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Concept
Zeta function regularization
Summary
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.
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 large Re(s) by
if this sum converges, and by analytic continuation elsewhere.
In the case when an = n, the zeta function is the ordinary Riemann zeta function. This method was used by Euler to "sum" the series 1 + 2 + 3 + 4 + ... to ζ(−1) = −1/12.
showed that in flat space, in which the eigenvalues of Laplacians are known, the zeta function corresponding to the partition function 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 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. 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(−ζ′A(0)). 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.
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