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Take, for example, the hydrogen atom. Both the classical and the quantum models are based on the same Hamiltonian, describing the Coulomb potential. The classical model however misses a lot of important properties like the discrete energy spectrum. The quantum model does the job right (of course, the simple Coulomb model only works well to some limit, but that is another story).

Apparently, to obtain the correct observables like the energy spectrum one only needs to know that the right description is quantum. No new parameters which are model-specific appear (Plank's constant is universal).

Speaking more generally and more loosely, the quantum description becomes relevant at a very small scale. It seems natural to expect that a lot more details are visible at this scale. However, the input of our model, the Hamiltonian, stays basically the same. Only the general theoretical framework changes.

Probably, the question may be rephrased as follows. Why do the quantization rules exist? By the quantization rules I mean the procedures that allow to go from the classical description to the quantum in a very uniform fashion that is applicable to many systems?

Most likely my question is not too firm and contains some wrong assumptions. However, if it were not for this confusion I would not be asking!

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  • $\begingroup$ What kind fo answer to this question could there possibly be that would not just provoke the follow-up question "And why is that?". Asking why that which describes nature describes nature is not really an answerable question. $\endgroup$ – ACuriousMind Sep 1 '16 at 19:45
  • $\begingroup$ What about when you go to higher energies than QED, for example QCD where you have to introduce non-classical properties like confinement? $\endgroup$ – jim Sep 1 '16 at 19:48
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    $\begingroup$ The answer is "by pure chance", just because you are looking at very particular systems. If the system were more complicated further information wold need. Think of QFT where renormalization needs much more information than in the classical realm. Even in QM, if you consider classical systems involving observables where products like $x^mp^n$ take place, their quatisation is ambiguous and one really needs further information. The quantum world is the real world and the classical world is just an approximation. No universal quantisation procedures exist therefore. $\endgroup$ – Valter Moretti Sep 1 '16 at 19:54
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    $\begingroup$ @ACuriousMind Of course, one can ask "why is that" forever. Each correct answer deepens our knowledge though. I expect that some perspectives exists, which shed light on my question. Maybe in the spirit of Valter Moretti's comment. $\endgroup$ – Weather Report Sep 1 '16 at 20:02
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    $\begingroup$ Renormalisation, in my view, is the symptom that the classical-like framework is not enough to describe an interacting quantum field: you must supply more information at each step than the one you encapsulated in the initial classical-like description. I am referring to the finite renormalization counter terms (those which remain after having subtracted infinities) which are ambiguous and have to be fixed by hand. $\endgroup$ – Valter Moretti Sep 1 '16 at 20:29
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The "reason" why the procedure of quantization works cannot be known. Asking why that which describes nature describes nature is not a question physics can answer.

However, the procedure of quantization does not work without extra knowledge. In fact, it's not even known in all cases what the "correct" procedure for quantization is. I'll list several hurdles (without any claim to completeness) that should convince you that there is additional information necessary to quantize a classical system:

  • The Groenewold-van Hove no-go theorem (see also this answer of mine) says that canonical quantization that just replaces Poisson brackets with commutators does not work in the generality in which we would like. There are several possible modifications to the Poisson bracket (or rather the product of classical observables on the phase space) that yield a consistent quantization procedure, but that choice is not unique. You are using additional information when you pick a particular modification. This is essentially the formal reflection of what is usually called an "ordering ambiguity": Given a classical observable $x^np^m = x^{n-1}p^m x = \dots = p^m x^n$, which of these classically equivalent expressions do you turn into the corresponding quantum operator, if you have the CCR between $x$ and $p$ making them all unequal in the quantum theory?

  • Quantum anomalies: For a general discussion of anomalies, see this excellent answer by DavidBarMoshe, for a formal derivation of the possibility of the appearance fo central charges in the passage from the classical to the quantum theory see this answer of mine. The bottom line is that in the course of quantization we can get our classical symmetry groups "enlarged", and formerly invariant objects may not be invariant anymore. This usually introduces a new parameter into the quantum theory, the central charge of the enlarged symmetry group, and again needs additional input to be determined, if it doesn't wreck the quantum theory altogether.

    In fact, this might be the most important aspect of such an anomaly: If you have an anomaly of a gauge or gravitational symmetry, you don't have a consistent quantum theory. In certain field theories, the anomaly term is naturally determined by the rest of the theory, so unless those "miraculously" cancel, the quantum theory of such field theories does not exist in the usual sense. No amount of additional information can fix this, we simply do not know a consistent quantization of such theories.

  • The lattice problem: Classically, it is rather uncontroversial that we can view comtinuum field theories as the limits of discretized theories. Quantumly, this becomes horrendously difficult: It is not known whether the continuum limit of a quantized lattice theory conincides with the quantization of the continuum theory; in fact, I believe this is not always the case, see, for example, the problem of triviality of the lattice $\phi^4$ theory. However, one might remark that this particular problem is due to the absence of a fully rigorous framework of quantum field theory in general.

Finally, let me remark that thinking of quantization as a fundamental operation has it the wrong way around if we take quantum mechanics seriously: It is the classical system that must be obtained from the quantum system in a certain limit, not the other way around. It is perfectly possible that there are quantum systeme without corresponding classical system - they just have no way to view them that would look classical to us. For a handwavy example, think about fermionic/spin-1/2 degrees of freedom: These are very hard to come by in a classical theory since there's simply no motivation to consider them, but they emerge rather naturally from the quantum viewpoint.

In this sense, it is remarkable how well quantization works as a general guiding principle, but it shouldn't surprise us that the "we don't need any extra knowledge" is not really accurate.

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    $\begingroup$ I can't agree with your first sentence: "The "reason" why the procedure of quantization works cannot be known. Asking why that which describes nature describes nature is not a question physics can answer." There could simply be an underlying principle that in turn leads to the need of applying quantization rules to Lagrangian-theories. That principle could in fact just be as reasonable as the principle of relativity or some experimental confirmable statement (e.g. constancy of light for relativistic mechanics). $\endgroup$ – image Sep 7 '16 at 12:58
  • $\begingroup$ I agree with your technical points but not with the general attitude. To quote your last sentence "it is remarkable how well quantization works as a general guiding principle". But that's what my question is about! Given a proper (quantum) theory aren't we able to explain this surprising universality of the classical limits which allow to reconstruct a lot of quantum behaviour without additional input, $f(\hbar)$ from $f(0)$? There are exceptions, of course, but they should not be surprising. It is the success of the naive approach which in my view deserves an explanation. $\endgroup$ – Weather Report Sep 7 '16 at 13:24
  • $\begingroup$ @WeatherReport: The naive approach is not as naive as it seems: Try it with polar coordinates, or action-angle variables (this was Bohr's and Sommerfeld's original attempt) and it goes wrong rather quickly. The canonical quantization we teach today is finely crafted to be as naive-looking as possible while getting as much right as possible. $\endgroup$ – ACuriousMind Sep 7 '16 at 13:28
  • $\begingroup$ @ACuriousMind Well, that's might be it, the textbooks fool us! I've always been suspecting that. Unfortunately, this would be hard to back up in detail. Let it stay a working assumption unless something better will show up. $\endgroup$ – Weather Report Sep 7 '16 at 13:58
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Some years ago I started with almost the same question: "What is it that makes us quantizate a system or what happens when we quantizate a system?"

Questions like this were asked approximately 90-50 years ago in a similar way by analyzing whether or not the description of quantum mechanics is complete and real (that is wheter all elements have a real counterpart).

The topic was setteled with the so called Copenhagen interpretation, the EPR paradox and finally with Bell's inequalites, which all together tell us that quantum mechanics is a bit strange. For example one shouldn't think of the wavefunction as a real particle unless currently meassured by some classical measurment apparatus and that such things are in absolut contradiction to a reasonable pictorial explanation of quantum mechanics.

I found all that a bit dissatisfying and went on to find a flaw in that view of quantum mechanics.


The first thing I stumbled across was bohmian mechanics which tries to explain the quantization procedure onto the fact that we indeed just didn't know the "right" classcial equations. One can show, that solving the Schrödinger equation (which one arives at by canonical quantization) $$ \left(-\frac{\hbar^2}{2m}\Delta + V(x)\right)\ \Psi = i\hbar \frac{\partial}{\partial t}\ \Psi $$ is equivalent to solving two equations \begin{align} (1)&\ \ \dot{\vec{p}} = \vec{F} - \vec{\nabla} Q\\ (2)&\ \ \frac{\partial R^2}{\partial t} + \vec{\nabla} \cdot \left(\frac{\vec{p}}{m} \cdot R^2\right) = 0 \end{align}

when one considers wavefunctions $\Psi = R \cdot \exp \left(i\frac{S}{\hbar}\right)$ which is no restriction to generality. Equation (2) is the continuity equation for a charge-denstiy $\rho = R^2$ whichs happens to be the probability distribution $\varrho = |\Psi|^2 = R^2$ in quantum mechanics. The first equation (1) is just usual classical mechanics extended by an additional potential $Q = -\frac{\hbar^2}{2m} \frac{\Delta R}{R}$ the so called quantum potential. This interpretation has some problems though. First and foremost, it can't explain (just axiomize) why a real charge-distribution $R^2$ governs the whole statistical behavior of a system regardless of the other acting forces $\vec{F}$.

The key to understanding quantum mechanics is understanding its statistical nature. So could it be that quantum mechanics is some kind of usual classical statistical mechanics (since both seem to be related by the same Lagrangians/Hamiltonians)?

Bell investigated this question by his famous Bell inequalities and came to the conclusion, that there are indeed expectation values in quantum mechanics (which are in agreement with experiment) that cannot be reproduced by any classical statistical mechanics (in the usual sense of non-instantanious action, e.g. relativistic mechanics). He was nominated for a Nobel prize which accounts for the credibility physicist put into these inequalities. As a result there should be no way of describing quantum mechanics onto the basis of classical statistical mechanics.

However, as far as my analysis goes there is a mayor flaw in the derivation of those inequalites, which make them devoid of meaning (e.g. classical systems can violate them too). I'm not the first to come to this conclusion, in fact there is a huge list of so called loopholes in Bell's theorem, which for the most part concentrate on the measurment process and on whether or not violations if found can be interpreted according to Bell's theorem.

Unfortunately, due to the philosophical nature of that question, that whole field of research has been drifted to the crackpot area. Only lately (last 10-20 years or so) it became a bit more popular again.

Now, if you accept my statement that Bell's theorem is wrong, there is no need to discard the possiblity of quantum mechanics being some kind of statistical mechanics. In fact, there might be a way to show that the process of quantizating a theory is just doing classical statistical mechanics with some further assumptions.

Still, this cannot account for the fact that usual classical statistical mechanics is an ensemble statistical mechanics while standard QM and experiments are usually about single particles. In ensemble mechanics, one calculates expectaion values on the basis of many similar and independant particles that have different initial values (e.g. place and momentum). In an experiment however, it seems a single particle mystically knows how to behave according to different and not present ensemble particles. This problem can be solved by the so called principle of Ergodicity, which states that for some systems the mean value over time is the same as the ensemble mean. Usually, this only holds for chaotical systems, which clearly we have counterexamples for (not every system we observe behaves chaotically).

The current pinnacle of quantum mechanics QFT gets rid of the description of nature onto the basis of particles. Everything becomes a field, whichs is an object with infinitely many degress of freedom. E.g., there is an electron field, as well as an photon field. Only later one introduces states, that are in close relation to particles as we know them. In context, of the classical statistical interpretation this means, that particles are just statistical artifacts of the theory, that is, the fields can be in states that "simulate" the behaviour of particles. Due to the infinity of degrees of freedom of such a field, it is quite possible that the principle of Ergodicity holds, such that a measurment within a certain finite time interval $\Delta t$ actually reflects the ensemble mean of the field!


As a result we have regained the following pictorial view of quantum mechanics:

Take for instance the hydrogen atom. It consists of an electron field, a photon field and a proton field (or rather quark and gluon fields, that form the proton). Those fields behave according to the non-quantizated equations of QFT-Lagrangians. Due to the infinity of degrees of freedom, the behaviour is highly chaotic. As a result, we are only interested in the mean behavior of such a system. One would then try to calculate the time mean of that system which is (due to the principle of Ergodicity) the same as the ensemble mean. The process of canonical quantization is now just the usage of usual ensemble statistical mechanics. We know that there are statistical states that correspond to our pictorial view of single particles and thus we can explain why experiments show that the hydrogen atom consists of particles that act differently than free particles. E.g. the electron doesn't radiate Bremsstrahlung and has a quantizated mean energy level because bound (statistical) partical states are ultimatly different to the ones formed by non-interacting fields (free statistical particle states).


So, to come back to your question: "Why is there no need in extra knowledge to go from the classical to the quantum desctiption of a system?"

Answer: We simply do statistical mechanics based on the classical equations.

This is a highly hypothetically standpoint, but it represent my currents views on the quantization process and quantum mechanics. It all falls and stand with the assumption: $\textrm{quantization} \leftrightarrow \textrm{statistical mechanics}$. There has been some work on this topic, e.g. in the form of Koopman–von Neumann classical mechanics which shows that statistical mechanics can be brought into a form of operators on Hilbert-spaces. Recently I also found a way to derive the quantization rule $\vec{p} \rightarrow -i\hbar \vec{\nabla}$ based on a classical statistical mechanical expectation value, but it's not yet in a form that can be published. So take all this with caution.

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  • $\begingroup$ I suspect that the down voter stopped at the place where Bohm was mentioned. Sorry, but that's my gut feeling, too. I would rather prefer the explanation from the traditional viewpoint, not the alternative. Of course, unless they are fully equivalent. And showing that should be quite problematic in your case, or maybe you do not expect that at all? As a side note, the counter examples to the naive quantization that other people point out do not bother me, but should really be a concern for you, given the generality of the answer you propose. $\endgroup$ – Weather Report Sep 7 '16 at 13:33
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    $\begingroup$ @WeatherReport: Actually I didn't explain it in context of bohmian mechanics, just mentioned it as a approach that has some difficulties. And yes, bohmian mechanics is completly equivalent to standard QM with the Schrödinger equation as far as any calculation goes. Only the interpretations are different, which is why most people reject it. However my main point doesn't thouch the topic of bohmian mechanics at all. Strictly speaking I say: Bell is wrong, there was some work to show that quantization could be statistical mechanics. All this is currently hypothetically. $\endgroup$ – image Sep 7 '16 at 13:42
  • $\begingroup$ @WeatherReport: Concerning the problems the others have mentioned: I can't comment so much on renormalization and how that is a problem for quantization. I do know that the current quantization procedure breaks down when trying to apply it to the equations of general relativity. Also it doesn't work with curvilinear coordinate systems, which I consider to be based on the same problem. My viewpoint is that the "statistical quantization" procedure can in fact give some general rules that reduce to "ordinary quantization" for flat spacetime and different ones for general relativity. $\endgroup$ – image Sep 7 '16 at 13:58
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    $\begingroup$ I'm not the downvoter, but I definitely don't think the OP's question has much to do with the question of "whether or not the description of quantum mechanics is complete and real." $\endgroup$ – Peter Shor Sep 7 '16 at 15:25
  • $\begingroup$ @PeterShor: quantization is part of the quantum-mechanical process of finding the right equations. The question whether or not all of the parts of quantum mechanics are real and complete (e.g not complete could mean that quantization is indeed to be extended on basis of a more general principle) thus touches also the topic of quantization. Furthermore, I have to counter Bell's Theorem, which is closely related to the topic of reality and completness, in order to make my arguments valid. $\endgroup$ – image Sep 7 '16 at 15:46

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