A method for assigning finite values to divergent spectral sums or infinite products by forming an associated zeta function ζ(s) (typically ζ(s)=∑ λ_i^{-s} for positive eigenvalues λ_i), analytically continuing ζ(s) to a neighborhood of a special point (often s=0), and defining regularized sums or determinants via values like ζ(0) or derivatives such as -ζ'(0).

Principle

Analytic continuation replaces a formally divergent expression with the analytic continuation of an auxiliary function whose value at a special parameter yields a canonical regularized value; the method relies on meromorphic continuation and choice of normalization consistent with spectral geometry or analysis.

Demonstration

For an elliptic operator with eigenvalues {λ_i>0}, define ζ(s)=∑ λ_i^{-s} for Re(s) large, continue ζ(s) meromorphically to s=0, and set log det' Δ = -ζ'(0) to obtain a finite regularized determinant used in spectral geometry and in certain quantum field computations.

Misapplication

Applying zeta regularization without ensuring the necessary analytic continuation, ignoring poles and residue contributions, or treating the regularized value as invariant under arbitrary manipulations (branch choices, noncanonical subtractions) can lead to inconsistencies or scheme-dependent results.

Consequence

Yields finite, often canonical invariants from divergent spectral data, enabling comparisons across geometries and facilitating computations in mathematical physics, though some ambiguities (finite renormalizations) may remain depending on choices made.

Reversal

Cutoff or dimensional regularizations approximate by truncation or by analytic continuation in dimension; whereas zeta regularization replaces divergence by analytic continuation of a spectral zeta function, cutoffs expose dependence on explicit regulators and subtraction schemes.

Boundary

Applicable when an associated zeta function can be defined and meromorphically continued to the relevant point; not applicable if no suitable spectral zeta exists or the continuation has essential singularities or uncontrolled residues.

Semantic Tension

Tension between formal analytic regularization and physically motivated renormalization: different regularization schemes may differ by finite terms that are physically relevant or require renormalization conditions; zeta regularization is elegant but sometimes scheme-dependent.

Synthesis

Associate a zeta function to the divergent expression, perform analytic continuation to a point where the formal expression would be evaluated, and extract finite values (e.g., ζ(0), -ζ'(0)) as canonical regularizations when the analytic structure permits, tracking any scheme-dependent ambiguities.