Local Zeta Regularization and the Scalar Casimir Effect

A General Approach based on Integral Kernels

Zeta regularization is a method to treat the divergent quantities appearing in several areas of mathematical physics and, in particular, in quantum field theory; it is based on the fascinating idea that a finite value can be ascribed to a formally divergent expression via analytic continuation with respect to a complex regulating parameter.

This book provides a thorough overview of zeta regularization for the vacuum expectation values of the most relevant observables of a quantized, neutral scalar field in Minkowski spacetime; the field can be confined to a spatial domain, with suitable boundary conditions, and an external potential is possibly present. Zeta regularization is performed in this framework for both local and global observables, like the stress-energy tensor and the total energy; the analysis of their vacuum expectation values accounts for the Casimir physics of the system. The analytic continuation process required in this setting by zeta regularization is deeply linked to some integral kernels; these are determined by the fundamental elliptic operator appearing in the evolution equation for the quantum field. The book provides a systematic illustration of these connections, devised as a toolbox for explicit computations in specific configurations; many examples are presented. A comprehensive account is given of the existing literature on this subject, including the previous work of the authors.

The book will be useful to anyone interested in a mathematically sound description of quantum vacuum effects, from graduate students to scientists working in this area.

Contents:

• General Theory:

  • Zeta Regularization for a Scalar Field
  • The Zeta Regularized Stress-Energy VEV in Terms of Integral Kernels
  • Total Energy and Forces on the Boundary
  • Some Variations of the Previous Schemes

• Applications:

  • A Massless Field on the Segment
  • A Massless Field Between Parallel Hyperplanes
  • A Massive Field Constrained by Perpendicular Hyperplanes
  • A Massless Field in a Three-Dimensional Wedge
  • A Scalar Field with a Harmonic Background Potential
  • A Massless Field Inside a Rectangular Box

• Appendices:

  • The "Improved" Stress-Energy Tensor
  • On the Regularity of Some Integral Kernels
  • A Contour Integral Representation for Mellin Transforms
  • Some Identities for the Dirichlet Kernel in a Slab Configuration
  • Derivation of Some Results on Boundary Forces
  • An Explicit Expression for the Renormalized Dirichlet Kernel of Half-Integer Order

Readership: Graduate students and researchers including academics in theoretical physics.
Key Features:

• Zeta regularization is used in a systematic way for both local and global aspects related to the vacuum state of a quantized field. A special emphasis is given to local aspects (such as the stress-energy tensor) and to the role of boundary conditions
• Explicit computations are carried out for several configurations, applying in a uniform way the general algorithms
• The book gives a more intuitive approach to the subject by implementing the regularization via canonical quantization in a Lorentzian framework