Contents -- Preface -- 1 Survey of Path Integral Quantization and Regularization Techniques -- 1.1 Path Integral and Regularization Techniques for Functional -- 1.2 Schwinger-Like Regularizations and Heat-Kernel Expansion -- 1.3 Logarithmic Terms in the Heat-Kernel Expansion -- 1.4 One-Loop Renormalization Group Equations -- 1.5 Static Spacetimes: Thermodynamic Effects -- 1.5.1 Static and ultrastatic spacetimes -- 1.5.2 Finite-temperature effects -- 1.5.3 The free energy -- 1.5.4 The thermodynamic potential -- 1.5.5 Regularization of the vacuum energy -- 1.5.6 A generalized vacuum energy formula -- 2 The Zeta-Function Regularization Method -- 2.1 Survey of the Chapter, Notation and Conventions -- 2.1.1 Feasibility of physical interpretation via Wick rotation -- 2.2 Heat-Kernel Expansion and Coefficients -- 2.2.1 The heat-kernel expansion on compact manifolds -- 2.2.2 The self-adjoint extension -- 2.2.3 Existence of the (differentiated) heat-kernel expansion -- 2.2.4 The heat-kernel coefficients -- 2.3 Local and Global Spectral Zeta Functions on Compact Manifolds -- 2.3.1 Weyl's asymptotic formulae -- 2.3.2 Spectral zeta functions -- 2.4 Effective Action, Effective Lagrangian and Green Functions -- 2.4.1 Comparison with the point-splitting regularization procedure -- 2.4.2 Green functions and zeta functions -- 2.4.3 Differential calculus of the heat kernel and local zeta functions -- 2.5 Noncompact Manifolds and Manifolds with a Boundary -- 2.6 The Stress-Energy Tensor and Field-Fluctuation Regularization -- 2.6.1 The stress-energy tensor -- 2.6.2 Zeta-function regularization of the stress-energy tensor and the field fluctuation -- 2.6.3 The regularized stress tensor and its properties -- 2.6.4 On the physical interpretation -- 3 Generalized Spectra and Spectral Functions on Non-commutative Spaces.

3.1 Extended Chowla-Selberg Formulae and Arbitrary Spectral Forms -- 3.2 Barnes and Related Zeta Functions -- 3.2.1 The two-dimensional case -- 3.2.2 The D-dimensional case -- 3.3 Spectral Zeta Functions for Scalar and Vector Fields on a Spacetime with a Non-commutative Toroidal Part -- 3.3.1 Poles of the zeta function -- 3.3.2 Explicit analytic continuation of ζα s) -- 3.4 Applications to Quantum Field Theory in Non-commutative Space -- 3.4.1 Finite-temperature partition function -- 3.4.2 The spectral zeta function and the regularized vacuum energy -- 3.4.3 The regularized vacuum energy -- 3.4.4 High-temperature expansion -- 4 Spectral Functions of Laplace Operator on Locally Symmetric Spaces -- 4.1 Locally Symmetric Spaces of Rank One -- 4.2 The Spectral Zeta Function -- 4.3 Asymptotics of the Heat Kernel -- 4.4 Product of Einstein Manifolds -- 4.4.1 The Kronecker sum of Laplace operators -- 4.4.2 The Selberg zeta function. Factorization formula -- 4.4.3 Meromorphic continuation -- 4.5 Real Hyperbolic Manifolds -- 4.5.1 Laplacian on forms -- 4.5.2 Simple complex Lie group -- 4.5.3 An example of functional determinant evaluation -- 4.5.4 Scalar fields in spacetime with spatial section of the form Γ\H3 -- 5 Spinor Fields -- 5.1 The Dirac Operator and Spectral Invariants -- 5.1.1 The eta invariant -- 5.1.2 Induced Chern-Simons terms by quantum effects -- 5.1.3 Another form for the eta invariant variation -- 5.2 The Massive Dirac Operator -- 5.3 One-Dimensional Example -- 5.4 The One-Loop Effective Action -- 5.5 Dirac Bundle and the Ray-Singer Norm -- 5.6 The Determinant Line Bundles -- 5.7 The Dirac Index of Hyperbolic Manifolds -- 6 Field Fluctuations and Related Variances -- 6.1 The First Variation of the Effective Action -- 6.1.1 Deformation of elliptic operators: the first variation -- 6.1.2 The vacuum expectation values.

6.2 The Second Variation of the Effective Action -- 6.2.1 Deformation of elliptic operators: the second variation -- 6.2.2 The relative variance -- 6.3 Some Examples -- 6.3.1 The Casimir slab -- 6.3.2 The D-dimensional torus -- 6.4 Remarks -- 7 The Multiplicative Anomaly -- 7.1 Introduction -- 7.2 Zeta Trace, Determinant and the Multiplicative Anomaly -- 7.2.1 The zeta determinant -- 7.2.2 Simple examples of infinite determinants -- 7.3 Perturbative Derivation of the Multiplicative Anomaly -- 7.3.1 Explicit expression for the multiplicative anomaly from perturbation theory -- 7.4 The Multiplicative Anomaly Formula -- 7.4.1 The multiplicative anomaly formula in lower dimensions -- 7.4.2 Heat-kernel coefficients from the multiplicative anomaly -- 8 Applications of the Multiplicative Anomaly -- 8.1 Anomalies for Dirac-like Operators -- 8.1.1 The one-dimensional case -- 8.1.2 Generalization to arbitrary dimensions -- 8.1.3 Harmonic oscillator in D-dimensions -- 8.2 The Massive Dirac Operator -- 8.3 Consistent, Covariant and Multiplicative Anomalies -- 8.4 Interacting Charged Scalar Model -- 8.4.1 Chemical potential in the non-interacting case revised -- 8.4.2 The interacting case in the one-loop approximation -- 8.4.3 The interacting 0(2) model -- 8.4.4 The free charged bosonic model at finite temperature -- 8.5 Concluding Remarks -- 9 The Casimir Effect -- 9.1 Introduction -- 9.2 The Casimir Energy -- 9.3 The Casimir Energy in the Ball -- 9.3.1 The method -- 9.3.2 An explicit example: the scalar field with Dirichlet boundary conditions -- 9.4 A Braneworld Computation -- 9.4.1 Casimir energy density for a dS brane in a 5D AdS background -- 9.4.2 The one-brane case -- 9.4.3 The two-brane case -- 9.4.4 The massive case -- Appendix A -- Useful Mathematical Relations -- A.l The Poisson Summation Formula -- A.2 The Mellin Transform.

A.3 The McDonald Functions -- A.4 The Riemann-Hurwitz Functions -- A.5 The Epstein Z-Function -- Appendix B -- The Wodzicki Residue -- Definitions and Conventions -- Bibliography -- Index.