Davide Fermi

Local zeta regularization and the scalar Casimir effect : A general approach based on integral kernels

This is the first one of a series of papers about zeta regularization of the divergences appearing in the vacuum expectation value (VEV) of several local and global observables in quantum field theory. More precisely we consider a quantized, neutral scalar field on a domain in any spatial dimension, with arbitrary boundary conditions and, possibly, in presence of an external classical potential. We analyze, in particular, the VEV of the stress-energy tensor, the corresponding boundary forces and the total energy, thus taking into account both local and global aspects of the Casimir effect. In comparison with the wide existing literature on these subjects, we try to develop a more systematic approach, allowing to treat specific configurations by mere application of a general machinery. The present Part I is mainly devoted to setting up this general framework; at the end of the paper, this is exemplified in a very simple case. In Parts II, III and IV we will consider more engaging applications, indicated in the Introduction of the present work.

Local Zeta Regularization and the Casimir Effect

The local zeta regularization allows to treat local divergences appearing in quantum field theory; these are renormalized by pure analytic continuation (in the parameter of the regulator), with no need to remove or subtract divergent terms. This approach can be applied to the stress-energy tensor of the Casimir effect, and works as well on curved space-times. It is not useless to illustrate the power and elegance of this method in a simple case. In the present paper, our attention is devoted to the case of a neutral, massless scalar field in flat space-time, on a space domain with suitable (e.g., Dirichlet) boundary conditions. After a general outline of the local zeta method for the Casimir effect, we exemplify it in the typical case of a (Dirichlet) field between two parallel plates, or outside them. The results agree with the ones obtained by more popular methods, such as point splitting regularization. Connections with the existing literature on this subject are indicated. Subject Index: 130, 132, 187

Local zeta regularization and the scalar Casimir effect IV. The case of a rectangular box

Applying the general framework for local zeta regularization proposed in Part I of this series of papers, we compute the renormalized vacuum expectation value of several observables (in particular, of the stress-energy tensor and of the total energy) for a massless scalar field confined within a rectangular box of arbitrary dimension.

Local zeta regularization and the scalar Casimir effect III. The case with a background harmonic potential

Applying the general framework for local zeta regularization proposed in Part I of this series of papers, we renormalize the vacuum expectation value of the stress-energy tensor (and of the total energy) for a scalar field in presence of an external harmonic potential.

Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

We study the relative zeta function for the couple of operators A0 and A α , where A0 is the free unconstrained Laplacian in L2(R d ) (d ≥ 2) and A α is the singular perturbation of A0 associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter α, which is related to the strength of the perturbation, is of the kind α = α(−Δ∥), where −Δ∥ is the free Laplacian in L2(R d−1). Thus α may depend on the components of the wave vector parallel to the hyperplane; in this sense A α describes a semitransparent hyperplane selecting transverse modes.

Local Casimir Effect for a Scalar Field in Presence of a Point Impurity

The Casimir effect for a scalar field in presence of delta-type potentials has been investigated for a long time in the case of surface delta functions, modelling semi-transparent boundaries. More recently Albeverio, Cacciapuoti, Cognola, Spreafico and Zerbini [9,10,51] have considered some configurations involving delta-type potentials concentrated at points of

A time machine for free fall into the past

\mathbb{R}^3

Local zeta regularization and the scalar Casimir effect II. Some explicitly solvable cases

; in particular, the case with an isolated point singularity at the origin can be formulated as a field theory on

On inverses of Krein's Q-functions.

\mathbb{R}^3\setminus \{\mathbf{0}\}

Scattering from local deformations of a semitransparent plane

, with self-adjoint boundary conditions at the origin for the Laplacian. However, the above authors have discussed only global aspects of the Casimir effect, focusing their attention on the vacuum expectation value (VEV) of the total energy. In the present paper we analyze the local Casimir effect with a point delta-type potential, computing the renormalized VEV of the stress-energy tensor at any point of