- Classical mechanics: t↦→x(t), the world is described by particle trajectories →x(t) or xμ(λ), i.e. the Hilbert vector is the particle coordinate function →x (or xμ), which is then projected into the space parametrized by the "coordinate" time t or the relativistic parameter λ (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: xμ↦ψ(xμ), (sometimes called "the first quantization") yields Quantum mechanics, where the Hilbert vector is the wave function (being a field) |Ψ⟩ that is for example projected into coordinate space so the parameters are (→x,t) or xμ.
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: ψ(xμ)↦Φ[ψ], (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 Φ as Hilbert vector projected into quantum field space parametrized by the wave functions ψ(xμ).
Interpretation: For each possible wave function, the (to my knowledge nameless) Φ 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 Φ Non-deterministic: The wave function ψ and the "particle" position
Now, could there be a third quantization Φ[ψ(xμ)]↦ξ{Φ}? What would it mean? And what about fourth, fifth, ... quantization? Or is second quantization something ultimate?
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