# Local Zeta Regularization And The Scalar Casimir Effect: A General Approach Based On Integral Kernels

@inproceedings{Fermi2015LocalZR, title={Local Zeta Regularization And The Scalar Casimir Effect: A General Approach Based On Integral Kernels}, author={Davide Fermi and Livio Pizzocchero}, year={2015} }

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…

## 19 Citations

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

- Physics
- 2015

In Part I of this series of papers we have described a general formalism to compute the vacuum effects of a scalar field via local (or global) zeta regularization. In the present Part II we exemplify…

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

- Physics
- 2015

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,…

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

- Physics
- 2015

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…

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

- MathematicsSymmetry
- 2018

The local Casimir effect with a point delta-type potential is analyzed, computing the renormalized VEV of the stress-energy tensor at any point of $\mathbb{R}^3\setminus \mathbf{0}\}$; to this purpose the zeta regularization approach is followed, in the formulation already employed for different configurations in previous works of the authors'.

### Algebraic Approach to Casimir Force Between Two $$\delta $$-like Potentials

- Physics, Mathematics
- 2021

We analyse the Casimir effect of two nonsingular centers of interaction in three space dimensions, using the framework developed by Herdegen. Our model is mathematically well-defined and all physical…

### Vacuum Polarization with Zero-Range Potentials on a Hyperplane

- PhysicsUniverse
- 2021

The quantum vacuum fluctuations of a neutral scalar field induced by background zero-range potentials concentrated on a flat hyperplane of co-dimension 1 in (d+1)-dimensional Minkowski spacetime are…

### Integrating Gauge Fields in the $\zeta$-formulation of Feynman's path integral

- Physics, Mathematics
- 2019

In recent work by the authors, a connection between Feynman’s path integral and Fourier integral operator ζ-functions has been established as a means of regularizing the vacuum expectation values in…

### Integrating Gauge Fields in the ζ-Formulation of Feynman’s Path Integral

- Physics
- 2020

In recent work by the authors, a connection between Feynman’s path integral and Fourier integral operator ζ-functions has been established as a means of regularizing the vacuum expectation values in…

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

- Mathematics
- 2017

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…

### On the Casimir Effect with $$\delta $$ δ -Like Potentials, and a Recent Paper by K. Ziemian (Ann. Henri Poincaré, 2021)

- PhysicsAnnales Henri Poincaré
- 2023

The local and global aspects of the Casimir effect for a scalar field in the presence of a point-like impurity were treated in our papers (Fermi and Pizzocchero in Symmetry 10(2):38, 2018; Fermi in…

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In Part I of this series of papers we have described a general formalism to compute the vacuum effects of a scalar field via local (or global) zeta regularization. In the present Part II we exemplify…

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

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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,…

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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…

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