# Converting momentum sums to integrals in curved spacetime

• A
• KariK
KariK
TL;DR Summary
Is Parker and Toms book wrong calculating number density of particle creation?
I am studying particle pair production using Parker and Toms book: Quantum Field Theory in Curved Spacetime. On page 48 they talk about converting the sum over momentum (k) into an integral. You assume boundary conditions so that k = 2*Pi*n/L, where n is an integer and L is the coordinate length of the cube, the volume of which is L^3. Then on page 60 he derives a formula for the number density of the created particles. There he has the volume as (L*a)^3, with a being the cosmological scale factor. But then in the next line (Equation 2.103) he uses the integral and uses (2*Pi^2*a)^-3 as the factor? (I understand that you get the 4*Pi from the angles integration.) In my mind if the side of the cube is L*a, then k = 2*Pi*n/(L*a), so we have a factor [L*a*dk/(2*Pi)]^3. So the integral in Equation 2.103 should be just (2*Pi)^-3 times the integral. Why do they have the additional a in there? Am I missing something or is the book wrong?

• Quantum Physics
Replies
1
Views
676
• Quantum Physics
Replies
4
Views
924
• Quantum Physics
Replies
3
Views
638
• Quantum Physics
Replies
4
Views
1K
• Quantum Physics
Replies
15
Views
2K
• Introductory Physics Homework Help
Replies
28
Views
558
• Quantum Physics
Replies
1
Views
621
• Quantum Physics
Replies
16
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
582
• Quantum Physics
Replies
12
Views
2K