Analytic Aspects of Quantum Fields

The following sections are included:

• Preface

• Contents

The following sections are included:

• Path Integral and Regularization Techniques for Functional Determinants

• Schwinger-Like Regularizations and Heat-Kernel Expansion

• Logarithmic Terms in the Heat-Kernel Expansion

• One-Loop Renormalization Group Equations

• Static Spacetimes: Thermodynamic Effects
  • Static and ultrastatic spacetimes
  • Finite-temperature effects
  • The free energy
  • The thermodynamic potential
  • Regularization of the vacuum energy
  • A generalized vacuum energy formula

The following sections are included:

• Survey of the Chapter, Notation and Conventions
  • Feasibility of physical interpretation via Wick rotation
  • Notation and general hypotheses

• Heat-Kernel Expansion and Coefficients
  • The heat-kernel expansion on compact manifolds
  • The self-adjoint extension
  • Existence of the (differentiated) heat-kernel expansion
  • The heat-kernel coefficients

• Local and Global Spectral Zeta Functions on Compact Manifolds
  • Weyl's asymptotic formulae
  • Spectral zeta functions

• Effective Action, Effective Lagrangian and Green Functions
  • Comparison with the point-splitting regularization procedure
  • Green functions and zeta functions
  • Differential calculus of the heat kernel and local zeta functions

• Noncompact Manifolds and Manifolds with a Boundary

• The Stress-Energy Tensor and Field-Fluctuation Regularization
  • The stress-energy tensor
  • Zeta-function regularization of the stress-energy tensor and the field fluctuation
  • The regularized stress tensor and its properties
  • On the physical interpretation

The following sections are included:

• Extended Chowla-Selberg Formulae and Arbitrary Spectral Forms

• Barnes and Related Zeta Functions
  • The two-dimensional case
  • The D-dimensional case

• Spectral Zeta Functions for Scalar and Vector Fields on a Space-time with a Non-commutative Toroidal Part
  • Poles of the zeta function
  • Explicit analytic continuation of ζ_α(s)

• Applications to Quantum Field Theory in Non-commutative Space
  • Finite-temperature partition function
  • The spectral zeta function and the regularized vacuum energy
  • The regularized vacuum energy
  • High-temperature expansion

The following sections are included:

• Locally Symmetric Spaces of Rank One

• The Spectral Zeta Function

• Asymptotics of the Heat Kernel

• Product of Einstein Manifolds
  • The Krönecker sum of Laplace operators
  • The Selberg zeta function. Factorization formula
  • Meromorphic continuation

• Real Hyperbolic Manifolds
  • Laplacian on forms
  • Simple complex Lie group
  • An example of functional determinant evaluation
  • Scalar fields in spacetime with spatial section of the form Γ \ℍ³

The following sections are included:

• The Dirac Operator and Spectral Invariants
  • The eta invariant
  • Induced Chern-Simons terms by quantum effects
  • Another form for the eta invariant variation

• The Massive Dirac Operator

• One-Dimensional Example

• The One-Loop Effective Action

• Dirac Bundle and the Ray-Singer Norm

• The Determinant Line Bundles

• The Dirac Index of Hyperbolic Manifolds

The following sections are included:

• The First Variation of the Effective Action
  • Deformation of elliptic operators: the first variation
  • The vacuum expectation values

• The Second Variation of the Effective Action
  • Deformation of elliptic operators: the second variation
  • The relative variance

• Some Examples
  • The Casimir slab
  • The D-dimensional torus

• Remarks

The following sections are included:

• Introduction

• Zeta Trace, Determinant and the Multiplicative Anomaly
  • The zeta determinant
  • Simple examples of infinite determinants

• Perturbative Derivation of the Multiplicative Anomaly
  • Explicit expression for the multiplicative anomaly from perturbation theory

• The Multiplicative Anomaly Formula
  • The multiplicative anomaly formula in lower dimensions
  • Heat-kernel coefficients from the multiplicative anomaly

• Multiplicative Anomaly for Locally Symmetric Spaces

The following sections are included:

• Anomalies for Dirac-like Operators
  • The one-dimensional case
  • Generalization to arbitrary dimensions
  • Harmonic oscillator in D-dimensions

• The Massive Dirac Operator

• Consistent, Covariant and Multiplicative Anomalies

• Interacting Charged Scalar Model
  • Chemical potential in the non-interacting case revised
  • The interacting case in the one-loop approximation
  • The interacting O(2) model
  • The free charged bosonic model at finite temperature

• Concluding Remarks

The following sections are included:

• Introduction

• The Casimir Energy

• The Casimir Energy in the Ball
  • The method
  • An explicit example: the scalar field with Dirichlet boundary conditions

• A Braneworld Computation
  • Casimir energy density for a dS brane in a 5D AdS background
  • The one-brane case
  • The two-brane case
  • The massive case

The following sections are included:

• Appendix A Useful Mathematical Relations
  • The Poisson Summation Formula
  • The Mellin Transform
  • The McDonald Functions
  • The Riemann-Hurwitz Functions
  • The Epstein Z-Function

• Appendix B The Wodzicki Residue

• Definitions and Conventions

• Bibliography

• Index