From observable to operators in QFT

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The discussion centers on the concept of second quantization in quantum field theory (QFT), which is often misunderstood as implying two steps of quantization. It clarifies that in QFT, the classical field equations, like the Klein-Gordon equation, are interpreted differently, treating fields as classical quantities before quantization. The transition from observables to operators is essential for describing quantum states, with the field operator representing the observable nature of the field. The conversation also touches on the relationship between classical fields and quantum mechanics, emphasizing that QFT can be viewed as a reformulation of standard quantum mechanics for many identical particles. Overall, the dialogue highlights the complexities and nuances of understanding QFT and its foundational principles.
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Yes, that's the non-relativistic case. Note that this is a bit incomplete. One has to carefully check that the suggested operators fulfill the operator algebra, i.e., the commutation relations following from their group-theoretical meaning according to Noether's theorem. For non-relativistic QT this is the Lie algebra of a central extension of the covering group of the Galilei group, and for relativistic QT that of the covering group of the Poincare group, SL(2,C).
 
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