Saturday, 22 October 2016

quantum field theory - Is a "third quantization" possible?



  • 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?




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...