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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.

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.

Edit: Parts b) & c) have been nicely answered, thus part a) is all that remains, & it's solution seems to lie in this article, which derives the time dependent Schrodinger equation (TDSE) from the TISE. In other words, the TISE is apparently derived from classical mechanical principles, as Schrodinger did it, then at some point in the complicated derivation from page 12 on the authors reach a point at which quantum mechanical assumptions become absolutely necessary, & apparently this is the reason one assumes tons of axioms & feels comfortable constructing Hilbert spaces etc... Thus elucidating how this derivation incontravertibly results in quantum mechanical assumptions should justify why quantization is necessary, but I cannot figure this out from my poorly-understood reading of the derivation. Understanding this is the key to QM apparently, unless I'm mistaken (highly probable) thus if anyone can provide an answer in light of this articles contents that would be fantastic, thank you!

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Would you slightly clarify your thread... actually, what is the question? (a) definition of quantization; (b) why do we need quantum mechanics; (c) what is the difference between classical and quantum mechanics; (d) combination of (a),(b),(c) or others. Quantum mechanics differs the classical least action principle, since the operators do not commute. The reason for using the non commuted operators, is simply, it fits experiments. –  user26143 Sep 12 '13 at 1:19
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Dear bolbteppa: you've been asking extremely probing questions lately so I'd be a fool to think I can grasp exactly what you're driving at in my first reading (I need to come back and read again to let it sink in) but I suggest the following might be helpful: [Johannes answer]( physics.stackexchange.com/a/46209/26076) to the Physics SE question Why quantum mechanics and also my answer to Heisenberg picture of QM as a result of Hamilton formalism where I describe how the .... –  WetSavannaAnimal aka Rod Vance Sep 12 '13 at 1:25
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... Hamilton's equation with Poisson bracket can be "continuously deformed" into the Heisenberg equation of motion with $\hbar$ as the deformation parameter. –  WetSavannaAnimal aka Rod Vance Sep 12 '13 at 1:29
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b) least action on Lagrangian will give a trajectory, which is inconsistent with the uncertain principle; c) suppose there is only real number, [x,p]=1, take a hermitian conjugation on the commutator, we get [p,x]=1. Therefore it is inconsistent... You could define complex number as a pair of real numbers, but that's essential the same as using complex number... –  user26143 Sep 12 '13 at 1:33
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Perhaps you would be more comfortable with the path integral formulation of quantum mechanics, wherein the amplitude for a process is the "integral" over all possible paths of $\exp(i S/\hbar)$ where $S$ is the classical action. The classical action principle arises from a saddle-point evaluation of the integral in the $\hbar\to 0$ limit. The best book is still the original "Quantum Mechanics and Path Integrals" by Feynman and Hibbs. Make sure you get the emended edition by Daniel Styer which fixes a lot of typos. –  Michael Brown Sep 12 '13 at 2:10

6 Answers 6

I make my comments into an answer:

In my opinion your confusion arises because you assume that Classical Mechanics is the underlying framework of physics, or at least the tool necessary to describe nature. People who are close to experimental results understand that it is the experimental results which require tools necessary to describe the measurements, formulate a theory and predict new measurements, they do not have this problem. @MichaelBrown 's answer is close to what I mean. It is Classical Mechanics that is derivative to Quantum mechanics and not the other way around. Classical Mechanics emerges from Quantum Mechanics and not the other way around

An example: think of the human body before the discovery of the microscope. There was a "classical" view of what a body was. The experiments could only see and describe macroscopic effects. When the microscope was discovered the theory of cells constituting the human body and the complex functions operating on it of course became the underlying framework, and the old framework a limiting case of this.

That the theories of physics use mathematics as tools is what is confusing you, because mathematics is so elegant. But it is the physical structure we are exploring and not mathematical elegance.

There is a relevant ancient greek myth, that of Procrustis:

he had an iron bed, in which he invited every passer-by to spend the night, and where he set to work on them with his smith's hammer, to stretch them to fit. In later tellings, if the guest proved too tall, Procrustes would amputate the excess length.

If we try to impose the mathematics of classical mechanics on the microscopic data we are using the logic of Procrustis, trying to fit the data to the bed and not find the bed that fits the data.

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Unfortunately Procrustes does not address any of my three questions, though I will definitely make use of that beautiful metaphor at some point in my life, thank you. –  bolbteppa Sep 12 '13 at 3:27
    
Well, imo it answers the impulse behind your questions. You assume the classical mechanics bed and try to fit quantum mechanical formulations to classical mechanics mathematics, instead of looking how the limit as hbar goes to 0 of quantum mechanical formulations become classical mechanics ones. This means that the set of values/variables on which quantum mechanics operates mathematically is much larger than the set of values/variables classical mechanics does. –  anna v Sep 12 '13 at 3:32
    
It is true that physicists formulating quantum mechanical mathematical tools used the ones they knew from classical mechanics. The framework of finding classical mechanics emerging from quantum mechanics came later, but it exists and cannot but be true, because qm fits experimental results. –  anna v Sep 12 '13 at 3:35
    
You may (?) have located the motivation behind my question, but unfortunately I still don't know why one can't model these problems by extremizing a Lagrangian, in fact I've been told I can't by someone else even though they are comfortable assuming a Hamiltonian exists, where a Hamiltonian is nothing more than a Lagrangian in new coordinates, & Poisson brackets of the H-J PDE corresponding to this apparently non-existent action are weapons in the quantization you tell me is more fundamental than the CM used to construct it in the first place. Any thoughts on any of this? Seems circular to me. –  bolbteppa Sep 12 '13 at 3:44
    
No thoughts of why it cannot be done, though I suspect it will hinge on the value of hbar . "is more fundamental than the CM used to construct it in the first place" . The classical mechanics mathematics was/is a tool as far as QM goes. Like derivatives and integrals. It was not CM that was used but the tools, which were not good enough for the job and were modified. –  anna v Sep 12 '13 at 3:51

Concerning point c), on how complex numbers come into quantum theory:

This has a beautiful conceptual explanation, I think, by applying Lie theory to classical mechanics. The following is taken from what I have written on the nLab at quantization -- Motivation from classical mechanics and Lie theory. See there for more pointers and details:


Quantization of course was and is motivated by experiment, hence by observation of the observable universe: it just so happens that quantum mechanics and quantum field theory correctly account for experimental observations where classical mechanics and classical field theory gives no answer or incorrect answers. A historically important example is the phenomenon called the “ultraviolet catastrophe”, a paradox predicted by classical statistical mechanics which is not observed in nature, and which is corrected by quantum mechanics.

But one may also ask, independently of experimental input, if there are good formal mathematical reasons and motivations to pass from classical mechanics to quantum mechanics. Could one have been led to quantum mechanics by just pondering the mathematical formalism of classical mechanics? (Hence more precisely: is there a natural Synthetic Quantum Field Theory?)

The following spells out an argument to this effect. It will work for readers with a background in modern mathematics, notably in Lie theory, and with an understanding of the formalization of classical/prequantum mechanics in terms of symplectic geometry.

So to briefly recall, a system of classical mechanics/prequantum mechanics is a phase space, formalized as a symplectic manifold (X,ω). A symplectic manifold is in particular a Poisson manifold, which means that the algebra of functions on phase space X, hence the algebra of classical observables, is canonically equipped with a compatible Lie bracket: the Poisson bracket. This Lie bracket is what controls dynamics in classical mechanics. For instance if H∈C ∞(X) is the function on phase space which is interpreted as assigning to each configuration of the system its energy – the Hamiltonian function – then the Poisson bracket with H yields the infinitesimal time evolution of the system: the differential equation famous as Hamilton's equations.

Something to take notice of here is the infinitesimal nature of the Poisson bracket. Generally, whenever one has a Lie algebra 𝔤, then it is to be regarded as the infinitesimal approximation to a globally defined object, the corresponding Lie group (or generally smooth group) G. One also says that G is a Lie integration of 𝔤 and that 𝔤 is the Lie differentiation of G.

Therefore a natural question to ask is: Since the observables in classical mechanics form a Lie algebra under Poisson bracket, what then is the corresponding Lie group?

The answer to this is of course “well known” in the literature, in the sense that there are relevant monographs which state the answer. But, maybe surprisingly, the answer to this question is not (at time of this writing) a widely advertized fact that has found its way into the basic educational textbooks. The answer is that this Lie group which integrates the Poisson bracket is the “quantomorphism group”, an object that seamlessly leads to the quantum mechanics of the system.

Before we spell this out in more detail, we need a brief technical aside: of course Lie integration is not quite unique. There may be different global Lie group objects with the same Lie algebra.

The simplest example of this is already one of central importance for the issue of quantization, namely, the Lie integration of the abelian line Lie algebra ℝ. This has essentially two different Lie groups associated with it: the simply connected translation group, which is just ℝ itself again, equipped with its canonical additive abelian group structure, and the discrete quotient of this by the group of integers, which is the circle group U(1)=ℝ/ℤ. Notice that it is the discrete and hence “quantized” nature of the integers that makes the real line become a circle here. This is not entirely a coincidence of terminology, but can be traced back to the heart of what is “quantized” about quantum mechanics.

Namely, one finds that the Poisson bracket Lie algebra 𝔭𝔬𝔦𝔰𝔰(X,ω) of the classical observables on phase space is (for X a connected manifold) a Lie algebra extension of the Lie algebra 𝔥𝔞𝔪(X) of Hamiltonian vector fields on X by the line Lie algebra: ℝ⟶𝔭𝔬𝔦𝔰𝔰(X,ω)⟶𝔥𝔞𝔪(X). This means that under Lie integration the Poisson bracket turns into an central extension of the group of Hamiltonian symplectomorphisms of (X,ω). And either it is the fairly trivial non-compact extension by ℝ, or it is the interesting central extension by the circle group U(1). For this non-trivial Lie integration to exist, (X,ω) needs to satisfy a quantization condition which says that it admits a prequantum line bundle. If so, then this U(1)-central extension of the group Ham(X,ω) of Hamiltonian symplectomorphisms exists and is called… the quantomorphism group QuantMorph(X,ω): U(1)⟶QuantMorph(X,ω)⟶Ham(X,ω). While important, for some reason this group is not very well known, which is striking because it contains a small subgroup which is famous in quantum mechanics: the Heisenberg group.

More precisely, whenever (X,ω) itself has a compatible group structure, notably if (X,ω) is just a symplectic vector space (regarded as a group under addition of vectors), then we may ask for the subgroup of the quantomorphism group which covers the (left) action of phase space (X,ω) on itself. This is the corresponding Heisenberg group Heis(X,ω), which in turn is a U(1)-central extension of the group X itself: U(1)⟶Heis(X,ω)⟶X. At this point it is worth pausing for a second to note how the hallmark of quantum mechanics has appeared as if out of nowhere simply by applying Lie integration to the Lie algebraic structures in classical mechanics:

if we think of Lie integrating ℝ to the interesting circle group U(1) instead of to the uninteresting translation group ℝ, then the name of its canonical basis element 1∈ℝ is canonically ”i”, the imaginary unit. Therefore one often writes the above central extension instead as follows: iℝ⟶𝔭𝔬𝔦𝔰𝔰(X,ω)⟶𝔥𝔞𝔪(X,ω) in order to amplify this. But now consider the simple special case where (X,ω)=(ℝ 2,dp∧dq) is the 2-dimensional symplectic vector space which is for instance the phase space of the particle propagating on the line. Then a canonical set of generators for the corresponding Poisson bracket Lie algebra consists of the linear functions p and q of classical mechanics textbook fame, together with the constant function. Under the above Lie theoretic identification, this constant function is the canonical basis element of iℝ, hence purely Lie theoretically it is to be called ”i”.

With this notation then the Poisson bracket, written in the form that makes its Lie integration manifest, indeed reads [q,p]=i. Since the choice of basis element of iℝ is arbitrary, we may rescale here the i by any non-vanishing real number without changing this statement. If we write ”ℏ” for this element, then the Poisson bracket instead reads [q,p]=iℏ. This is of course the hallmark equation for quantum physics, if we interpret ℏ here indeed as Planck's constant. We see it arises here merely by considering the non-trivial (the interesting, the non-simply connected) Lie integration of the Poisson bracket.

This is only the beginning of the story of quantization, naturally understood and indeed “derived” from applying Lie theory to classical mechanics. From here the story continues. It is called the story of geometric quantization. We close this motivation section here by some brief outlook.

The quantomorphism group which is the non-trivial Lie integration of the Poisson bracket is naturally constructed as follows: given the symplectic form ω, it is natural to ask if it is the curvature 2-form of a U(1)-principal connection ∇ on complex line bundle L over X (this is directly analogous to Dirac charge quantization when instead of a symplectic form on phase space we consider the the field strength 2-form of electromagnetism on spacetime). If so, such a connection (L,∇) is called a prequantum line bundle of the phase space (X,ω). The quantomorphism group is simply the automorphism group of the prequantum line bundle, covering diffeomorphisms of the phase space (the Hamiltonian symplectomorphisms mentioned above).

As such, the quantomorphism group naturally acts on the space of sections of L. Such a section is like a wavefunction, except that it depends on all of phase space, instead of just on the “canonical coordinates”. For purely abstract mathematical reasons (which we won’t discuss here, but see at motivic quantization for more) it is indeed natural to choose a “polarization” of phase space into canonical coordinates and canonical momenta and consider only those sections of the prequantum line bundle which depend only on the former. These are the actual wavefunctions of quantum mechanics, hence the quantum states. And the subgroup of the quantomorphism group which preserves these polarized sections is the group of exponentiated quantum observables. For instance in the simple case mentioned before where (X,ω) is the 2-dimensional symplectic vector space, this is the Heisenberg group with its famous action by multiplication and differentiation operators on the space of complex-valued functions on the real line.

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This is absolutely stunning stuff, amazing motivation for studying universal covers & lie groups in more depth. On a basic level this ties in exactly to my original question, here you have the 'quantomorphism group' which arises by integrating the lie algebra structure of the Poisson brackets & 'seamlessly' leads to quantum mechanics. Similarly Schrodinger, at least in the time-independent situation, assumes the lie algebra structure in his variational derivation of extremizing the H-J equation & ends up with quantum mechanics. The derivation of the TISE from the TDSE is the last question mark –  bolbteppa Sep 12 '13 at 11:07
    
+1 Nice, but some stuff render as boxes; maybe you should use boxes. –  Dimensio1n0 Sep 12 '13 at 16:02
    
This answer seems interesting: quantum mechanics arising from geometry. However, this is extremely obscure for non-mathematicians like myself. Is there someway you could put it in terms a broader public could understand? –  fffred Sep 12 '13 at 22:55
    
@fffred: The summary statement is: the Poisson bracket which controls classical mechanics is what is called a Lie bracket, and Lie brackets are infinitesimal approximations to smooth symmetry groups. The smooth symmetry group which corresponds to the Poisson bracket Lie algebra contains the Heisenberg group of exponentiated quantum operators. Moreover, this symmetry group is naturally acting on the space of quantum states. This story of what is called "geoemtric quantization" is the key to understanding the universe :-) –  Urs Schreiber Sep 12 '13 at 23:18
    
Thank you for trying to simplify. Unfortunately, this is still too mathematical for me. What do the symmetry groups physically represent? Are you saying that the quantum and classical operators are contained in the same group? Then how do they differ? What makes them geometrically different? Where does quantization appear? Sorry, maybe I just need to learn more math ... –  fffred Sep 13 '13 at 0:10

Concerning point b):

Quantum mechanics can be formulated by extremizing an action and using Hamilton-Lagrange-Jacobi theory.

This is a simple but certainly underappreciated fact: the Schrödinger equation defines a Hamiltonian flow on complex projective space. A quick exposition of this fact was once posted here:

  • Scott Morrison, Quantum mechanics and geometry, November 2009 (web post)

More details on this are in

  • Abhay Ashtekar, Troy A. Schilling, Geometrical Formulation of Quantum Mechanics (arXiv:gr-qc/9706069)

and

  • L. P. Hughston, Geometry of Stochastic State Vector Reduction, Proceedings of the Royal Society (web)
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Thanks, I just found this also in Landau & Lifshitz section 20 where they re-frame QM from the point of view of extremizing an action involving complex-valued functions. In L&L the functional they use is $J[q] = \smallint \psi^*(\hat{H} - E)\psi dq$, where $\hat{H}$ & $E$ are interpreted as operators. I find projective spaces too difficult unfortunately, at the moment, but I will definitely come back to this point so thank you. –  bolbteppa Sep 12 '13 at 10:47
    
At least in the time independent scenario it seems one can justify complex functions & there is no necessity for operators if Schrodinger's original derivation holds any weight, thus the functional may be constructed in terms of standard functions as well. The time-dependent case is another story altogether, the article I've linked to holds the answers, I can't find the point at which the derivation of the TDSE completely shakes off classical ideas (which the TISE implicitly assumes, this has to be true based off of Schrodinger's derivation) & incontravertibly incorporates QM, if it exists... –  bolbteppa Sep 12 '13 at 10:50
    
@bolbteppa: am not sure what you mean here and where you are headed. A warning: while there just happen to be these formulations of Schrödinger evolution as Hamiltonian flows, there is no indication that this is more than a curiosity and that it points to something deep about quantum mechanics. Rather, I think is important to face the mathematics of quantum mechanics as what it is. If you care about deep conceptual understanding of quantum mechanics, then look at its best mathematical formulation, which is geometric quantization, see here: ncatlab.org/nlab/show/geometric%20quantization –  Urs Schreiber Sep 12 '13 at 18:30

i think the fact that indeed quantum mechanics can be formulated as a langrangian extremum (as almost any differential equation, just reverse the process of the Euler-Lagrange differemtial equation and theorem) is already answered well.

Another facet of the "quantization" process is this:

How can we take a "static" relation / equation and transform it into a process.

Difficult? Think of i like this: How can we find the solution to this equation: F(x) = x ?

if direct solving is difficult, one can always use the equation as a "process" (assuming f() function is "Lipschich" )

  1. start form an inital x1
  2. compute x2 = f(x1)
  3. goto to 1 until x1-x2 < epsilon

This made the equation into a process/algorithm, how is this related to quantum mechanics and quantization?

Well quantum mechanics does just this (at a great extent). Takes a "classical" euqaton and makes the "static variables" into "operators" (processes)

So this part of the question can have this answer.

A more ineteresting question is why this works (infact only for certain choice of coordinate systems)?

How did they (the pioneers of quantum mechanics) think of it, is it because it retains the same "classical" relations (probably)?

Can this be generalized or re-cast into sth less confusing?

PS For a further analysis of quantum mechanics and its relations to other processes, see also this other post of mine http://math.stackexchange.com/a/782596/139391

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It seems Schrodinger's original derivation was of the time-independent Schrodinger equation, & in his paper he makes no mention of the time-dependent Schrodinger equation. Thus as far as I can see this process does not apply to the time-dependent version & the problem is apparently insoluble :( A nice discussion of this is given here.

Edit: I'm not sure this is correct anymore, that article I've linked to has made my question immensely more complicated. The article assumes Schrodinger's derivation as valid, & derives the time-dependent SE from it, so apparently all of the reasons why one must assume axioms etc... are justified by that derivation, or made redundant - I have no idea, this is now the central focus of my thread it seems.

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I've made an edit to this response, one that has radically changed the thrust of my thread & made things immensely more complicated, if anybody has an idea how to deal with it - fantastic! –  bolbteppa Sep 12 '13 at 10:07

In classical mechanics the solutions of the equations of motion are the deterministic trajectory of the system. In quantum mechanics if $\Psi$ is the solution of the EOM then $\int _a^b\Psi^*(x)\Psi(x)dx$ is the probability of finding the particle between $a$ and $b$. To have QM you need to supplement the EOM with with this (and hermicity of observables).

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