- 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?
Best Answer
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:
[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:
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).