Local $\zeta$-function techniques vs point-splitting procedure: a few rigorous results

Abstract

Some general properties of local $\zeta$-function procedures to renormalize some quantities in $D$-dimensional (Euclidean) Quantum Field Theory in curved background are rigorously discussed for positive scalar operators $-\Delta +V(x)$ in general closed $D$-manifolds, and a few comments are given for nonclosed manifolds too. A general comparison is carried out with respect to the more known point-splitting procedure concerning the effective Lagrangian and the field fluctuations. It is proven that, for $D>1$, the local $\zeta$-function and point-splitting approaches lead essentially to the same results apart from some differences in the subtraction procedure of the Hadamard divergences. It is found that the $\zeta$ function procedure picks out a particular term $w_0(x,y)$ in the Hadamard expansion. Also the presence of an untrivial kernel of the operator $-\Delta +V(x)$ may produce some differences between the two analyzed approaches. Finally, a formal identity concerning the field fluctuations, used by physicists, is discussed and proven within the local $\zeta$-function approach. This is done also to reply to recent criticism against $\zeta$-function techniques.

40 pages, latex, no figures, shortened version, some previous Comments and minor errors corrected, final version accepted for publication in Commun. Math. Phys

Cited by in corpus (28)

• Hadamard renormalization of the stress-energy tensor for a quantized scalar field in a general spacetime of arbitrary dimension
• Quantum Scalar Fields on Anti-de Sitter Spacetime
• Scalar field instability in de Sitter space-time
• Fluctuations of quantum fields via zeta function regularization
• One-loop stress-tensor renormalization in curved background: the relation between $\zeta$-function and point-splitting approaches, and an improved point-splitting procedure
• Standard and geometric approaches to quantum Liouville theory on the pseudosphere
• Quantum Scalar Field on the Massless (2+1)-Dimensional Black Hole Background
• Euclidean Scalar Green Function in a Higher Dimensional Global Spacetime
• Quantum effects in a rotating spacetime
• The Casimir effect for parallel plates in Friedmann-Robertson-Walker universe
• Universal definition of the non-conformal trace anomaly
• Hadamard renormalization for a charged scalar field
• On the quantization of continuous non-ultralocal integrable systems
• Anomaly and the self-energy of electric charges
• Local zeta regularization and the Casimir effect
• An approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions
• Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory
• Dynamical residues of Lorentzian spectral zeta functions
• Analytic continuation of the Hurwitz Zeta Function with physical application
• Scalar Casimir densities and forces for parallel plates in cosmic string spacetime
• Quantum scalar field in D-dimensional static black hole space-times
• Pauli-Villars regularization of Kaluza-Klein Casimir energy with Lorentz symmetry
• Local $\zeta$-functions, stress-energy tensor, field fluctuations, and all that, in curved static spacetime
• UV sensitivity of Casimir energy
• On the uniqueness of the expected stress-energy tensor in renormalizable field theories
• Wick squares of the Gaussian Free Field and Riemannian rigidity
• A note on "Algebraic approach to Casimir force between two $\delta$-like potentials" (K. Ziemian, Ann. Henri Poincaré, Online First, 2021)
• Equivalence of zeta function technique and Abel-Plana formula in regularizing the Casimir energy of hyper-rectangular cavities