The Mathematics of the Casimir Effect - School of Mathematical ...

The Casimir Effect
Making Sense of Infinity - Infinity
Conclusion
Regularising Divergent Series
The Mathematical Setting
Divergent Series
Euler-Maclaurin Formula
Abel-Plana Formula
If a divergent series cannot be summed, physicists like to remove infinity
Regularisation of
∞∑
f (n) (in particular, f (n) = n)
n=0
∞X
Heat kernel regularisation ˜f (s) = f (n) e −sn
n=0
∞X
in particular, n e −sn e s
=
(e s − 1) = 1 2 s − 1
2 12 + O(s2 )
n=0 Z ∞
Compare with t e −st dt = 1 : divergent terms cancel
0
s2 Zeta function regularisation ˜f
∞X
(s) = f (n) n −s
in particular,
∞X
n=0
n=0
n n −s = ζ(s − 1) , ζ(−1) = − 1
12
Digression
Thomas Prellberg
The Mathematics of the Casimir Effect