A Quantum Field Theory: 3-4 Equation Steps Explained

[Adwit]

TL;DR Summary

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 ?

 

[PeterDonis]

@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.

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 $p^\prime$.)

 

[Adwit]

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 ?

 

[PeroK]

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?

 

[Dr.AbeNikIanEdL]

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.