Quantum Field Theory: 3-4 Equation Steps Explained

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Discussion Overview

The discussion revolves around the derivation of the fourth equation from the third equation in the context of Quantum Field Theory. Participants are seeking clarification on the steps involved in this transition, particularly focusing on the application of delta functions and the handling of integrals.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant expresses understanding of the transition from the first two equations to the third but struggles with the transition to the fourth equation.
  • Another participant emphasizes the importance of using LaTeX for posting equations to facilitate discussion and understanding.
  • A participant notes that the integral in the third equation should be a double integral, involving both ##p## and ##p'##.
  • One participant suggests that the transition from the third to the fourth equation is a "simple" application of the delta function, implying that this step may be more straightforward than previous steps.
  • Another participant raises a concern about potential issues with factors of ##2\pi## in the equations, suggesting that this might be a point of confusion that needs clarification.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the specific steps required to derive the fourth equation from the third. Multiple viewpoints and uncertainties regarding the application of delta functions and integral handling remain present.

Contextual Notes

There are unresolved mathematical steps related to the application of the delta function and the treatment of integrals, particularly concerning the factors of ##2\pi## mentioned by participants.

Adwit
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TL;DR
I had been trying to understand second quantization for few months from an easy book "Quantum Field Theory Demystified". But couldn't succeed. So, I have posted my problem.
I understand how do 3 no. equation come from 1 & 2 no. equation. But I am struggling to understand how do 4 no. equation come from 3 no. equation. Will anyone do the steps between 3 no. equation and 4 no. equation, please ?
oOAfyizG.jpg
 
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@Adwit, we normally do not allow equations posted as images; that means they cannot be quoted usefully in responses. The PF LaTeX Guide (link at the bottom left of each post window) will help you to use LaTeX to format equations directly in your post.

That said, these equations are admittedly pretty gnarly so I can understand the temptation to use images.

Adwit said:
Will anyone do the steps between 3 no. equation and 4 no. equation, please ?

Look at the delta functions in the third equation. What do they tell you about the integral over ##p^\prime##? (Note that the integral in the third equation should really be a double integral; there is an integral over ##p## and an integral over ##p^\prime##.)
 
Ok, this is the last time I posted image of calculation. For now, I will write the calculation. Now, can you do the steps between 3 no. equation and 4 no. equation ?
 
Adwit said:
Ok, this is the last time I posted image of calculation. For now, I will write the calculation. Now, can you do the steps between 3 no. equation and 4 no. equation ?

Step 3 to step 4 is a "simple" application of the delta function. That is much simpler than understanding the equations that have gone before.

You must know what the Dirac delta function ##\delta(p-p')## is?
 
For what its worth, as far as I can tell there is indeed something funny going on with the factors of ##2\pi## here. If that is indeed the problem the best is probably if you show us what you got.
 
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