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