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-   -   Quantization = construction of quantum theories based on the classical limit? (http://www.physicsforums.com/showthread.php?t=424860)

tom.stoer Aug28-10 06:54 AM

Quantization = construction of quantum theories based on the classical limit?
 
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 Aug28-10 07:19 AM

Re: Quantization = construction of quantum theories based on the classical limit?
 
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 Aug28-10 09:04 AM

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

Quote by dx (Post 2856265)
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.

Quote:

Quote by dx (Post 2856265)
...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 Aug28-10 09:13 AM

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

Quote by tom.stoer (Post 2856342)
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/...809.0305v2.pdf

julian Aug28-10 05:12 PM

Re: Quantization = construction of quantum theories based on the classical limit?
 
Mathematical physicists have asked this question and I think "geometric quantization" might be the answer to this

julian Aug28-10 05:21 PM

Re: Quantization = construction of quantum theories based on the classical limit?
 
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 Aug28-10 05:32 PM

Re: Quantization = construction of quantum theories based on the classical limit?
 
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 Aug28-10 07:28 PM

Re: Quantization = construction of quantum theories based on the classical limit?
 
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 Aug28-10 07:48 PM

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

Quote by tom.stoer (Post 2856342)
[...] 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 Aug29-10 02:02 AM

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

Quote by julian (Post 2856894)
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 havent 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 Aug29-10 02:41 AM

Re: Quantization = construction of quantum theories based on the classical limit?
 
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 Aug29-10 05:32 AM

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

Quote by tom.stoer (Post 2857405)
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 Aug29-10 07:25 AM

Re: Quantization = construction of quantum theories based on the classical limit?
 
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 Aug29-10 01:02 PM

Re: Quantization = construction of quantum theories based on the classical limit?
 
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 Aug29-10 02:19 PM

Re: Quantization = construction of quantum theories based on the classical limit?
 
well, i dont 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 thats 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 dont 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 dont need to prove the lagragian for all situations.

Fredrik Aug29-10 04:01 PM

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

Quote by tom.stoer (Post 2856254)
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?

Quote:

Quote by tom.stoer (Post 2856342)
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 Aug29-10 04:07 PM

Re: Quantization = construction of quantum theories based on the classical limit?
 
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 Aug29-10 04:49 PM

Re: Quantization = construction of quantum theories based on the classical limit?
 
OK sorry the original question was constructing quantum theories without resorting to classical reasoning? A valid question as classical theories dont 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.


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