From observable to operators in QFT

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SUMMARY

The discussion centers on the concept of second quantization in Quantum Field Theory (QFT) and its relationship with traditional Quantum Mechanics (QM). Participants clarify that second quantization does not imply two steps of quantization but rather a transition from wave functions to field operators. The Klein-Gordon equation serves as a classical field equation, and the quantization process involves replacing classical fields and their conjugate momenta with operators, adhering to commutation relations. This shift allows for the treatment of particle creation and annihilation within the framework of QFT, emphasizing the importance of understanding these concepts through foundational texts such as Ballentine's "Quantum Mechanics" and Susskind's works.

PREREQUISITES
  • Understanding of Quantum Mechanics principles, particularly the Schrödinger equation.
  • Familiarity with classical field theory and the Klein-Gordon equation.
  • Knowledge of Lagrangian mechanics and its role in field theory.
  • Basic concepts of operators and observables in quantum mechanics.
NEXT STEPS
  • Study the implications of the Klein-Gordon equation in classical field theory.
  • Learn about the quantization process in QFT, focusing on the role of operators.
  • Explore the relationship between QFT and many-particle quantum mechanics.
  • Read "Quantum Mechanics" by Ballentine for foundational concepts in quantum theory.
USEFUL FOR

Physicists, graduate students in quantum mechanics, and researchers in theoretical physics seeking to deepen their understanding of Quantum Field Theory and its foundational principles.

  • #61
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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