Bogoliubov transformations in QFT

[pleasehelpmeno]

Homework Statement

I am trying to teach myself QFT and reproduce cosmological equations from papers.

Given the bogoliubov transformations:

i) a(conformal time η, k) = α[/k](η)a(k)+β[/k](η)b^\dagger
ii) b[\dagger] = -β*[/k](η)a(k)+α*[/k](η)b^\dagger

find the ground state number density N/V, where a|0> = b|0> = 0.

Homework Equations

<0|N/V|0>=1/(2∏)^3 ∫d^3 k (a[/r](η,r)a^\dagger[/r](η,r))

The Attempt at a Solution

I have then taken the complex conjugate of i) and done i) X i)* then made us of a|0> = 0 and thus therefore <0|a^\dagger = 0.

I then get 1\(2∏)^3 ∫ (b|β|^2 b^\dagger d^3 k but am supposed to find it equal to 1\(∏)^2 ∫ (|β|^2 k^2 dk.
I am not sure why it becomes ∏^2 or where the creation/annihilation operators disappear.

I am not sure if i am missing something obvious like a table integral or am miss understanding basic principles.

Please help

 

[mark726]

me understand where i am going wrong and how i can get to the correct solution.



Firstly, it is commendable that you are taking the initiative to teach yourself QFT and reproduce cosmological equations from papers. It is a challenging subject, but with persistence and determination, you can achieve your goal.

Regarding your question, it seems like you have made a small error in your calculations. Let me guide you through the correct steps:

1. First, let's rewrite the given equations in a more compact form for easier understanding:

i) a = αa + βb^\dagger
ii) b^\dagger = -β*a + α*b^\dagger

2. Taking the complex conjugate of equation (i), we get:

a^\dagger = α*a^\dagger + β*b

3. Now, substituting the values of b and b^\dagger from equations (i) and (ii) into the expression for the ground state number density, we get:

<0|N/V|0> = 1/(2∏)^3 ∫d^3 k (a*a^\dagger + b*b^\dagger)

4. Using the values of a^\dagger and b^\dagger from step 2, we get:

<0|N/V|0> = 1/(2∏)^3 ∫d^3 k [α*a*a^\dagger + β*b*b + α*b*b^\dagger + β*a*a]

5. Since a|0> = b|0> = 0, the terms with b and b^\dagger will disappear from the integral, leaving us with:

<0|N/V|0> = 1/(2∏)^3 ∫d^3 k α*a*a^\dagger

6. Now, using the fact that <0|a^\dagger = 0, we can simplify the expression further:

<0|N/V|0> = 1/(2∏)^3 ∫d^3 k α*a*a^\dagger = 0

7. Therefore, the final result for the ground state number density is N/V = 0.

I hope this helps you understand where you went wrong in your calculations. Remember to always double check your steps and equations to avoid making small errors. Keep up the good work and don't hesitate to ask for help when