QFT: Bogolyiubov transformations and KG inner product

[Joey21]

(I hope this post goes in this part of the forum)

Hi,

I was wondering if someone could help me with the following:

I have a (1+1) scalar field decomposed into two different sets of modes. One set corresponds to a Minkowski frame in (t,x) coordinates, the other to a Rinder frame in conformal ($\tau,\bar{\xi}$) coordinates. I know that I need to calculate the bogoliubov coefficients using the Klein Gordon invariante inner product

$(\phi_1,\phi_2)=i\int dx(\phi_1^*\frac{\partial{\phi_2}}{\partial{dx^0}}-\frac{\partial{\phi_1^*}}{\partial{dx^0}}\phi_2)$

How should approach this calculation in my case, where the modes are expressed in different coordinates?

I am not after an easy answer, just some guidance, so thanks in advanced. Any references I could get some furthur reading would be great too! Anything that helps furthur my understanding of QFT.

Thanks again,

Joe.

 

[strangerep]

Have you tried Birrell & Davies, sect 4.5?

 

[Joey21]

Hi,

Thanks for the reply. I checked it out, but my problem is slightly different. The field is constrained to a Dirichlet box so I'm not sure it null coordinates are the most convenient for the problem, that's why I was going to try and compute the coefficients via de inner product.

I will give it another read.

Thanks again!

 

[strangerep]

The field is constrained to a Dirichlet box

Not exactly sure what you mean.

Disclaimer: I have not worked through this (nor the ordinary Rindler space) problem pen-in-hand, but only skim-studied B&D. It could be worthwhile to work through this problem properly, if you have the stamina to type a lot more latex in detail...

 

[Joey21]

OK, let me explain.

With a the filed being confined to a Dirichlet box I mean the field is constrained to a 1D box with boundary conditions such that the field is 0 at the edges.

I managed to calculate the integral, its as simple as using the chain rule when you calculate the derivative. I was overworked at the time of posting, so sorry for that.

Thanks again for the help.

P.S If anyone is interested I can type up or scan the solution.

 

[strangerep]

If anyone is interested I can type up or scan the solution.

I am interested -- but I don't have much spare time for close study right now. I.e., I'd be interested in returning to this later, but I don't expect you to put excessive extra effort into this unless you feel inspired to do so...