A confusion from David tong's notes on QFT

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Discussion Overview

The discussion revolves around a specific step in the derivation presented in David Tong's notes on Quantum Field Theory (QFT), particularly focusing on the insertion of the vacuum state |0><0| in the context of normal ordering and Wick's theorem. The scope includes theoretical reasoning and technical clarification related to QFT conventions and operator manipulations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the validity of inserting |0><0| in the derivation, seeking clarification on the reasoning behind it.
  • Another participant asserts that the insertion is justified through the application of Wick's theorem and the correlations of contractions in the vacuum state |0>.
  • A different participant argues that the expression is already normal ordered, suggesting that Wick's theorem may not have been applied and expresses uncertainty about the insertion of |0><0|.
  • One participant provides a detailed reasoning based on their understanding of normal ordering, explaining that the annihilation operators act on the right and that the only contributions come from terms with two particle annihilation operators, leading to the vacuum state.

Areas of Agreement / Disagreement

Participants express differing views on the application of Wick's theorem and the justification for inserting |0><0|, indicating that there is no consensus on these points. The discussion remains unresolved regarding the conventions and reasoning applied in the derivation.

Contextual Notes

Participants mention uncertainties regarding conventions used in the notes, which may affect their interpretations. There is also a lack of clarity on the assumptions underlying the application of normal ordering and the role of annihilation operators.

kof9595995
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He didn't just insert |0><0|. He's applying Wick's theorem, and his contractions correlations in the |0> state.
 
It's already normal ordered, so I don't think he applied Wick's theorem. And I still don't see why we can insert |0><0|.
 
I just took a quick look and I'm not sure about his conventions, but from the conventions I'm used to here is my reasoning:

The expression : \psi_1^\dagger \psi_1 \psi_2^\dagger \psi_2 : is normal ordered. So in any terms in the creation/annihilation-operator expansion of this, all annihilation operators act on the state to the right before any creation operators do.

Since the incoming state on the right contains two particles and no anti-particles, the only contribution from this comes from terms where there are two particle annihilation operators (no anti-particle annihilation operators). This comes only from the \psi, not the \psi^\dagger.

The only possible result from the particle annihilation operators acting on the right is the vacuum state. Therefore he can factorize as he does, and put in |0><0| in the middle.
 

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