Zeta function regularization

Renormalization and regularization

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.

Contents

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(−1) if this is defined, where the zeta function is defined for Re(s) large by

\zeta_A(s) = \frac{1}{a_1^s}%2B\frac{1}{a_2^s} %2B\cdots

if this sum converges, and by analytic continuation elsewhere. In the case when an = n the zeta function is the ordinary Riemann zeta function, and this method was used by Euler to "sum" the series 1 + 2 + 3 + 4 + ... to ζ(−1) = −1/12.

Another method defines the possibly divergent infinite product a1a2.... to be exp(−ζ′A(0)). 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. Hawking (1977) suggested using this idea to evaluate path integrals in curved spacetimes. 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. So for such operators one can define the determinant using zeta function regularization.

Example

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. More generally, the zeta-function approach can be used to regularize the whole energy-momentum tensor in curved spacetime. [1] [2]

The unregulated value of the energy is given by a summation over the zero-point energy 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 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 \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 regularized 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 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, renormalization and in fixing 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 f(n). Such sums are known as Dirichlet series. The regularized form

\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

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

F(t)=\frac{a_N}{t^N} %2B \frac{a_{N-1}}{t^{N-1}} %2B \cdots

one finds that the zeta-series has the structure

\tilde{f}(s) = \frac{a_N}{s-N} %2B \cdots. \,

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:

\Gamma(s%2B1)=\int_0^\infty x^s e^{-x} \, dx

which lead to the identity

\Gamma(s%2B1) \tilde{f}(s%2B1) = \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 heat kernel. In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the Laplace–Stieltjes transform, in that

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

where \alpha(t) is a step function, 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 [3]

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 uniformly convergent on every compact subset 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 G.H. Hardy and J. E. Littlewood in 1916[4] 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.

See also

References