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. 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

For example

${\displaystyle 1+2+3+\cdots =-{\frac {1}{12}}(\Re )}$

This infinite sum appears in the calculation of the Casimir force, and the regularized value of

${\displaystyle -{\frac {1}{12}}(\Re )}$

gives the correct physically measurable result.

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An analytic continuation gives the desired result.

The most well-known example is the sum

1+2+3+4+5+ ... = -1/12

whose value may be calculated as zeta(-1). Such a regularization and the corresponding result are guaranteed to preserve various symmetries of the physical system such as conformal symmetry. This particular result also occurs in the Casimir effect and the critical spacetime dimension of string theory.