An Introduction to Hilbert Space: Physically Explained

[Sirius24]

I'm in Quantum 1, and the professor briefly mentioned Hilbert Space. I'm having a difficult time finding a non-technical description of what Hilbert Space is. Could someone give me a brief description of what it is physically, rather than mathematically?

 

[micromass]

I'll try to give an intuitive definition: a Hilbert space is a set where it is possible to perform geometry.

In a vector space, it is possible to talk about lines, planes, hyperplanes. But that's not enough to do geometry. You'll need some notion of distance.

In a normed vector space, it is possible to talk about about a distance. But that's not enough to do geometry. You'll need some notion of "angles".

In a pre-Hilbert space is a special normed vector space. In such a space you can talk about angles, orthogonality, the Pythagorean theorem,...

Finally, a Hilbert space is a special pre-Hilbert space. In a Hilbert space, there are no holes. Formally: a Hilbert space is a complete pre-Hilbert space.

 

[Hurkyl]

And to see the difference between "pre-Hilbert space" and "Hilbert space", you have to consider infinite-dimensional spaces.


Now, Hilbert space pretty much is a technical concept -- I don't think you can really get around that fact. But what you can ask is the motivation for the concept rather than the detail of what it is -- the point of a Hilbert space is more or less that:

1. We can do linear algebra with it
2. We can do calculus with it
3. Calculus is well-behaved

In other words, "Hilbert space" is the technical condition that says it's okay to do the sorts of things you're going to do in your class, and that it all behaves more or less as you would expect.


The question "What is a Hilbert space physically?" really doesn't make sense -- what you are really asking is "What sort of physical thing can be described via Hilbert spaces?" Learning some answers to that question is the whole point of the class you're taking.

 

[Careful]

The question "What is a Hilbert space physically?" really doesn't make sense -- what you are really asking is "What sort of physical thing can be described via Hilbert spaces?" Learning some answers to that question is the whole point of the class you're taking.

There is a way do derive Hilbert space from scratch simply by imposing six axioms on the set of yes and no questions you can ask. This is the program of Von Neumann, later developped by the Brussels-Geneva group; so, one can derive these structures from very elementary (but sometimes wrong) considerations.

 

[micromass]

There is a way do derive Hilbert space from scratch simply by imposing six axioms on the set of yes and no questions you can ask. This is the program of Von Neumann, later developped by the Brussels-Geneva group; so, one can derive these structures from very elementary (but sometimes wrong) considerations.

Really? That sounds interesting! Can you give some kind of reference for that?

 

[Careful]

Really? That sounds interesting! Can you give some kind of reference for that?

http://www.vub.ac.be/CLEA/aerts/publications/2004QuoVadisQM.pdf and references therein. The author is aware that standard QM falls short and tries to find more general axioms than those which merely derive QM and classical physics.

Btw. I see we come from the same country; where do you study?

Careful

 

[Fredrik]

Really? That sounds interesting! Can you give some kind of reference for that?

Assuming that Careful is talking about quantum logic, the basic idea is to look for a mathematical structure that can represent the set of "experimentally verifiable statements" (e.g. "if I measure the energy, the result will be in the interval [a,b]"). What they end up with is a lattice that's isomorphic to the lattice of closed subspaces of a complex separable Hilbert space. So from a physicist's point of view, you might as well start right there. But you don't have to mention Hilbert spaces right away. You can start with an abstractly defined lattice, and specify its properties.

Geometry of quantum theory by V.S. Varadarajan is a pretty comprehensive treatment of QM from this point of view. It's very difficult to read for a typical physics student, but perhaps not for you.

Edit: I wrote that before I saw Careful's answer. Hm, the article starts with

We propose a general operational and realistic framework that aims at a generalization of quantum mechanics and relativity theory, such that both appear as special cases of this new theory.​

That sounds very ambitious. Certainly a lot more ambitious than quantum logic.

 

[micromass]

http://www.vub.ac.be/CLEA/aerts/publications/2004QuoVadisQM.pdf and references therein. The author is aware that standard QM falls short and tries to find more general axioms than those which merely derive QM and classical physics.

Btw. I see we come from the same country; where do you study?

Careful

Ah, Dirk Aerts I actually considered doing my PhD with him. His research area is very interesting...

I studied at the VUB. But now I'm doing my PhD in Antwerp. But next year I'll probably go back to the VUB...

 

[dextercioby]

Really? That sounds interesting! Can you give some kind of reference for that?

There are books on this issue as well, the most famous is probably Varadarajan's <Geometry of Quantum Theory> (first volume, iirc) . I think there's also a book by the Italian mathematician F. Strocchi <The Logics of Quantum Mechanics>.

 

[Careful]

Ah, Dirk Aerts I actually considered doing my PhD with him. His research area is very interesting...

I studied at the VUB. But now I'm doing my PhD in Antwerp. But next year I'll probably go back to the VUB...

Ah, I know him quite (''very'') well personally. But as far as I know, nobody in Antwerp is doing something even close to this. What are you playing then with? Path integrals ? ...

 

[Careful]

Assuming that Careful is talking about quantum logic, the basic idea is to look for a mathematical structure that can represent the set of "experimentally verifiable statements" (e.g. "if I measure the energy, the result will be in the interval [a,b]"). What they end up with is a lattice that's isomorphic to the lattice of closed subspaces of a complex separable Hilbert space. So from a physicist's point of view, you might as well start right there. But you don't have to mention Hilbert spaces right away. You can start with an abstractly defined lattice, and specify its properties.

Geometry of quantum theory by V.S. Varadarajan is a pretty comprehensive treatment of QM from this point of view. It's very difficult to read for a typical physics student, but perhaps not for you.

Edit: I wrote that before I saw Careful's answer. Hm, the article starts with

We propose a general operational and realistic framework that aims at a generalization of quantum mechanics and relativity theory, such that both appear as special cases of this new theory.​

That sounds very ambitious. Certainly a lot more ambitious than quantum logic.

Yes, it is much more ambitious. Actually what he first discusses is a framework which originated by Piron in which as well classical as quantum physics is retrieved. But still this is not enough and (I discussed this with the author) in my opinion at least 4 out of 6 axioms are false while he only discussed two of them (at that time !).

 

[micromass]

Ah, I know him quite (''very'') well personally. But as far as I know, nobody in Antwerp is doing something even close to this. What are you playing then with? Path integrals ? ...

Well, due to circumstances, I was involved in something completely different then I was wanting to do. Right now I'm playing with string theory and mirror symmetry from a mathematical point of view of course...

 

[Careful]

Well, due to circumstances, I was involved in something completely different then I was wanting to do. Right now I'm playing with string theory and mirror symmetry from a mathematical point of view of course...

I don't know any string theorist in Antwerp... I am aware Lieven Lebruyn and Fred Van Ostayen have some tangential interests in these things, but that is at the mathematics departement. Actually, since you are mathematically inclined you might be interested in the draft of my book http://arxiv.org/abs/1101.5113 which I spoke about with Dirk (and the concrete quantum theory I develop suggests a logic in the generalized framework he is looking for, but we are not quite there yet).

 

[Fredrik]

Yes, it is much more ambitious. Actually what he first discusses is a framework which originated by Piron in which as well classical as quantum physics is retrieved. But still this is not enough and (I discussed this with the author) in my opinion at least 4 out of 6 axioms are false while he only discussed two of them (at that time !).

In that case, I'm a little confused. If at least 4/6 of the axioms are false (which I assume means that theories of physics based on them will make predictions that disagree with experiments), should we even bother to read the article?

 

[micromass]

I don't know any string theorist in Antwerp... I am aware Lieven Lebruyn and Fred Van Ostayen have some tangential interests in these things, but that is at the mathematics departement. Actually, since you are mathematically inclined you might be interested in the draft of my book http://arxiv.org/abs/1101.5113 which I spoke about with Dirk (and the concrete quantum theory I develop suggests a logic in the generalized framework he is looking for, but we are not quite there yet).

Well, I am a mathematician, so I study at the math department. I don't study string theorists like the physicists, but I rather try to approach it from a mathematical point of view. It is rather hard, since I don't know much about physics

 

[Careful]

In that case, I'm a little confused. If at least 4/6 of the axioms are false (which I assume means that theories of physics based on them will make predictions that disagree with experiments), should we even bother to read the article?

You don't understand. The axioms which are written there give standard QM + classical mechanics. But the author is smart enough to realize that standard QM doesn't work well enough and he has written dozens of papers about these ''failing axioms'' since the 1970 ties with amongst others Ingrid Daubechies (the president of the international mathematical society). Furthermore your reasoning is not very ''sophisticated'', saying that an assumption fails does not imply you get in contradiction to experiment !

 

[Careful]

Well, I am a mathematician, so I study at the math department. I don't study string theorists like the physicists, but I rather try to approach it from a mathematical point of view. It is rather hard, since I don't know much about physics

Ah so you are with Fred ? Well I had the luck to study mathematical physics first and then relativity and QFT in my PhD time. I did not suffer from the pressure to do strings ... but I guess you get a fairly unique perspective upon the matter in this way. I thought Antwerp had connections with IHES in Paris where Connes is doing his stuff.

 

[Fredrik]

Furthermore your reasoning is not very ''sophisticated'', saying that an assumption fails does not imply you get in contradiction to experiment !

What? Then what did you mean when you said that the axioms are false? Mathematical axioms are always a part of a definition of something, and in this case, without having read the article, I can only assume that what's being defined is either a theory of physics or a mathematical structure that the authors would like to use in a theory of physics. The only problems that a bad choice of axioms can cause are logical inconsistencies or disagreement with experiments.

I don't doubt that the authors are smart guys. I skimmed one of those Aerts & Daubechies articles last year (the one that justifies the use of tensor product spaces in QM), and it looked very solid.

 

[Careful]

What? Then what did you mean when you said that the axioms are false? Mathematical axioms are always a part of a definition of something, and in this case, without having read the article, I can only assume that what's being defined is either a theory of physics or a mathematical structure that the authors would like to use in a theory of physics. The only problems that a bad choice of axioms can cause are logical inconsistencies or disagreement with experiments.

I don't doubt that the authors are smart guys. I skimmed one of those Aerts & Daubechies articles last year (the one that justifies the use of tensor product spaces in QM), and it looked very solid.

With false we (Diederik, I and probably Ingrid too) mean that the logical structure of a theory which involves as well QM as relativity is going to be based on axioms which subtly contradict those in the paper and slightly extend them as well. This means we move away from Hilbert space; Piron's primary concern was to find axioms which would retrieve a slight extension of real, complex and quaternionic quantum mechanics.

The original Aerts-Daubechies paper about the tensor product was written in the same vein. We know that the tensor product construction is not adequate for quantum gravity (and it gets actually violated in my theory). So when I use false, I always mean physically wrong even though the logical structure is coherent and agreement with experiment is rather good (because of the weak gravitational fields).

Careful