# Method of calculating the vacuum energy divergence

• A
• Haorong Wu
Haorong Wu
TL;DR Summary
When calculating the vacuum energy divergence, an author said, "This divergence can be usefully analysed by performing the integral with n continued away from integral values". What does it mean?
In the book, quantum fields in curved space, when calculating the vacuum energy divergence for scalar fields, it reads:
$$\sum_{\mathbf k}\omega/2=(L^2/4\pi)^{(n-1)/2}\frac 1 {\Gamma ((n-1)/2)} \int_0^\infty (k^2+m^2)^{1/2}k^{n-2}dk .$$ This divergence can be usefully analyzed by performing the integral with ##n## continued away from integral values to obtain $$-L^{n-1}2^{-n-1}\pi^{-n/2}m^n \Gamma(-n/2).$$

I could get the answer by letting ##k=m\tan t ## and using the properties of Beta functions and Gamma functions, but I still do not understand what it means by saying "with ##n## continued away from integral values".

Haorong Wu said:
TL;DR Summary: When calculating the vacuum energy divergence, an author said, "This divergence can be usefully analysed by performing the integral with n continued away from integral values". What does it mean?
Just after your quotation from pg. 16 of Birrell & Davies, Quantum field theory in curved space, it states:
"This method of temporarily making divergent quantities finite by continuing the dimension of spacetime away from integer values forms the basis of dimensional regularization (see chapter 6)."

Thanks, @renormalize. I mistakenly thought the integral value was somehow related to the integration in the equation.

renormalize

Replies
1
Views
686
Replies
18
Views
2K
Replies
1
Views
845
Replies
7
Views
984
Replies
1
Views
813
Replies
0
Views
409
Replies
3
Views
764
Replies
2
Views
1K
Replies
16
Views
1K
Replies
24
Views
2K