| zeta function regularization | |
|---|---|
| Name | Zeta function regularization |
| Invented by | Bernhard Riemann (foundational work on zeta functions); developed in physics by John Archibald Wheeler and formalized by Stephen Hawking and Raymond^1 (see history) |
| Introduced | 1970s |
| Field | Quantum field theory; Mathematical physics |
zeta function regularization
Zeta function regularization is a technique that assigns finite values to divergent products or sums by using analytic continuation of associated zeta functions. In Quantum Physics it provides a rigorous tool to compute otherwise ill-defined determinants and vacuum energies, playing a central role in computations such as the Casimir effect and one-loop effective actions. Its power lies in combining spectral theory with complex analysis to respect symmetries of the underlying physical system.
Divergent series and products routinely appear in quantum field theory (QFT) when computing vacuum expectation values, functional determinants, or partition functions. For example, the formal product of eigenvalues of an elliptic operator on a manifold defines a determinant that is typically infinite. Zeta function regularization replaces such divergent constructs with values obtained from a spectral zeta function, preserving diffeomorphism and gauge symmetries better than naive cutoffs. This method is widely used in contexts ranging from semiclassical approximations in path integral formulation to spectral geometry problems encountered in studies by researchers at institutions such as CERN and Princeton University.
At the core is the spectral zeta function ζ(s)=Σ_n λ_n^{-s}, where {λ_n} are nonzero eigenvalues of an elliptic operator like the Laplacian or Dirac operator on a compact manifold. The series converges for Re(s) large and admits analytic continuation to a meromorphic function on the complex plane, a property rooted in ideas originating with Bernhard Riemann's study of the Riemann zeta function ζ_R(s). Methods to extend ζ(s) include the Mellin transform of the heat kernel trace and the use of complex contour integration developed in spectral analysis by authors such as John von Neumann and Atle Selberg (Selberg trace formula). The value and derivative at specific points, notably s=0, yield regularized determinants via det' A = exp(-ζ'(0)), connecting to the theory of functional determinants and index theorems like the Atiyah–Singer index theorem.
The procedure begins by associating to an operator A its spectral zeta ζ_A(s)=Σ λ_n^{-s}. One computes ζ_A(s) for Re(s)≫0, performs analytic continuation to s=0, and defines the regularized determinant as det_ζ A := exp(-ζ_A'(0)). For operators with zero modes a projection or modified zeta is used to exclude null eigenvalues. The approach uses the asymptotic expansion of the heat kernel K(t)=Tr(e^{-tA}) as t→0+, linking heat kernel coefficients (Minakshisundaram–Pleijel or Seeley–DeWitt coefficients) to pole structure of ζ_A(s). This formalism respects gauge invariance and general covariance when applied to covariant operators on curved backgrounds, and it interfaces with regularized partition functions in the path integral via functional determinants computed at one-loop order.
Zeta regularization is commonly applied to compute one-loop effective actions, determinants of fluctuation operators around classical solutions (instantons, solitons), and vacuum energies. In the calculation of the Casimir effect, the sum over zero-point energies of quantized fields between boundaries is converted to a spectral zeta, and analytic continuation produces a finite pressure prediction between conducting plates, first experimentally measured in setups refined by groups at Bell Labs and later precision experiments at Stanford University. In curved spacetime, zeta-regularized determinants contribute to trace anomalies and the conformal anomaly computations used in semiclassical gravity and studies by Stephen Hawking and G. W. Gibbons on black hole thermodynamics. It is also used in studies of compactification in string theory and in thermal field theory to evaluate finite-temperature determinants.
Zeta function regularization differs from cutoff, dimensional, and Pauli–Villars regularizations by relying on analytic continuation instead of an explicit regulator parameter. While dimensional regularization analytically continues spacetime dimension and preserves gauge invariance, zeta regularization works directly with operator spectra and often yields scheme-independent finite parts of determinants. For renormalization, zeta regularization supplies a finite baseline; counterterms computed in a chosen renormalization scheme (e.g., minimal subtraction) are still required to absorb physical divergences consistent with renormalization group flow. Connections have been established between zeta-regularized effective actions and anomalies computed via Fujikawa's method, and comparisons with heat kernel regularization clarify relations among subtraction schemes used by practitioners at institutions like Imperial College London and the Institute for Advanced Study.
Concrete examples include the computation of the determinant of the Laplacian on the circle S^1, where ζ(s) reduces to a multiple of the Riemann zeta function and yields the finite product of eigenvalues related to the Dedekind eta function. On manifolds with boundary, explicit mode summation combined with contour integral techniques produce zeta functions for simple geometries (parallel plates, sphere, cylinder), used in Casimir calculations by authors such as K. A. Milton and M. Bordag. The heat kernel expansion K(t) ~ Σ_{k} a_k t^{(k-n)/2} links Seeley–DeWitt coefficients a_k to ultraviolet behavior and pole structure of ζ(s); computational tools include spectral geometry software and asymptotic analysis methods promoted in texts by Gilkey and Vassilevich. Numerical implementations often combine spectral cutoff with analytic continuation to evaluate ζ'(0) for complex geometries in condensed matter and cosmology applications.
Category:Mathematical physics Category:Quantum field theory