whatever other set of mathematical objects. Why do we want that the mathematical objects that represent the states should be linear, complete and have an inner product?

4. May 12, 2012

### strangerep

Try Ballentine chapters 1-3.

(Sorry, I don't have time for a detailed reply right now, but I think Ballentine will take you at least part-way towards an answer...)

5. May 12, 2012

### Staff: Mentor

The basic and fundamental postulate of QM - the Superposition Principle - implies it must be a vector space. It is a mathematical convenience to require that the bras and kets can be put into one to one correspondence which implies it is a Hilbert Space (Riesz Representation Theorem) - and some important and powerful theorems such as Gleason's Theroem hold for Hilbert Spaces.

But you do not have to do that and you can extend it to be something more general called a Rigged Hilbert Space which has mathematical advantages in allowing the existence of things like Dirac Delta functions ruled out of a Hilbert Space.

Thanks
Bill

Last edited: May 12, 2012
6. May 12, 2012

### martinbn

You can start from the observables, motivated by the experimental results you can deduce the mathematical structure of the set of the observables. Then some mathematical theorems will force on you the use of Hilbert spaces.

7. May 12, 2012

### the_pulp

Some mathematical theorems? Which ones? What is the line of reasoning that, through those theorems, I will be able to derive that the only choice is to work with elements of Hilbert Spaces?

Thanks for all!!

Ps: I will check Ballentine!

8. May 12, 2012

### Staff: Mentor

Well if you want rigor then it can all in fact be derived from 5 reasonable axioms that on the surface have nothing to do with Hilbert Spaces - check out:
http://arxiv.org/pdf/quant-ph/0101012v4

The key Axiom is the requirement of continuous transformations between pure states and while the proof it leads to Hilbert Spaces is long (it took me about a week to go through checking each step) the actual math is basically linear algebra.

Thanks
Bill

9. May 12, 2012

### the_pulp

Ive already read that one. I did not like those axioms, especially the 2nd one (K=K(N)) but I didnt give them so much time as you. Perhaps if I give them another try I would find them different.

I am reading ballentine right now. Its a pitty I havent read it before.

I have another question related to this topic. Why does the Hilbert Space that we use in QM should be complex. What would happen if it were a real Hilbert Space? Would we arrive to classical physics?

Last edited: May 12, 2012
10. May 12, 2012

### meopemuk

Search for "quantum logic" and "Piron's theorem".

Eugene.

11. May 12, 2012

### Staff: Mentor

Its not really a question of if you like the axioms or not - I personally prefer others - its a question do they imply a Hilbert Space - they do.

Its by far the best book on QM I know - my go to book

Well at a physical level its required for interference effects - you can't model those in a Real Hilbert Space. At a mathematical level you require it because you want to be able to continuously go from one state to another via other states which would seem to be a key requirement for how physical systems should evolve - you must go to complex numbers to allow that.

For a bit more detail check out:
http://www.scottaaronson.com/democritus/lec9.html

Thanks
Bill

12. May 12, 2012

### the_pulp

Yes, its a question of if I like it. In fact I started this post because I didnt like the Hilbert axiom.

Yes! that was the answer I was looking.

Yes, it is the introduction I needed when I started reading this things.

What I didnt like about Ballentine is that it assumes the Born Rule. I saw a paper that demonstrates it assuming certain basic assumptions (such as representating with Hilbert Spaces -This is the reason of my post-, coherence of rules of calculating probabilities from equivalent Hilbert representations and not much more). It is:

Derivation of the Born Rule from Operational Assumptions, Simon Saunders
http://users.ox.ac.uk/~lina0174/born.pdf

I strongly recommend it (and if you can give me any feedback about it, would be very helpful).

13. May 12, 2012

### Staff: Mentor

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 [Broken]

My route to QM is as follows. Suppose you have a system with n possible states. Map them to the orthonormal basis vectors of an n dimensional vector space. Now apply coordinate invariance to deduce the physics remains the same if you transform to any other orthonormal basis meaning any vector in the vector space must also be a possible state. Use the requirement for continuous transformations between states to make it a complex vector space. Apply Gleasons Theorem and bingo - you have QM. It also explains why QM is a statistical theory - for a deterministic one you should be able to map the states to 0 or 1 - but Gleasons theorem shows you can't do that - the continuity assumption basically fouls you up.

Had a peek at the link. Yea - know that one - its basically an update of David Deutsches argument. Its fine as far as I am concerned - I just prefer Gleasons Theorem for a few reasons. First the objection in the paper it only applies to space of dimension 3 and above has now been rectified with a simpler modern version based on POVM's instead of resolutions of the identity. Secondly it immediately shows why QM must be statistical in a very elegant way. And finally in my route to QM coordinate invariance (strongly related to and implying non contextuality) is central and Gleasons Theroem follows from that. In fact Gleasons Theorem really shows what a powerfull assiumption non contextuality really is.

But that is simply a personal reference. Its good two methods lead to the same result.

Thanks
Bill

Last edited by a moderator: May 6, 2017
14. May 12, 2012

### atyy

Is there really a proof that the structure of QM is required, ie. emergent quantum mechanics is doomed to fail?

Isn't it more that QM is our most fundamental theory at the moment, extremely successful, with some poorly understood bits like measurement, so we accept it for the time being?

15. May 13, 2012

### Staff: Mentor

Of course there is no proof the structure of QM is required - that is an experimental matter. All I am suggesting is a way to present QM exists that makes it seem more reasonable and clarifies its basic assumptions and structure. And no it is indeed possible QM can emerge from another theory eg Primary State Diffusion, although I personally believe it is fundamental - but my personal favorite interpretation, the ensemble interpretation, does whisper in your ear there is more to it.

The poorly understood areas are being clarified further all the time eg we know a lot more about decoherence and its effects than when Bohr and Einstein were around and having their truly magnificent debates - and how it resolves a number of the issues. I believe it will all eventually be completely clarified.

Thanks
Bill

16. May 13, 2012

### the_pulp

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 Ive got the ideas a little bit more ordered.

Thanks!

17. May 13, 2012

### atyy

Do these derivations show that states aren't elements of Hilbert spaces, but rays?

18. May 13, 2012

### Staff: Mentor

The reason it is a vector space depends on mapping the states to basis vectors and applying invariance - its logically equivalent to saying its a vector space to begin with - which is logically equivalent to the principle of superposition. It just depends on what appeals to your intuition more - that's all. To me the principle of superposition seems like a rabbit pulled out of a hat but applying invariance is something that is very intuitive from everyday experience.

Thanks
Bill

19. May 13, 2012

### the_pulp

What do yo mean by "applying invariance"? invariance of what under change of what? perhaps applying invariance of probability under change of equivalent states representation as in "Saunders"?

20. May 13, 2012

### Ken G

The only modification I'd make to that clear and nice structure is that I think one should always base the foundations on empiricism-- what can be measured. So we should not start with possible states, we should start with possible outcomes to measurements, and then simply associate states to those outcomes. It's not a trivial difference-- we are making some kind of stand when we claim there is a "state" associated with an outcome to a measurement, and indeed it is generally true that there is a whole family of degenerate states that all lead to that outcome. So we may need to combine different measurements to resolve the degeneracies and generate a concept of individual "states", and if all our efforts to separate two states fail to break their degeneracy, then we can assert those states were never different in the first place. But then we have to hope that statistical mechanics will back us up-- if we find a given state tends to be multiply populated relative to other states of the same energy (or worse, when dealing with identical fermions), then we have a real problem-- either we have states that don't obey the statistics, or we have states that we are going to claim are different even though we cannot resolve their differences using any known measurements. Those kinds of issues have apparently not been encountered, but it shows the kinds of issues that could be lurking if we are careful to separate eigenvalues of measurements from rationalistic descriptions of states. To be sure, this is all just the eigenvalue/eigenstate structure of quantum mechanics, which allows us to essentially equate the concepts, but we always have to recognize which one we are anchoring our physics on in case we encounter surprises.

Also, the continuity principle looks a bit different when you think about eigenvalues instead of states-- it then means that if we can do a measurement that locates the particle in some box centered at A, or a box centered at B, we should be able to do a measurement that can locate the particle in a box centered anywhere between A and B. This sounds like a more physically reasonable assumption than assuming that the states themselves be continuous, which to me sounds like another kind of "rabbit."

Last edited: May 13, 2012
21. May 13, 2012

### Staff: Mentor

The idea is you map the states of a system to the orthonormal basis vectors of a vector space. Now you make a change of basis to another orthonormal basis (ie you rotate your coordinate system) so that your states now have a different representation. The laws of physics should not depend on your choice of basis (ie that you have rotated your coordinate system) so you conclude that any normalised vector in the vector space can be a possible system state.

I am not the guy to come up with it - Victor Stenger did (and he probably was not the only one):

'This is also called the superposition principle and is responsible for much of the difference between quantum and classical mechanics, in particular, interference effects and so-called entangled states. As we saw in chapter 4, linearity and thus the superposition principle are required to maintain point-of-view invariance. That is, point of-view invariance can be used as a motivation for the superposition principle, which, in conventional developments, is simply a hypothesis that almost seems to be pulled out of a hat.'

Thanks
Bill

Last edited by a moderator: May 6, 2017
22. May 13, 2012

### strangerep

It's been a while since I looked at any of them, but I recall always getting the feeling that they slipped normalization in for convenience.

IMHO, the ray approach is more fundamental -- and concentrating on angles between rays, and the dynamical group action which maps rays into other rays. In this regard, there are papers which emphasize the use of a Fubini-Study metric topology rather than the usual Hilbert norm topology, but they always seem (iirc) to use an underlying Hilbert space with normalizable vectors. But for unbounded operators and continuous spectra we already know that some sort of triplet space (rigged Hilbert, or partial inner product) is a more natural fit. I don't know of any papers which use Fubini-Study topology on rays exclusively, but this seems to me an interesting line of research.

23. May 13, 2012

### strangerep

Underlying any (class of) dynamical system is a dynamical group. (Think canonical transformations in Hamiltonian mechanics, for example.) The observables for that system come from the generators of that group (or higher products thereof).

I still don't have enough spare time to write a more detailed explanation, but I mentioned a little about a related point in this thread:

(See my post #15, as modified by post #26.)

Actually, it might be worth your time to read that whole thread if you haven't done so before.

24. May 15, 2012

### the_pulp

Sorry I come back with the same topic but Ive been trying to clear up my ideas but there is a little something that I still dont know why we are taking as obvious.

1) Someone said (and I agree) that first there are experiments that can take certain values and we associate those values with states. So, until here, an experiment is an association of certain values to certain states
2) To make 1) mean something we have to define what is an state. We know that set of states has to accomplish 2 assumptions
a) Continuity (Continuity of experiments or continuity of states I think its more or less equivalent -I can be wrong- and this in some point will lead us to the use of complex numbers)
b) Metric Space (because we know that there is a probability between two states, there is a notion of distance between two states).
The main ingredient that I am not assuming is Linearity, but linearity (and some bits more such as scalar product) can lead us to Born Rule (through Gleasons or Saunders). What is the justification of Linearity?
Perhaps you have already explained that to me. In that case I couldnt detect that so I would be very thankful if you can enlight me once again!

Ps: and, again I dont think that "superposition principle" is the answer because to me, saying "Linearity" occurs because of "Superposition Principle" is like saying "Linearity" occurs because of "Linearity".

25. May 15, 2012

### Fredrik

Staff Emeritus
I don't think there is one, other than "it's the simplest possibility, and our best experiments haven't given us a reason to consider another one".