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  • Classical mechanics: $t\mapsto \vec x(t)$, the world is described by particle trajectories $\vec x(t)$ or $x^\mu(\lambda)$, i.e. the Hilbert vector is the particle coordinate function $\vec x$ (or $x^\mu$), which is then projected into the space parametrized by the "coordinate" time $t$ or the relativistic parameter $\lambda$ (which is not necessarily monotonous in $t$).
    Interpretation: For each parameter value, the coordinate of a particle is described.
    Deterministic: The particle position itself
  • Quantum mechanics: $x^\mu\mapsto\psi(x^\mu)$, (sometimes called "the first quantization") yields Quantum mechanics, where the Hilbert vector is the wave function (being a field) $|\Psi\rangle$ that is for example projected into coordinate space so the parameters are $(\vec x,t)$ or $x^\mu$.
    Interpretation: For each coordinate, the quantum field describes the charge density (or the probability of measuring the particle at that position if you stick with the non-relativistic theory).
    Deterministic: The wave function
    Non-deterministic: The particle position
  • Quantum Field Theory: $\psi(x^\mu)\mapsto \Phi[\psi]$, (called the second quantization despite the fact that now the wave field is quantized, not the coordinates for a second time) basically yields a functional $\Phi$ as Hilbert vector projected into quantum field space parametrized by the wave functions $\psi(x^\mu)$.
    Interpretation: For each possible wave function, the (to my knowledge nameless) $\Phi$ describes something like the probability of that wave function to occur (sorry, I don't know how to formulate this better, it's not really a probability). One effect is for example particle generation, thus the notion "particle" is fishy now
    Deterministic: The functional $\Phi$ Non-deterministic: The wave function $\psi$ and the "particle" position

Now, could there be a third quantization $\Phi[\psi(x^\mu)] \mapsto \xi\{\Phi\}$? What would it mean? And what about fourth, fifth, ... quantization? Or is second quantization something ultimate?

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As far as I know -- "second quantization" is just a deprecated term, used traditionaly. See e.g. here. –  Kostya Nov 13 '10 at 9:05
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8 Answers

Wow, that's a very good question. Unfortunately, I can't write down a question, because a haven't got one.

Nevertheless, I tried to found something related to third quantization in arxiv, and surprisingly (or not so surprisingly), you can find some papers related to this new step.

Just to name a few:

http://arxiv.org/abs/gr-qc/0606021

http://arxiv.org/abs/hep-th/9212044

I really hope that someone can get a full answer here.

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+1 That's some nice findings. Then I also stumbled upon nth quantization (Baez) and Strominger, Third Quantization (and Discussion), the latter suggesting (after a brief glance only) the third quantization would quantize spacetime as a result of String theory... –  Tobias Kienzler Nov 12 '10 at 12:35
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Baez's functorial description of quantization, while nice, doesn't include classical mechanics as one of the steps. My personal take on this question is that "first quantization" and "second quantization" are actually misnomers in that they describe very different mathematical processes. If no one else comes by, I might try to expand this into an answer. –  j.c. Nov 12 '10 at 14:36
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I agree with Kostya that these names are deprecated and, in this sense, should be avoided (A. Zee's, "QFT in a Nutshell", book makes this point pretty straightforwardly).

Now, if you think of the process of "quantization" as a functor, you get to Baez's constructions. But, note that the objects being acted upon by this 'quantization functor' get progressively different from what you may be expecting.

An example that comes to mind is the quantization of gerbes, which does make an appearance in high energy physics (see section 3 of Geometric Langlands From Six Dimensions). But these objects are very non-intuitive from the Physics point-of-view: you don't even get an Action associated to this construction.

So, at this point, the furthest we've moved in this direction is String Field Theory. But, in some sense, "quantization" is still a mystery…

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One more answer against “second quntization”, because I think it is a good demonstration of how a lame notation can obscure a physical meaning.

The first statement is: there is no second quantization. For example, here is citation from Steven Weinberg's book “The Quantum Theory of Fields” Vol.I:

It would be a good thing if the misleading expression ‘second quantization’ were permanently retired.

[I would even say that there is no quantization at all, as a procedure to pass from classical theory to quantum one, because (for example) quantum mechanics of single particle is more fundamental than the classical mechanics, therefore you can derive all “classical” results from QM but not vice versa. But I understand that it is a too speculative answer.]

There is a procedure called “canonical quantization”, which is used to construct a quantum theory for a classical system which has Hamiltonian dynamics, or more generally, to construct a quantum theory which has a certain classical limit.

In this case, if by the “canonical quantization” of a Hamiltonian system with finite number of degrees of freedom (classical mechanics) you imply quantum mechanics (QM) with fixed number of particles, then quantum field theory (QFT) is the “canonical quantization” of a classical Hamiltonian system with infinite number of degrees of freedom - classical field theory, not quantum mechanics. For such procedure, there is no difference between quantization of the electro-magnetic field modes and quantization of vibrational modes of the surface of the droplet of superfluid helium.

One more citation from Weinberg's book:

The wave fields $\phi$, $\varphi$, etc, are not probability amplitudes at all...

It is useful to keep in mind the following analogy: the coordinates are the “classical configuration” of a particle. QM wave function $\psi(x)$ corresponds to the “smearing” of a quantum particle over all possible “classical configurations”. QFT wave function $\Psi(A)$ corresponds to “smearing” of a quantum field over all possible configurations of a classical field $A$. Operator $\hat{A}$ corresponds to the observable $A$ in the same way as observable $x$ is represented by Hermitian operators $\hat{x}$ in QM.

The second statement is: “canonical quantization” is irrelevant in the context of fundamental theory. QFT is the only way to marry quantum mechanics to special relativity and can be contracted without a reference to any "classical crutches"

Conclusion: There is not any sequence of “quantizations” (1st, 2nd,.. nth).

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I really didn't intend this to be a revival of the "second quantization" term, I know it's "bad". But still, why shouldn't there be a next step? QFT is the quantum theory of fields, but also its "classical" Lagrangian describes the equation of motion for QM-fields, e.g. the Dirac equation. In QM, the wave function is usually used as the basic description to then obtain e.g. expectation values, while in QFT one usually starts directly with correlation functions / expectation values. Yet the is also a wave functional and a quantum-Lagrangian possible, what happens if you quantize that again? –  Tobias Kienzler Nov 15 '10 at 12:12
    
@Tobias: But why would you do that? More importantly, which object would you get if you did that? Let me try to make a long story short, amputated: the Jacobi Metric is given by $\tilde{g}_E = \sqrt{2 (E - V(q))}$, where $V$ is the Potential energy for your system (be it particles, fields, etc). Once you re-write your Lagrangian in terms of Jacobi's metric, you map the Hamiltonian flow into the Geodesic one. The bottom line is that the eqs of motion, now, have a very clear geometrical meaning. (continues…) –  Daniel Nov 15 '10 at 13:20
    
(continuing…) This geometrical meaning is given by noting that Curvature is really the relevant quantity in this game. The question, then, is the following: What would you get if you did what you want? Fine, you go ahead and quantize again… what kinds of structures do you get? What do they represent? I hinted at this in my answer above… –  Daniel Nov 15 '10 at 13:24
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Tobias, you don't understand my (actually Weinberg's) point of view. In the context of "canonical quantization", "classical Lagrangian" of QFT describes the equation of motion for classical fields, not probability amplitudes $\psi(x)$. In fact, it is not possible to perform "canonical quantization" for Dirac equation in strict mathematical way (all attempts are some sort of cheating), that is why we should consider Dirac QF in the context of the second statement I made. There aren't two steps, only one - quantum theory. –  Grisha Kirilin Nov 15 '10 at 14:10
    
One more citation from Weinberg (I really like his "The Quantum Theory of Fields", because it is quite simple and consistent): From the point of view adopted here, the free-particle Dirac equation is nothing but a Lorentz-invariant record of the convention that we have used in putting together the two irreducible representations of the proper orthochronous Lorentz group to form a field that transforms simply also under space inversion. –  Grisha Kirilin Nov 15 '10 at 14:13
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In the context of quantum field theory Weinberg's advice to ignore the term "second quantisation" is good advice. However, to go beyond quantum field theory anything goes and some people have promoted the idea of multiple quantization as a speculative idea that could be fruitful. It's not a popular idea as you can see from the other answers, but the response to this question would not be complete without mentioning it.

Beware that the term "third quantization" is used in the context of quantum cosmology and does not really mean an extra quantization after second quantization. If you want to learn about the real thing try searching for terms like "multiple quantization", "iterated quantization", "repeated quantization", "fourth quantization" or "infinite quantization" (and ignore anything about data compression.)

You will find that the results are speculative, varied and incomplete, but not always totally mad. I don't think people should get overexcited about the idea but it should not be blithely dismissed either. It's just something to keep in the back of your mind if you are trying to understand the structure of theories about quantum gravity for example.

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Second quantization is a way of recasting things. Second quantization defines fields over the Fock space so formerly waves are now parameters of field amplitudes. I have heard string theory called “third quantization,” but as I see it this is probably an abuse of language. At one time when membranes were first considered the term fourth quantization was raised a few times, though I think more in jest.

In the end it is all just quantization, and Weinberg is probably right in ignoring numberical ordering of quantization. Writing nonrelativistic QM according to $a$ and $a^\dagger$ is called second quantization by some, but really nothing much has changed.

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There's a pretty interesting article where they use a trick that they call "Third Quantization" to study open fermi systems.

http://iopscience.iop.org/1367-2630/10/4/043026 (open access no less!)

It's not exactly what you have in mind, but as clearly illustrated by all of these other answers, "third quantization" is not really canon among physicists.

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3rd Quant IS not only possible, but is now being employed to develop a quantum theory of the Multiverse. First invented 60 yrs ago by Nambu, it was first employed in string theory (Strominger), as necessary to describe topology change, in analogy to 2nd quant, which is used to explain particle creation/annihilation.

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sounds interesting, can you give my a reference to that? –  Tobias Kienzler Feb 14 '12 at 17:41
    
I agree with @Tobias Kleinzier, it would be nice if you gave a reference, so that we can believe this wvery exotic idea. –  Dimensio1n0 Sep 7 '13 at 5:34
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As far as I understand, string theory is the quantization of a conformal quantum field theory, treated as a classical theory - apparently in precisely the same way as a spinor quantum field is the quantization of the Dirac particle, treated as a classical field. Thus it is a prominent example of third quantization.

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