Quantization = construction of quantum theories based on the classical limit?

[tom.stoer]

I have a question regarding quantization.

In most cases one never starts with a quantum theory, but always writes down a classical expression, goes through quantization, implementation of constraints (Dirac, BRST, ...), construction of Hilbert space, inner product, measure of an path integral etc. to arrive at a viable quantum theory. Hopefully the theory is anomaly free, finite / renormalizable etc.

I would like to question this approach which is based on the classical limit and constructs a quantum theory via ad-hoc rules. It's like starting with a drawing and derive from it how the final building shall look like; w/o having ever seen a building, experience as an architect or with construnction this will never work.

So my question is if there is another approach, a research program, ..., to write down or "construct" quantum theories w/o using classical expressions as a starting point?

 

[dx]

To guess sensible quantum theories directly is very difficult. The relativistic invariance of the action or the Lagrangian density makes it easiest to start with S or L and then find the Hamiltonia; even if we start from a relativistic action to obtain the Hamiltonian, it is not guaranteed that the quantum theory obtained from it will be relativistic. Also, Dirac's and Bohr's writings on quantum theory suggest that deep quantum theories always arise out of the quantization of classical theories (e.g. quantum theory of relativistic strings), so in a sense the process of 'quantization' is conceptually more fundamental than the resulting 'quantum theory'.

 

[tom.stoer]

To guess sensible quantum theories directly is very difficult. The relativistic invariance of the action or the Lagrangian density makes it easiest to start with S or L and then find the Hamiltonia; even if we start from a relativistic action to obtain the Hamiltonian, it is not guaranteed that the quantum theory obtained from it will be relativistic.

I have explained exactly this in another thread here; that's why I came to the conclusion that I should ask for alternatives.

...so in a sense the process of 'quantization' is conceptually more fundamental than the resulting 'quantum theory'.

If this is the case, then one should be able to say what quantization really IS (except for an ad-hoc collection of rules)

 

[dx]

If this is the case, then one should be able to say what quantization really IS (except for an ad-hoc collection of rules)

Yes, I agree, but unfortunately this is an unsolved problem as far as I know. A precise formulation of quantization would involve conceptual as well as mathematical issues. The deepest discussion of the former that I know of are Bohr's papers. Some recent work on the mathematical aspects by Gukov and Witten: http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.0305v2.pdf

 

[julian]

Mathematical physicists have asked this question and I think "geometric quantization" might be the answer to this

 

[julian]

geometric quantization provides a general framework starting with a general symplectic manifold (and with a few technical assumptions) gives rigorous quantization rules. In fact it can be thouight of as an investigation into the whole question of the quantization procedure.

 

[julian]

Also take a look at Ashtekar's refined algebraic quantization.


I once explained geo-quantization to my friend down the pub - intend to write it all up at some point. At the mooment I'm stuck on some maths to do with it.

 

[naturale]

It is possible to build such a theory. See for instance the thread "new quantization method" (arXiv:0903.3680 [hep-th]). There quantum mechanics emerge from relativistic (not quantized) waves with periodic boundary conditions, in a very intuitive way.

 

[strangerep]

[...] one should be able to say what quantization really IS (except for an ad-hoc collection of rules)

I would have said that it's the imposition of a mapping from a space of observable quantities
to a space of numbers which can be used to form a sensible probability distribution.
This is description applies equally well to both the classical and quantum contexts.
The difference is that in the quantum context we pay attention to the noncommutativity
of the algebra of observables. Viewed in this way, classical and quantum are much
closer than is usually recognized.

The tricky bit is knowing what algebra of observables to start from. I.e., one must
choose a dynamical group and representations thereof. This is at least as difficult as
choosing a Lagrangian with interaction term, except that the passage from dynamical
group to quantum theory is somewhat cleaner -- via the method of generalized
coherent states.

But which dynamical group should one start from? Depends on the physical situation.

 

[tom.stoer]

Also take a look at Ashtekar's refined algebraic quantization.

What is this?

I have studied a lot of Ashtekar's LQG papers but I haven't seen this in detail. To me the quantization in LQG is only a mathematical adjustment due to specific details of diff.-inv. systems like GR. Is there anything more?

 

[tom.stoer]

Anyway - it seems that constructing quantum theories always starts with classical concepts and goes through some procedures. We are not able to write down a quantum theory!

 

[naturale]

Anyway - it seems that constructing quantum theories always starts with classical concepts and goes through some procedures. We are not able to write down a quantum theory!

Yes we are! In arXiv:0903.3680 "Compact time and determinism: foundations" (published in Found. Phys.) it is shown that starting from classical objects such as relativistic waves and boundary conditions quantum mechanics an exact matching with quantum mechanics emerges.
For instance it is possible to obtain:
1) Energy spectrum of the relativistic fields
2) Hilbert space
3) Schrodinger equation
4) Commutation relation
5) Path Integral
...

 

[Haelfix]

As for figuring out a general way to write down consistent quantum theories without starting from a classical starting point. Well get in line, that's one of the oldest and hardest tasks in theoretical and mathematical physics, and has been an issue since the beginning.. Not just a prosaic or academic issue either, as it was known from the beginning that the standard quantization methods (canonical quantization, BV quantization , path integral etc) have uniqueness problems -- it is often the case that a unique classical theory can lead to multiple or even infinite quantum theories.

Going the other way has never worked in generality as far as I know.. The only examples of quantum theories that exist without a classical limit that I know off, are certain rarefied examples of conformal field theories. They were kind of stumbled upon by accident, and some of them are very weird (like they don't necessarily have a lagrangian description).

In some ways, it is a bit of a miracle that Dirac's program - a set of adhoc prescriptions- has worked so well and conformed to so many experiments. But it has, and at this time no one has much of a better idea on how to proceed.

 

[julian]

refrined algebraic quatization is a rigorous quantization scheme which is applied to LQG. I think you can find a precurser to RAQ in "lectures on non-perturbative canonical gravity" by ashtekar et al.

 

[sancharsharma]

well, i don't think we start with classical physics...
look at this approach, we start with quantum amplitudes.

postulate1: probability is mod of amplitude squared

then we call everything that changes amplitudes as operators.

we define energy, momentum and all...

till now was something that's going to be true for a mathematician...

now we resort to experiments to find what the hamiltonian is for all the fundamental situations... here comes the nature finally. we don't need to call them classical. they are experimentally obtained even when we started classical.

for instance, can we prove that (force x distance) or p^2/2m is the same energy as eV??
so we cannot or atleast we don't need to prove the lagragian for all situations.

 

[Fredrik]

So my question is if there is another approach, a research program, ..., to write down or "construct" quantum theories w/o using classical expressions as a starting point?

I'd say that there are lots of ways to write down quantum theories of non-interacting systems, and that the real problem is interactions. In the Hilbert space approach to QM, a theory is defined by a specification of a projective representation of the symmetry group of spacetime. Only two spacetimes are relevant if we ignore gravity, so the theories we find this way are the non-relativistic and special relativistic single-particle theories. (Each irreducible representation defines the theory of a specific particle species). In the algebraic approach to QM, a theory is defined by specifying a C*-algebra of observables instead. There's probably a similar statement about the quantum logic approach to QM too. Edit: I should probably have said that a theory in the algebraic approach is defined by a representation of a C*-algebra. I'm not sure. I only know a little about this stuff.

As for interactions, I don't know. Isn't this the sort of thing they're trying to deal with in mathematically rigorous QFT?

If this is the case, then one should be able to say what quantization really IS (except for an ad-hoc collection of rules)

John Baez wrote some interesting comments about that on this web page.

 

[Physics Monkey]

In some sense, writing down quantum theories is an easy task. Pick a finite dimensional complex vector space with inner product, and, for time evolution, pick a Hermitian operator. That's a quantum system. I would call this a quantum computing perspective. One doesn't have to care about locality, relativistic invariance, etc. (these are in some sense classical notions anyway).

If you want, you can try to add additional structure to your quantum system. For example, you might require that the Hilbert space arise in some natural away as a composite of many smaller systems and that the Hamiltonian consists of terms involving only a few of these smaller systems. This is a very primitive notion of locality.

Perhaps not a very satisfying point of view, but there it is. I was personally never bothered by the fact that a classical system can have many "quantizations".

 

[julian]

OK sorry the original question was constructing quantum theories without resorting to classical reasoning? A valid question as classical theories don't exist, just quantum ones.

I know that rovelli starts off with information as fundamental, Isham does something "similar". Motivated by what is the fundamental entity of reality and a set of rules to go along with it.

 

[Fra]

Here is my view in case anyone missed it:

Fredrik pointed to baez notes, where Baez writes

"Quantum theory can be thought of as the generalization of classical mechanics you get by dropping the assumption that observable quantities like position and momentum commute. In quantum theory one thus learns to like noncommutative, but still associative, algebras. "

This is good, but the question is still what structure to assume on our information of the observables, and what properties does this algebra have? And how can we understand this beyond merely postulated axiomatic systems (which in themselves have no explanatory power; as the axioms are essentially arbitrary choices)

So my question is if there is another approach, a research program, ..., to write down or "construct" quantum theories w/o using classical expressions as a starting point?

I find it instructive to compare a normaly statistical reasoning, say computation of a bayesian expectation; with feynmanns path integral constrction.

In simple classical probabilistic inference where we considers information to be "stored" in a single prior in a single probability space, you some simple things like thermodynamics.

But what if we insist that information can in fact be stored in parts, parts that are defined as "priors" in different probability spaces, that furthermore has particular relations, but where we by relations between spaces can still define logical operations. An implications is that in general it doesn't commute. So there may be a first principle understanding of WHY non-commutativity is a more general case, that can still be understood within an inference perspective.

The corresponding evolution is far more complex than thermodynamics. It's not longer just dissipative stuff, we can easily get cyclic phenomena for example, which can simply be understood as oscillations between datastructurs in the inference process, as opposed to simple diffusion in case where there is just one type of datastructure.

So in my view, the essens of QM is a "measurement theory" where I associate measurement to inference. The particular structure of this theory, depends on which data structures we have. And these in turn might possible be envisions to be a result of evolution, in that they represent the most efficient overall structure for given datastreams; where the datastream is determined by the environment.

I think the future will enlighten us in more here. There are SOME ideas in this direction, but none so far that's satifsying. Ariel Caticha for example has "derived" QM as a special case of inference. However, he still sneaks in the core points as assumptions, but the general direction is good and I think it can and will be further developed; It just for some reason doesn't seem a popular area of research, because progress is extremely slow.

/Fredrik

 

[Fra]

But what if we insist that information can in fact be stored in parts

Why this would be the case, is then just because it's more efficient. And those structures in nature that did this has survived and remained stable.

Although the specific form of division can be an infinite of mathematically possible, not all of them are equally fit in a given environment.

/Fredrik

 

[Fredrik]

This is good, but the question is still what structure to assume on our information of the observables, and what properties does this algebra have?

They are C*-algebras. I don't know the algebraic approach well enough to explain it, but the books I intend to learn it from are: "An introduction to the mathematical structure of quantum mechanics", by F. Strocchi and "Mathematical theory of quantum fields", by H. Araki (supplemented by "Functional analysis: spectral theory", by V.S. Sunder).

And how can we understand this beyond merely postulated axiomatic systems (which in themselves have no explanatory power; as the axioms are essentially arbitrary choices)

Both books I mentioned contain interesting discussions about "observables" in theories of physics (in the first chapter of each book, so they're readable even if you don't know functional analysis), which give some motivation for why the set of observables is postulated to have the structure of a C*-algebra. But I think the best motivation is probably what DarMM mentioned in another thread: Abelian C*-algebras give us probability theories, i.e. they reproduce the axioms of probability theory, and non-abelian C*-algebras give us quantum theories. So quantum theories can be thought of as a generalization of the concept of probability (which was originally defined to deal only with probabilities due to ignorance).

 

[strangerep]

They are C*-algebras.
[...]
But I think the best motivation is probably what DarMM mentioned in another thread: Abelian C*-algebras give us probability theories, i.e. they reproduce the axioms of probability theory, and non-abelian C*-algebras give us quantum theories. So quantum theories can be thought of as a generalization of the concept of probability (which was originally defined to deal only with probabilities due to ignorance).

One thing that still puzzles me about the C*-algebra approach is the precise reason why it's
physically motivated to assume the algebra is normed. Only later, after introducing linear
functionals over the algebra (ie states) we get to the physically meaningful notion of
probability amplitudes, and one can define (say) a supremum norm in terms of the dual
pairing between algebra and states. But the original norm on the algebra seems adhoc.

The norm on the algebra just seems to be there so we can do analysis/calculus on the
algebra, but without further physical motivation.

Or have I missed something??

 

[tom.stoer]

Abelian C*-algebras give us probability theories, i.e. they reproduce the axioms of probability theory, and non-abelian C*-algebras give us quantum theories. So quantum theories can be thought of as a generalization of the concept of probability.

This sounds rather interesting as the Kochen-Specker theorem says that quantum mechanics does not admit certain structures known from boolean algebras on the Hilbert space (I forgot the details, so I have to check J. Bubb's book :-). So again one finds a structure that is related to probability theory.

 

[Fra]

Yes, QM uses C*-algebras. But the question was more a rethorical question in the sense that, we know of course what all these things are for normal QM.

But the question is, to question how confidence are we really in this structure, and how do we infer these structures physically?

But I think the best motivation is probably what DarMM mentioned in another thread: Abelian C*-algebras give us probability theories, i.e. they reproduce the axioms of probability theory, and non-abelian C*-algebras give us quantum theories. So quantum theories can be thought of as a generalization of the concept of probability (which was originally defined to deal only with probabilities due to ignorance).

I agree Indeed I this is the best direction to think of it as a generalization of probabilites!

This is exactly what I mean to say in my post. But in order to understand what we are doing I think we must pay some attention to the logic used, and not just make an arbitrary matehmatical generalization and think that it explains anything.

To seek an inference perspective is the way I envision this genaralisation. AS probability theory, is somehow the mathematical framework for regular probabilistic induction; which is a certain form of inference. So the probabiltiy framework, provides an inference machinery, with expectations, update rules etc.

But I speak of inference, and that we seek a physical framework that is in the form of inference, then current models is indeed a possible framework, but it's easy to see that it is not an intrinsic construction. And searching for intrinsic frameworks is a similar guide that led us from extrinsic to intrinsic formulations of geometry, and ultimately GR.

I the mainstream understanding still ignores a really important point here. The inference of QM, gives "expectations" that are somehow extrinisic. Ie. there is an external information sink that contains this. But if we scale observers down, or considers cosmological models it's IMO not really hard to see that current QM; seen as an inference framwork (that yields "probabilities" according the a new quantum logic) can't be right. The timeless hilbert spaces for example are problematic. So is the knowledge of hamiltonians.

This is really NOT just technical detrails, it really does matter to the simple foundations. Simple yes, still we seem to have a very poor understanding of these "simple" things.

/Fredrik