• 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: $x^\mu\mapsto\psi(x^\mu)$, 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: $\psi(x^\mu)\mapsto \Phi[\psi]$, the second quantization 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?

• 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?

• 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: $x^\mu\mapsto\psi(x^\mu)$, 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: $\psi(x^\mu)\mapsto \Phi[\psi]$, the second quantization 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?

• 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: $x^\mu\mapsto\psi(x^\mu)$, 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: $\psi(x^\mu)\mapsto \Phi[\psi]$, the second quantization 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?