# What is Quantization?

1. Sep 11, 2013

### bolbteppa

In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & symmetric"]), then extremizing it gives the equations of motion. Alternatively one can find a first order PDE for the action as a function of it's endpoints to obtain the Hamilton-Jacobi equation, & the Poisson bracket formulation is merely a means of changing variables in your PDE so as to ensure your new variables are still characteristics of the H-J PDE (i.e. solutions of the EOM - see No. 37). All that makes sense to me, we're extremizing a functional to get the EOM or solving a PDE which implicitly assumes we've already got the solution (path of the particle) inside of the action that leads to the PDE. However in quantum mechanics, at least in the canonical quantization I think, you apparently just take the Hamiltonian (the Lagrangian in canonical coordinates) & mish-mash this with ideas from changing variables in the Hamilton-Jacobi equation representation of your problem so that you ensure the coordinates are characteristics of your Hamilton-Jacobi equation (i.e. the solutions of the EOM), then you put these ideas in some new space for some reason (Hilbert space) & have a theory of QM. Based on what I've written you are literally doing the exact same thing you do in classical mechanics in the beginning, you're sneaking in classical ideas & for some reason you make things into an algebra - I don't see why this is necessary, or why you can't do exactly what you do in classical mechanics??? Furthermore I think my questions have some merit when you note that Schrodinger's original derivation involved an action functional using the Hamilton-Jacobi equation. Again we see Schrodinger doing a similar thing to the modern idea's, here he's mish-mashing the Hamilton-Jacobi equation with extremizing an action functional instead of just extremizing the original Lagrangian or Hamiltonian, analogous to modern QM mish-mashing the Hamiltonian with changes of variables in the H-J PDE (via Poisson brackets).

What's going on in this big Jigsaw? Why do we need to start mixing up all our pieces, why can't we just copy classical mechanics exactly - we are on some level anyway, as far as I can see... I can understand doing these things if they are just convenient tricks, the way you could say that invoking the H-J PDE is just a trick for dealing with Lagrangians & Hamiltonians, but I'm pretty sure the claim is that the process of quantization simply must be done, one step is just absolutely necessary, you simply cannot follow the classical ideas, even though from what I've said we basically are just doing the classical thing - in a roundabout way. It probably has something to do with complex numbers, at least partially, as mentioned in the note on page 276 here, but I have no idea as to how to see that & Schrodinger's original derivation didn't assume them so I'm confused about this, thanks!

To make my questions about quantization explicit if they aren't apparent from what I've written above:

a) Why does one need to make an algebra out of mixing the Hamiltonian with Poisson brackets?

(Where this question stresses the interpretation of Hamiltonian's as Lagrangian's just with different coordinates, & Poisson brackets as conditions on changing variables in the Hamilton-Jacobi equation, so that we make the relationship to CM explicit)

b) Why can't quantum mechanics just be modelled by extremizing a Lagrangian, or solving a H-J PDE?

(From my explanation above it seems quantization smuggles these idea's into it's formalism anyway, just mish-mashing them together in some vector space)

c) How do complex numbers relate to this process?

(Are they the reason quantum mechanics radically differs from classical mechanics. If so, how does this fall out of the procedure as inevitable?)

Apologies if these weren't clear from what I've written, but I feel what I've written is absolutely essential to my question, thank you.

Last edited: Sep 11, 2013
2. Sep 11, 2013

### rubi

Quantization basically means constructing a quantum theory corresponding to some theory that has your theory as it's classical limit.

In principle, there are no rules on how to do this. In practice however, several methods have emerged (like canonical quantization or path-integral quantization). There is no deep reason why these methods work. They are just heuristics. If they don't work, you just discard the theory and try to construct another one.

Unfortunately, nobody in the world knows how to find the right theory. You just have the guiding principle that your quantum theory must have the right classical limit (see correspondence principle). In the end, you always need to compare the predictions with experimental data.

3. Sep 11, 2013

### bolbteppa

In other words there very well may be an answer to my question, which is a research question... ?

4. Sep 11, 2013

### rubi

I'm not sure what your question is exactly.

The most general answer to the question "What is quantization?" is given by the first line of my previous answer in my opinion. Quantization isn't a well-defined procedure that one can follow in order to get the correct theory. It's more like witchcraft. That doesn't mean that there are no well-defined procedures to quantize a theory. It's just not guaranteed that they yield quantum theories that describe physics correctly.

If your question is "Why can't we use the same methods in quantum theory that we use in classical mechanics?", then the answer is that classical mechanics and quantum mechanics are formulated within very different frameworks. Whereas classical systems are described by differential equations, quantum theories always involve Hilbert spaces and operators.

There is no reason to believe that there is a well-defined procedure to quantize a theory that automatically yields the correct predictions. That would mean that you can construct the general theory already, just from knowing it's behavior in a special case. It might be this way, but it's highly unlikely in my opinion.

5. Sep 11, 2013

### atyy

Quantum mechanics is fundamentally formulated as a Hamiltonian theory, and is defined via commutation relations and the Hilbert space.

There is a "Lagrangian" version which is not as fundamental, but is calculationally more powerful in many cases. This is Feynman's path integral. The "saddle point approximation" to the path integral recovers the classical "extremize the action" principle. Then deviations from classical mechanics can be thought of as corrections to the saddle point approximation. Because the Hamiltonian version is more fundamental, conditions on the Lagrangian must be satisfied if the Lagrangian is to correspond to a quantum theory - for relativistic quantum field theories, some conditions go by the name of the Osterwalder-Schrader axioms.

http://www.einstein-online.info/spotlights/path_integrals

6. Sep 11, 2013

### bolbteppa

I'm really sorry guys, I've made my questions more explicit, adding them to the end of my OP.

Apologies for not being clearer before, my question is partially as to why one needs to invoke Hilbert spaces & operators? Are they a mathematical trick (the way H-J theory is a trick for working with Lagrangians & Hamiltonians)? Why do people claim this approach is absolutely essential & incontravertibly different from classical mechanics when Schrodinger's derivation was completely classical, albeit that the resulting eigenfunctions are complex (why that is I don't yet know).

I think this is a nice aspect of my question. According to Schrodinger's original derivation of his equation, which is very much in the classical Lagrange-Hamilton-Poisson framework as is very very apparent from the derivation I've linked to above, one can see that you get the Schrodinger equation which yields correct predictions. As far as I can see quantization is some algebraic formalism that results in the same results you get from solving Schrodinger, & my question is merely about the necessity of this, about why people claim this approach is so radically different when one sees it springs naturally from classical mechanics when you consider Schrodinger's derivation I've linked to.

I don't know much about path integrals & what I do know shows that they involve complex numbers from the get-go. I wonder if you'd know whether one can naturally arrive at the path integral formulation starting from Schrodinger's derivation I've linked to above? I would be really happy if a textbook were to explicitly say that "yes, we involve complex numbers in our path integral because eigenfunctions of Schrodinger's equation show complex numbers to be necessary", as opposed to just postulating that it works or something. I'm not sure what one actually does, but I'd love to do it in a way that follows what I'm talking about in this thread if you know of anything

7. Sep 11, 2013

### atyy

It's pretty standard to derive the path integral from Schroedinger's equation, try the first or second link below. The tricky thing is how to evaluate it - usually one goes to imaginary time, calculates the integral, then goes back to real time - and figuring out when this is legitimate from the point of view of Hailtonians and Hilbert spaces is part of the Osterwalder-Schrader axioms which are mentioned in the third link. It's best to regard the Hamiltonian version as fundamental, and the path integral as less fundamental tool, but a very powerful one.

http://arxiv.org/abs/quantph/0004090
http://www.blau.itp.unibe.ch/lecturesPI.pdf
http://www.einstein-online.info/spotlights/path_integrals

8. Sep 11, 2013

### bhobba

Hmmmm.

I don't quite get your issue, but I SUSPECT is how do we form a quantum theory from a classical one.

That actually is a VERY VERY deep issue and to really get to grips with it you need to study the most mathematically developed version of QM we have:

BEWARE. Such books are euphemistically said to be mathematically non trivial - translation - its bloody hard even for mathematics graduates like myself.

Also I would warm up with chapter 3 of Ballentine - QM - A Modern Development - which gives a similar sort of view but is much more accessible. Still if you are interested in this very deep issue Varadarajan is the book. Good luck - you will need it. Yea I have the book and one day may get around to a full study of it rather than a the cursory reading which is all I have so far done.

One thing I will mention is the need for complex numbers - its got to do with Wigners Theorem:
http://en.wikipedia.org/wiki/Wigner's_theorem

Unless you go to complex vector spaces it doesn't hold. This means you cant necessary find unitary transformations for symmetries which is the real deep foundation of dynamics in QM. If you do symmetry transformations you want the transformation to leave orthogonal vectors orthogonal and superpositions to still be in superposition ie unitary - there is only one way to guarantee it - Wigners Theorem. But that is only one aspect of the deep study of this stuff - it aren't easy - not easy at all.

Based on what you said in your clarifying post Varadarajan is the book you want.

Also check out the following:
http://arxiv.org/pdf/quant-ph/0101012v4.pdf
http://arxiv.org/pdf/0911.0695v1.pdf

In modern times many people such as myself think of QM as the most reasonable generalized probability model that allows for entanglement. That however doesn't get to the issue of dynamics which seems to be your concern.

Thanks
Bill

Last edited: Sep 11, 2013
9. Sep 11, 2013

### rubi

(First of all: A Hamiltonian isn't just a Lagrangian with different coordinates. It's more than that.)
As I pointed out above, we want to construct the quantum theory in such a way that it might have the right classical limit. How can we do this? The most obvious way to see this is to compare the classical Hamilton's equations with the quantum Heisenberg equations of motion:

Hamiltons equations of motion in classical mechanics: $\frac{\mathrm d}{\mathrm d t} f(t) = \{f,H\}$
Heisenbergs equations of motion in quantum mechanics: $\mathrm i\hbar\frac{\mathrm d}{\mathrm d t} \hat f(t) = [\hat f,\hat H]$

Now if you want the expectation values of the quantum dynamics to approach the classical dynamics for sharply peaked states and $\hbar\rightarrow 0$, you could postulate $[\widehat A,\widehat B] = \mathrm i\hbar\widehat{\{A,B\}} + O(\hbar^2)$. Then the idea is:
$$\frac{\mathrm d}{\mathrm d t} \left<\hat f(t)\right> = \frac{1}{\mathrm i\hbar}\left<[\hat f,\hat H]\right> = \left<\widehat{\{f,H\}}\right> + \frac{1}{\mathrm i\hbar}\left<O(\hbar^2)\right> \approx \{f,H\} + O(\hbar) \rightarrow \{f,H\}$$
So we can at least hope to get some sort of correct classical limit by transforming the classical Poisson brackets into commutators of observables on the quantum side. For canonical conjugate variables, you even get the Heisenberg uncertainty principle.

Because quantum variables aren't real-valued functions. Instead, they are operators on some Hilbert space without any canonical ordering. You can say $3 < 5$, but you can't say $\frac{\partial}{\partial x} < \frac{\partial^2}{\partial x^2}$ (this expression just doesn't make any sense). You can't "minimize" anything here.

Complex numbers aren't the reason why quantum mechanics is so drastically different. In fact, it might be possible to construct some reasonable quantum theories on Hilbert spaces over the real numbers. What makes QM so different from CM is the fact that it's formulated in an entirely different framework. States are (by definition) vectors in some Hilbert space, whereas in CM, they are just a list of real numbers and observables are self-adjoint operators (by definition), whereas in CM, they are just real-valued functions.

10. Sep 11, 2013

### rubi

Hilbert spaces and operators are the way quantum theory is defined. If you don't have a Hilbert space, you are not doing quantum theory, but rather something else. Also nobody says that quantum theory is inevitable. It's just that our best theories are quantum theories, so at the moment, there is no reason to abandon it.

By the way: You don't derive Schrödingers equation. You can motivate it, but in the end, it's an axiom.

You can't derive quantum theories. What you actually do is to use as much physical input as you have available and then try to find a theory that is consistent with it. Schrödinger's derivation isn't water-proof either. It involves a large amount of heuristics.

11. Sep 11, 2013

### bhobba

Sorry mate - you can derive it eg see Chapter 3 of Ballentine or for an even deeper look the book by Varadarajan.

Its physical basis is space-time symmetry invarience and is not an axiom per-se. Specifically it's Galilean symmetries that leads to Schrodinger's equation. In fact of course its wrong because Galilaen relativity is wrong - the correct symmetries are relativistic and leads to the relativistic equations such as the Dirac equation.

That's one reason why I always recommend Ballentine as the textbook to learn QM from - he does the treatment correctly.

Thanks
Bill

12. Sep 11, 2013

### rubi

I know this. You can also view the unitary representation of time translations as the axiom if you want to. For strongly continuous representations, they are equivalent. You can also have quantum theories with neither Galilean nor Lorentz symmetry though and still have a Schrödinger equation. Just think about a spin in a magnetic field for example). The point is that you can either view the Schrödinger equation or the strongly continuous unitary representation of time translations (possibly embedded into some larger group of symmetries) as the axiom; it doesn't really matter.

13. Sep 11, 2013

### bhobba

Hmmmm.

Yes and no.

Yes one requires postulated axioms that are motivated by observation etc etc, and even seat of the pants guesses. But from those you can derive it. Also most exposed to such axiomatic approaches recognize more fundamental approaches over others.

Of relevance to dynamics it has long been known from the work of Wigner and others its true basis is symmetries. While no one can prove it, because its not the type of thing that you can prove, I suggest anyone that has been exposed to the symmetry approach recognizes its more fundamental than simply postulating things like Schrodinger's equation. I could be wrong of course - there may be someone out there who doesn't see its elegance and power - but to be blunt - I think you would have to have a funny view of what physics is about in the sense of discovering fundamental principles if you don't see it.

Thanks
Bill

14. Sep 11, 2013

### rubi

What I mean by "you can't derive quantum theories" is that you can't derive the correct representation of the symmetries that corresponds to a classical system. Of course you need to represent the symmetries due to Wigner's theorem, but there are usually infinitely many inequivalent representations and there is no way to choose the right one without additional physical input and experimental testing.

15. Sep 11, 2013

### bhobba

Really. You don't think symmetries is the more fundamental concept?

If you don't I cant prove you wrong - but I suspect most exposed to it would disagree - I certainly do.

Thanks
Bill

16. Sep 11, 2013

### bhobba

That goes right to the heart of what I think the original post was about - exactly how one derives a quantum from a classical system - you cant uniquely do it - but whats going on is very deep and to really get to the bottom of you need the more advanced approaches such as that found in Varadarajan's book.

Thanks
Bill

17. Sep 11, 2013

### bolbteppa

If you take the calculus of variations as your starting point, then yes it is exactly just that, (refer to the Gelfand quote I linked to in my OP if you have an issue with this statement). If you take the theory of first order pde's as your starting point, then we can say something which ultimately leads us to the exact same statement (c.f. chapter 16 of the Chester book in the No. 37 link I linked to in my OP if interested). If you want to deny all this then that's fine, but it is a different conversation because Schrodinger assumed what I'm talking about in his original derivation of his equation & got the correct result, so this is something that can't be ignored in the context of my questions.

This is interesting, if you take Schrodinger's derivation as holding, & factor in the fact that solutions are complex-valued eigenfunctions, then yes I can see that to get some form of Poisson bracket formulation you'll have to slightly modify things by throwing in complex numbers, & the rest of what you've written makes vagues sense to my uneducated self. None of this requires Hilbert spaces or any assumptions deviating from classical mechanics, it just factors in the reality that we have complex-valued solutions & modifies things in light of this. Do you have any problem with that? I don't see anything crazy going on here, I don't see any departure from classical mechanics apart from a complex function solution...

I don't know much quantum mechanics, so this very well may be a good reason in terms of the formalism you know, however I'm just going off the basics, I'm not assuming any of that formalism I'm trying to find out about the motivation that leads to such formalism as being essential.

I'm not trying to be flippant or aggressive but fundamentally I just doubt that this could be the reason, mainly because you're just extremizing a functional defined on a space of functions. Furthermore, if you can't construct an action & a Lagrangian for a system then you can't even talk about a Hamiltonian for the system, just doesn't make sense if you know anything about the calculus of variations. This really is fundamental, & if I'm wrong about any of this then Schrodinger was also just as wrong as I am in his original derivation, he was never allowed to write a Hamilton-Jacobi equation for a particle in his derivation (remember this is completely derivable from an action in terms of a Lagrangian in turn expressed in terms of a Hamiltonian c.f. Landau Mechanics Section 43 & 47), yet he did & he got his Schrodinger equation which gave the correct result. I'm just trying to make sense of this, why people would ignore this & go on to all the formalism so hastily, especially when it seems to implicitly encode classical mechanics?

If you read my OP carefully you'll see that I'm questioning this very process. I've spoken about how you're using classical ideas to construct this new space, & I'm asking why one needs to do this in the first place? Why does one ignore the fact that Schrodinger derived his equation using classical principles & instead chooses to mish-mash a bunch of classical ideas together in a vector space & simply assert you can't get it from classical principles, when Schrodinger went & got it from classical principles in his original derivation?

One of my questions is about this statement - if you accept Schrodinger's equation as an axiom, & Schrodinger's equation is derivable from completely & utterly basic classical mechanics principles as in Schrodinger's original derivation, then are you not saying quantum mechanics derives from classical mechanics, albeit with eigenfunction solutions that are complex-valued?

18. Sep 11, 2013

### rubi

They are more fundamental (in case they are there), but that's not relevant here. In case you have two equivalent statements, you are free to choose which one is the axiom and which one is to be derived. I was just trying to say "you have to postulate something that is equivalent to Schrödinger's equation" in a simple way in order not to confuse the OP too much. I didn't want to reformulate his statement into something he possibly doesn't understand.

The problem is that in many cases, the number of possible candidate theories is so huge that advanced approaches don't help either and you can't help but wait for experimental data. I'm arguing that you can't derive the correct quantum theory from purely theoretical considerations.

19. Sep 11, 2013

### atyy

bolbteppa - rubi is giving you quite precise answers. I just want to add put a rough statement that may help you see that QM is really different, and as far as we know Hilbert spaces and commutation relations for operators are fundamental. The way to see that QM is really different from the Lagrangian or path integral point of view is that in classical mechanics we only take the extremal path, but in QM we sum over all paths, and those that are not extremal are very important in the sum. In QM, using only the extremal path in the path integral is a classical approximation, also called the "saddle point" approximation.

Again, the formulation with Hilbert spaces and commutation relations is fundamental. The path integral is not as fundamental, but is a very powerful tool.

Last edited: Sep 11, 2013
20. Sep 11, 2013

### bolbteppa

That may very well be true in the formalism you know, but if I try to stick to what I know & use Schrodinger's original derivation based on completely classical principles I then arrive at complex numbers naturally, & apparently from my last post we can see how one arrives at a modified Poisson bracket formulation involving complex numbers. Based on this I can happily see why one would end up with something like Wigner's theorem but to say that Wigner's theorem is the actual reason seems to me to implicitly assume a QM formalism which, to me, seems to implicitly encode classical mechanics in it's build up, which is circular to my eyes at present. Schrodinger's derivation based on classical mechanics seems pretty solid to me, & nobody has pointed out flaws or assumptions in it, why does one leave all this & start constructing hilbert spaces, postulating axioms & claiming CM derives from QM when apparently it's the other way around...?

I can tell this book is crazy, & I unerstand enough of the absolute basics of the mathematics inside to tell it's far too advanced & far too left-field as regards my questions. Based on what you know of it though, do you think it addresses my concerns about Schrodinger's equation implicitly encoding classical mechanics since it was derived from CM, & naturally results in discrete energy levels & complex eigenfunctions? I mean based on this you can see why rubi's description of quantization looks natural (described in my last response to rubi), but it seems to me to fall out of completely classical principles.

That's a nice article, & I'll definitely use it in the future, but at the moment I'm not willing to assume axioms for QM when it appears to derive naturally from classical mechanics according to Schrodinger's original derivation. I could accept axioms if they were axioms on a Hilbert space that was constructed as a mathematical trick for getting answers the same way the H-J equation is a trick for dealing with lagrangian's & hamiltonians, & I'd be happy using that safe in the knowledge that it's just math, but at the moment I'm not sure what those axioms would be formalizing, it appears to me that they are formalizing classical mechanics allowing for complex-valued solutions & nothing more... What are your thoughts on what I'm saying about it all deriving from classical mechanics?

I heard a great comment from a lecturer that QM is a c-star algebra representation of non-commutative probability or something, potentially what you are talking about :tongue: