In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators.
The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.
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 path integral 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.
Ray & Singer (1971) used this to define the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifold in their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace of A−s.
Minakshisundaram & Pleijel (1949) showed that if A is the Laplacian of a compact Riemannian manifold then the Minakshisundaram–Pleijel zeta function converges and has an analytic continuation as a meromorphic function to all complex numbers, and Seeley (1967) extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds.
Hawking (1977) 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 Mellin transformation to the trace of the kernel of heat equations.
In this case we must calculate the value of Riemann zeta function at –3, which diverges explicitly.
A detailed example of this regularization at work is given in the article on the detail example of the Casimir effect, where the resulting sum is very explicitly the Riemann zeta-function (and where the seemingly legerdemain analytic continuation removes an additive infinity, leaving a physically significant finite number).
More generally, the zeta-function approach can be used to regularize the whole energy–momentum tensor both in flat and in curved spacetime.
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
For large, real s greater than 4 (for three-dimensional space), the sum is manifestly finite, and thus may often be evaluated theoretically.
Zeta-function regularization gives an analytic structure to any sums over an arithmetic function f(n).
The regularized form 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 This is sometimes called the Z-transform of f, where z = exp(−t).
In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging is known as an Abelian mean.
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 G. H. Hardy and J. E. Littlewood in 1916[6] and is based on the application of the Cahen–Mellin integral.
The effort was made in order to obtain values for various ill-defined, conditionally convergent sums appearing in number theory.
[7] Emilio Elizalde and others have also proposed a method based on the zeta regularization for the integrals