Zeta function regularization

In and,  regularization is a type of  or  that assigns finite values to   or products, and in particular can be used to define s and s of some s. The technique is now commonly applied to problems in , but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in.

Definition
There are several different summation methods called zeta function regularization for defining the sum of a possibly divergent series a1 + a2 + ....

One method is to define its zeta regularized sum to be ζA(&minus;1) if this is defined, where the zeta function is defined for Re(s) large by


 * $$ \zeta_A(s) = \frac{1}{a_1^s}+\frac{1}{a_2^s} +\cdots$$

if this sum converges, and by elsewhere.

In the case when an = n, the zeta function is the ordinary, and this method was used by to "sum" the series  to ζ(&minus;1) = &minus;1/12.

showed that in flat space, in which the eigenvalues of Laplacians are known, the corresponding to the  can be computed explicitly. Consider a scalar field φ contained in a large box of volume V in flat spacetime at the temperature T = β−1. The partition function is defined by a over all fields φ on the Euclidean space obtained by putting τ = it which are zero on the walls of the box and which are periodic in τ with period β. In this situation from the partition function he computes energy, entropy and pressure of the radiation of the field φ. In case of flat spaces the eigenvalues appearing in the physical quantities are generally known, while in case of curved space they are not known: in this case asymptotic methods are needed.

Another method defines the possibly divergent infinite product a1a2.... to be exp(&minus;ζ&prime;A(0)). used this to define the of a positive  A (the  of a  in their application) with s a1, a2, ...., and in this case the zeta function is formally the trace of A&minus;s. showed that if A is the Laplacian of a compact Riemannian manifold then the  converges and has an analytic continuation as a meromorphic function to all complex numbers, and  extended this to s A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization. See "."

suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta function regularization in order to calculate the partition functions for thermal graviton and matter's quanta in curved background such as on the horizon of black holes and on de Sitter background using the relation by the inverse ation to the trace of the kernel of s.

Example
The first example in which zeta function regularization is available appears in the Casimir effect, which is in a flat space with the bulk contributions of the quantum field in three space dimensions. In this case we must calculate the value of Riemann zeta function at -3, which diverges explicitly. However, it can be to s=-3 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 detail example of the, where the resulting sum is very explicitly the (and where the seemingly legerdemain analytic continuation removes an additive infinity, leaving a physically significant finite number).

An example of zeta-function regularization is the calculation of the of the  of a particle field in. More generally, the zeta-function approach can be used to regularize the whole in curved spacetime.

The unregulated value of the energy is given by a summation over the of all of the excitation modes of the vacuum:


 * $$\langle 0|T_{00} |0\rangle = \sum_n \frac{\hbar |\omega_n|}{2}$$

Here, $$T_{00}$$ is the zeroth 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 $$\omega_n$$; the absolute value reminding us that the energy is taken to be positive. This sum, as written, is usually infinite ($$\omega_n$$ is typically linear in n). The sum may be by writing it as


 * $$\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. For large, s greater than 4 (for three-dimensional space), the sum is manifestly finite, and thus may often be evaluated theoretically.

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. Zeta-function regularization is used in, and in fixing the critical  dimension of.

Relation to other regularizations
We can ask if there are any relations to the originated by the Feynman diagram. But now we may say they are equivalent to each other, see. However the main advantage of the zeta regularization is that it can be used whenever the dimensional regularization fails, for example if there are matrices or tensors inside the calculations $$ \epsilon _{i,j,k} $$

Relation to Dirichlet series
Zeta-function regularization gives an analytic structure to any sums over an f(n). Such sums are known as. The regularized form


 * $$\tilde{f}(s) = \sum_{n=1}^\infty f(n)n^{-s}$$

converts divergences of the sum into s 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


 * $$F(t)=\sum_{n=1}^\infty f(n) e^{-tn}.$$

This is sometimes called the of f, where z = exp(&minus;t). The analytic structure of the exponential and zeta-regularizations are related. By expanding the exponential sum as a


 * $$F(t)=\frac{a_N}{t^N} + \frac{a_{N-1}}{t^{N-1}} + \cdots$$

one finds that the zeta-series has the structure


 * $$\tilde{f}(s) = \frac{a_N}{s-N} + \cdots. $$

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


 * $$\Gamma(s+1)=\int_0^\infty x^s e^{-x} \, dx$$

which lead to the identity


 * $$\Gamma(s+1) \tilde{f}(s+1) = \int_0^\infty t^s F(t) \, dt$$

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

Heat kernel regularization
The sum
 * $$f(s)=\sum_n a_n e^{-s|\omega_n|}$$

is sometimes called a heat kernel or a heat-kernel regularized sum; this name stems from the idea that the $$\omega_n$$ can sometimes be understood as eigenvalues of the. In mathematics, such a sum is known as a generalized ; its use for averaging is known as an. It is closely related to the, in that


 * $$f(s)=\int_0^\infty e^{-st} \, d\alpha(t)$$

where $$\alpha(t)$$ is a, with steps of $$a_n$$ at $$t=|\omega_n|$$. A number of theorems for the convergence of such a series exist. For example, by the Hardy-Littlewood Tauberian theorem, if


 * $$L=\limsup_{n\to\infty} \frac{\log\vert\sum_{k=1}^n a_k\vert}{|\omega_n|}$$

then the series for $$f(s)$$ converges in the half-plane $$\Re(s)>L$$ and is on every  of the half-plane $$\Re(s)>L$$. In almost all applications to physics, one has $$L=0$$

History
Much of the early work establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods was done by and  in 1916 and is based on the application of the. The effort was made in order to obtain values for various ill-defined, sums appearing in.

In terms of application as the regulator in physical problems, before, J. Stuart Dowker and Raymond Critchley in 1976 proposed a zeta-function regularization method for quantum physical problems. Emilio Elizalde and others have also proposed a method based on the zeta regularization for the integrals $$ \int_{a}^{\infty}x^{m-s}dx $$, here $$ x^{-s} $$ is a regulator and the divergent integral depends on the numbers $$ \zeta (s-m) $$ in the limit $$ s \to 0 $$ see. Also unlike other regularizations such as and analytic regularization, zeta regularization has no counterterms and gives only finite results.