A. Zee, Quantum Field Theory in a Nutshell · Problem I.9.1

Field Theory Redux

Physics — QFT

Problem

Choose the damping function g(v)=1/(1+v)2g(v) = 1/(1 + v)^2 instead of the one in the text. Show that this results in the same Casimir force. [Hint: To sum the resulting series, pass to an integral representation: H(ξ)=−∑n=1∞1/(1+nξ)=−∑n=1∞∫0∞dte−(1+nξ)t=∫0∞dte−t/(1−e−ξt)H(\xi) = -\sum_{n=1}^\infty 1/(1 + n\xi) = -\sum_{n=1}^\infty \int_0^\infty dt e^{-(1+n\xi)t} = \int_0^\infty dt e^{-t}/(1 - e^{-\xi t}). Note that the integral blows up logarithmically near the lower limit, as expected.]

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