Zeta-function regularization, the multiplicative anomaly and the Wodzicki residue

Abstract

The multiplicative anomaly associated with the zeta-function regularized determinant is computed for the Laplace-type operators $L_1=-\lap+V_1$ and $L_2=-\lap+V_2$, with $V_1$, $V_2$ constant, in a D-dimensional compact smooth manifold $M_D$, making use of several results due to Wodzicki and by direct calculations in some explicit examples. It is found that the multiplicative anomaly is vanishing for $D$ odd and for D=2. An application to the one-loop effective potential of the O(2) self-interacting scalar model is outlined.

LaTeX, 17 pages, 3 figures. Small corrections, two new formulas, and an addition to the references. To appear in Commun. Math. Phys

Cited by in corpus (67)

• Heat kernel expansion: user's manual
• One-loop f(R) gravity in de Sitter universe
• String-inspired Gauss-Bonnet gravity reconstructed from the universe expansion history and yielding the transition from matter dominance to dark energy
• Euler's constant: Euler's work and modern developments
• Graviton 1-loop partition function for 3-dimensional massive gravity
• Quantum field theory on toroidal topology: algebraic structure and applications
• One-loop effective action for non-local modified Gauss-Bonnet gravity in de Sitter space
• On summability of distributions and spectral geometry
• One-loop corrections to the photon propagator in the curved-space QED
• A noncommutative approach to the cosmological constant problem
• The Dimensional-Reduction Anomaly
• Applications in physics of the multiplicative anomaly formula involving some basic differential operators
• Matching the observational value of the cosmological constant
• One-loop Modified Gravity in de Sitter Universe, Quantum Corrected Inflation, and its Confrontation with the Planck Result
• Fluctuations of quantum fields via zeta function regularization
• A different kind of string
• Regularization schemes and the multiplicative anomaly
• One-loop Effective Potential for a Fixed Charged Self-interacting Bosonic Model at Finite Temperature with its Related Multiplicative Anomaly
• Zeta Function Methods and Quantum Fluctuations
• Partition function on spheres: how (not) to use zeta function regularization
• Heat Kernel Coefficients for Chern-Simons Boundary Conditions in QED
• Physically Sound Hamiltonian Formulation of the Dynamical Casimir Effect
• Product Einstein Manifolds, Zeta-Function Regularization and the Multiplicative Anomaly
• There is no new physics in the multiplicative anomaly
• On the issue of imposing boundary conditions on quantum fields
• Renormalizable, asymptotically free gravity without ghosts or tachyons
• 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions
• Quantum Topologically Massive Gravity in de Sitter Space
• On the Dimensional Reduction Procedure
• The Dimensional-Reduction Anomaly in Spherically Symmetric Spacetimes
• Functional Determinant of the Massive Laplace Operator and the Multiplicative Anomaly
• Zeta function regularization in Casimir effect calculations and J.S. Dowker's contribution
• Multiplicative anomaly and zeta factorization
• Is the multiplicative anomaly relevant ?
• Zeta Determinant for Laplace Operators on Riemann Caps
• Singular perturbations with boundary conditions and the Casimir effect in the half space
• Is the multiplicative anomaly dependent on the regularization ?
• The Conformal Anomaly in General Rank 1 Symmetric Spaces and Associated Operator Product
• Universality and ambiguity in fermionic effective actions
• Functional Determinants in Higher Derivative Lagrangian Theories
• Notes on peculiarities of Schwinger--DeWitt technique: one-loop double poles, total-derivative terms and determinant anomalies
• Quantum probing of null-singularities
• Operator Product on Locally Symmetric Spaces of Rank One and the Multiplicative Anomaly
• Exact sum rules for inhomogeneous strings
• Inclusion of Chemical Potential for Scalar Fields
• Dynamical Casimir Effect with Semi-Transparent Mirrors, and Cosmology
• Odd Dimensional Nonlocal Liouville Conformal Field Theories
• One loop partition function of six dimensional conformal gravity using heat kernel on AdS
• Four-Fermion Interaction Model on ${\mathcal{M}}^{D-1}\otimes S^1$
• Abelian Duality
• Topology, Mass and Casimir energy
• Finite cut-off JT and Liouville quantum gravities on the disk at one loop
• Quantum gravity: unification of principles and interactions, and promises of spectral geometry
• Equivalence of the Self-Dual Model and Maxwell-Chern-Simons Theory on Arbitrary Manifolds
• The Multiplicative Anomaly of Regularized Functional Determinants
• Scaling Dimensions in AdS/QCD and the Gluon Field Strength Propagator
• Gauge fixing and BRST formalism in non-Abelian gauge theories
• Gravitino condensation, supersymmetry breaking and inflation
• Generalized Zeta Function Regularization and the Multiplicative Anomaly
• Adelic Amplitudes and Intricacies of Infinite Products
• Spectral Zeta Functions in Non-Commutative Spacetimes
• Anomalies and Analytic Torsion on Hyperbolic Manifolds
• Gravitational Zero Point Energy and the Induced Cosmological Constant
• Finite temperature quantum field theory on non compact domains and application to delta interactionsinteractions in three dimensions
• Zeta function regularization, anomaly and complex mass term
• Form factors and non-local Multiplicative Anomaly for fermions with background torsion
• Divergences of the scalar sector of quadratic gravity