# Bogoliubov transformations in QFT

In summary, the ground state number density in this scenario is found to be N/V = 0. This is obtained by taking the complex conjugate of the given equations and substituting the values of b and b^\dagger into the expression for the ground state number density. Then, using the fact that <0|a^\dagger = 0, the final result is simplified to N/V = 0. It is important to double check all steps and equations to avoid making small errors.

## 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.

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

## 1. What is a Bogoliubov transformation in QFT?

A Bogoliubov transformation is a mathematical operation used in quantum field theory (QFT) to transform the creation and annihilation operators of one quantum field into those of another quantum field, while preserving the commutation relations between these operators. This transformation is often used to study the effects of changes in the underlying physical system on the quantum field theory.

## 2. What is the purpose of using Bogoliubov transformations in QFT?

The main purpose of using Bogoliubov transformations in QFT is to study the behavior of quantum fields in different physical systems. By transforming the creation and annihilation operators, one can investigate the effects of changes in the system on the dynamics of the quantum field, such as changes in particle production and interactions.

## 3. How are Bogoliubov transformations related to the concept of particle number in QFT?

Bogoliubov transformations are intimately related to the concept of particle number in QFT. In fact, the transformation of creation and annihilation operators allows for the creation and annihilation of particles in the quantum field, which directly affects the number of particles present in the system. This is crucial in understanding the behavior of quantum fields in different physical systems.

## 4. Are there any limitations or assumptions associated with using Bogoliubov transformations in QFT?

Like any mathematical operation, there are limitations and assumptions associated with using Bogoliubov transformations in QFT. One of the main assumptions is that the quantum fields under consideration are linear, meaning that the superposition principle holds. Additionally, these transformations are most effective when used in perturbation theory, where small changes in the system can be studied.

## 5. Can Bogoliubov transformations be extended to other fields of physics?

While Bogoliubov transformations were originally developed for use in QFT, they have since been extended to other fields of physics, such as condensed matter physics and quantum optics. In these fields, the transformations are used to study the effects of changes in the physical system on the behavior of quantum fields, similar to their use in QFT. However, the specific details of the transformations may vary depending on the field of study.

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