Zeta Function Regularization - Definition

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

if this sum converges, and by analytic continuation elsewhere. In the case when a_n = 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. 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. See "analytic torsion."

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.