Zeta function regularization

In mathematics and theoretical physics, zeta function regularization is a summability method assign finite values to superficially divergent sums.

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:

${\displaystyle \langle 0|T_{00}|0\rangle =\sum _{n}{\frac {\hbar |\omega _{n}|}{2}}}$

Here, ${\displaystyle T_{00}}$ 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 ${\displaystyle \omega _{n}}$; the absolute value reminding us that the energy is taken to be positive. This sum, as written, is clearly infinite. However, it may be regularized by writing it as

${\displaystyle \langle 0|T_{00}(s)|0\rangle =\sum _{n}{\frac {\hbar |\omega _{n}|}{2}}|\omega _{n}|^{-s}}$

where s is some parameter, taken to be a complex number. For large, 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 at s=4, due to the bulk contributions of the quantum field in three space dimensions. However, it may be analytically continued to s=0 where hopefully there is no pole, thus giving a finite value to the expression. A detailed example of this regularization at work is given in the article on the Casimir effect, where the resulting sum is very explicitly the Riemann zeta function.

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.