Zeta function regularization

In mathematics and theoretical physics, zeta-function regularization is a type of regularization or summability method that assigns finite values to superficially divergent sums.

Definition

An example of zeta-function regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. The unregulated value 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. The sum 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.

Relation to other regularizations

Zeta-function regularization gives a nice analytic structure to any sums over an arithmetic function ${\displaystyle f(n)}$ . Such sums are known as Dirichlet series. The regularized form

${\displaystyle {\tilde {f}}(s)=\sum _{n=1}^{\infty }f(n)n^{-s}}$ 

converts divergences of the sum into simple poles on the complex s-plane. In numerical calculations, the zeta-function regularization is inappropriate, as it is extremely slow to converge. For numerical purposes, a more rapidly converging sum is the exponential regularization, given by

${\displaystyle F(t)=\sum _{n=1}^{\infty }f(n)e^{-tn}}$ 

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

${\displaystyle F(t)={\frac {a_{N}}{t^{N}}}+{\frac {a_{N-1}}{t^{N-1}}}+\ldots }$ 

one finds that the zeta-series has the structure

${\displaystyle {\tilde {f}}(s)={\frac {a_{N}}{s-N}}+\ldots }$ 

The structure of the exponential and zeta-regulators are related by means of the Mellin transform. The one may be converted to the other by making use of the integral representation of the Gamma function:

${\displaystyle \Gamma (s+1)=\int _{0}^{\infty }x^{s}e^{-x}dx}$ 

which lead to the identity

${\displaystyle \Gamma (s+1){\tilde {f}}(s)=\int _{0}^{\infty }t^{s}F(t)dt}$ 

relating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurent series.

See also

• Generating function