Zeta function regularization: Difference between revisions

Revision as of 15:30, 16 December 2005

Revision as of 23:49, 27 December 2005

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In [[mathematics]] and [[theoretical physics]], '''zeta function regularization''' is a [[summability method]] thatassign allows one to give some meaningfulfinite values to seeminglysuperficially meaninglessdivergent expressionssums. using the [[Riemann zeta function|zeta function]].

As an example, the [[vacuum expectation value]] of the [[energy]] is given by a summation over the [[zero-point energy]] of all of the excitation modes of the vacuum:

:<math>\langle 0|T_{00} |0\rangle = \sum_n \frac{\hbar |\omega_n|}{2}</math>

Here, <math>T_{00}</math> is the zero'th component of the [[energy-momentum tensor]] and the sum (which may be an integral) is understood to extend over all (positive and negative) energy modes. This sum, as written, is clearly infinite. However, it may be [[regularization (physics)|regularized]] by writing it as

:<math>\langle 0|T_{00}(s) |0\rangle =

\sum_n \frac{\hbar |\omega_n|}{2} |\omega_n|^{-s}</math>

where ''s'' is some parameter, taken to be a [[complex number]]. For large, [[real number|real]] ''s'' greater than 4 (for three-dimensional space), the sum is manifestly finite, and thus may often be evaluated theoretically.

Such a sum will typically have a [[pole (mathematics)|pole]] at ''s''=4, due to the bulk contributions of the quantum field in three space dimensions. However, it may be [[analytic continuation|analytically continued]]

For example

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An [[analytic continuation]] gives the desired result.

In [[theoretical physics]], '''zeta regularization''' is a method to assign finite values to superficially divergent sums. The method is based on generalizing the sum to the case of more general values of an exponent and treating the sum as an [[analytic function]] of the exponent -- namely a function related to the [[Riemann zeta function]]. An [[analytic continuation]] gives the desired result.

The most well-known example is the sum