Zeta function regularization: Difference between revisions

Revision as of 15:02, 25 January 2006

Revision as of 15:16, 25 January 2006

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The zeta regularization is useful as it can often be used in a way such that the various symmetries of the physical system are preserved. Besides the Casimir effect, zeta function regularization is used in [[conformal field theory]] and in fixing the the critical [[spacetime]] dimension of [[string theory]].

==Relation to numericalother methodsregularizations==

Zeta function regularization gives a nice analytic structure to any sums over an [[arithmetic function]] <math>f(n)</math>. The regularized form

:<math>\tilde{f}(s) = \sum_{n=1}^\infty f(n)n^{-s}</math>

converts divergences of the sum into [[simple pole]]s on the complex ''s''-plane. In numerical calculations, the zeta function regularization is inappropriate, as it is extremely slow to converge. A

For farnumerical purposes, a more useful sum is anthe exponential regularization, given by

:<math>F(t)=\sum_{n=1}^\infty f(n) e^{-tn}</math>

This is sometimes called the [[Z-transform]] of ''f'', where ''z''=\exp(-''t''). The analytic structure of the exponential and zeta regularizations are related. By expanding the exponential sum as a [[Laurent series]]

The analytic structure of these two functions is related.

:<math>F(t)=\frac{a_N}{t^N} + \frac{a_{N-1}}{t^{N-1}} + \ldots</math>

one finds that the zeta series has the structure

:<math>\tilde{f}(s) = \frac{a_N}{s-N} + \ldots</math>

The structure of one may be converted to the other by making use of the integral representation of the [[gamma function]]:

:<math>\int x^n e^{-tn}</math>

[[Category:Quantum field theory]]