Numerical Regularization Procedures for Estimating the Casimir Energy of a Massless Scalar Field in a Square

2011 ◽  
Vol 80 (9) ◽  
pp. 094002
Author(s):  
Norio Inui
Keyword(s):  
Scalar Field ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Numerical Regularization

scholarly journals Radiative correction to the Casimir energy for massive scalar field on a spherical surface

2017 ◽  
Vol 32 (24) ◽  
pp. 1750128 ◽  
Author(s):  
M. A. Valuyan
Keyword(s):  
Scalar Field ◽  
Radiative Correction ◽  
Spherical Surface ◽  
Perturbation Expansion ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Massive Scalar ◽  
Massive Scalar Field ◽  
Free Theory

In this paper, the first-order radiative correction to the Casimir energy for a massive scalar field in the [Formula: see text] theory on a spherical surface with [Formula: see text] topology was calculated. In common methods for calculating the radiative correction to the Casimir energy, the counter-terms related to free theory are used. However, in this study, by using a systematic perturbation expansion, the obtained counter-terms in renormalization program were automatically position-dependent. We maintained that this dependency was permitted, reflecting the effects of the boundary conditions imposed or background space in the problem. Additionally, along with the renormalization program, a supplementary regularization technique that we named Box Subtraction Scheme (BSS) was performed. This scheme presents a useful method for the regularization of divergences, providing a situation that the infinities would be removed spontaneously without any ambiguity. Analysis of the necessary limits of the obtained results for the Casimir energy of the massive and massless scalar field confirmed the appropriate and reasonable consistency of the answers.

Thermal Casimir effect in the Kerr spacetime with quintessence

2019 ◽  
Vol 34 (16) ◽  
pp. 1950125
Author(s):  
V. B. Bezerra ◽  
J. M. Toledo
Keyword(s):  
Black Hole ◽  
Scalar Field ◽  
Equatorial Plane ◽  
Casimir Effect ◽  
Kerr Black Hole ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Kerr Spacetime

We calculate thermal corrections to the Casimir energy of a massless scalar field in the Kerr black hole surrounded by quintessence, taking into account the metrics derived by Ghosh [S. G. Ghosh, Eur. Phys. J. C 76, 222 (2016)] and Toshmatov et al. [B. Toshmatov, Z. Stuchlík and B. Ahmedov, Eur. Phys. J. Plus 132, 98 (2017)]. We compare both results and show that they are almost the same, except very close to the horizons. At [Formula: see text], equatorial plane, the results are the same, as should be expected, due to the fact that the metrics coincide in this region.

Remarks on a gravitational analogue of the Casimir effect

2016 ◽  
Vol 25 (09) ◽  
pp. 1641018 ◽  
Author(s):  
V. B. Bezerra ◽  
H. F. Mota ◽  
C. R. Muniz
Keyword(s):  
Scalar Field ◽  
Cosmic String ◽  
Vacuum Energy ◽  
Casimir Effect ◽  
String Tension ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Quantum Field ◽  
Einstein Universe

We consider the Casimir effect, by calculating the Casimir energy and its corrections for nonzero temperatures, of a massless scalar field in the spacetime with topology [Formula: see text] (Einstein universe) containing an idealized cosmic string. The obtained results confirm the role played by the identifications imposed on the quantum field by boundary conditions arising from the topology of the gravitational field under consideration and illustrate a realization of a gravitational analogue of the Casimir effect. In this backgorund, we show that the vacuum energy can be written as a term which corresponds to the vacuum energy of the massless scalar field in the Einstein universe added by another term that formally corresponds to the vacuum energy of the electromagnetic field in the Einstein universe, multiplied by a parameter associated with the presence of the cosmic string, namely, [Formula: see text], where [Formula: see text] is a constant related to the cosmic string tension, [Formula: see text].

scholarly journals SCALAR CASIMIR EFFECT BETWEEN TWO CONCENTRIC SPHERES

2012 ◽  
Vol 27 (16) ◽  
pp. 1250082 ◽  
Author(s):  
MUSTAFA ÖZCAN
Keyword(s):  
Scalar Field ◽  
Casimir Effect ◽  
Casimir Energy ◽  
Outer Radius ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Parallel Plates ◽  
New Approach ◽  
Concentric Spheres ◽  
Spherical Boundary

The Casimir effect giving rise to an attractive force between the closely spaced two concentric spheres that confine the massless scalar field is calculated by using a direct mode summation with contour integration in the complex plane of eigenfrequencies. We developed a new approach appropriate for the calculation of the Casimir energy for spherical boundary conditions. The Casimir energy for a massless scalar field between the closely spaced two concentric spheres coincides with the Casimir energy of the parallel plates for a massless scalar field in the limit when the dimensionless parameter η, ([Formula: see text] where a(b) is inner (outer) radius of sphere), goes to zero. The efficiency of new approach is demonstrated by calculation of the Casimir energy for a massless scalar field between the closely spaced two concentric half spheres.

scholarly journals Comments on “Casimir effect in the Kerr spacetime with quintessence”

2017 ◽  
Vol 32 (21) ◽  
pp. 1775001 ◽  
Author(s):  
Bobir Toshmatov ◽  
Zdeněk Stuchlík ◽  
Bobomurat Ahmedov
Keyword(s):  
Black Hole ◽  
Scalar Field ◽  
Casimir Effect ◽  
Kerr Black Hole ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Kerr Spacetime ◽  
The Difference

This comment is devoted to the recalculation of the Casimir energy of a massless scalar field in the Kerr black hole surrounded by quintessence derived in [B. Toshmatov, Z. Stuchlík and B. Ahmedov, Eur. Phys. J. Plus 132, 98 (2017)] and its comparison with the results recently obtained in [V. B. Bezerra, M. S. Cunha, L. F. F. Freitas and C. R. Muniz, Mod. Phys. Lett. A 32, 1750005 (2017)] in the spacetime [S. G. Ghosh, Eur. Phys. J. C 76, 222 (2016)]. We have shown that in the more realistic spacetime which does not have the failures illustrated here, the Casimir energy is significantly bigger than that derived in [V. B. Bezerra, M. S. Cunha, L. F. F. Freitas and C. R. Muniz, Mod. Phys. Lett. A 32, 1750005 (2017)], and the difference becomes crucial especially in the regions of near horizons of the spacetime.

scholarly journals A Global Approach with Cutoff Exponential Function, Mathematically Well Defined at the Outset, for Calculating the Casimir Energy: The Example of Scalar Field

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
M. S. R. Miltão ◽  
Franz A. Farias
Keyword(s):  
Scalar Field ◽  
Spherical Shell ◽  
Bessel Function ◽  
Bessel Functions ◽  
Casimir Energy ◽  
The Other ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Global Approach ◽  
Exponential Functions

A global approach with cutoff exponential functions is used to obtain the Casimir energy of a massless scalar field in the presence of a spherical shell. The proposed method, mathematically well defined at the outset, makes use of two regulators, one of them to make the sum of the orders of Bessel functions finite and the other to regularize the integral involving the zeros of Bessel function. This procedure ensures a consistent mathematical handling in the calculations of the Casimir energy and allows a major comprehension on the regularization process when nontrivial symmetries are under consideration. In particular, we determine the Casimir energy of a scalar field, showing all kinds of divergences. We consider separately the contributions of the inner and outer regions of a spherical shell and show that the results obtained are in agreement with those known in the literature, and this gives a confirmation for the consistence of the proposed approach. The choice of the scalar field was due to its simplicity in terms of physical quantity spin.

Rotational effects on the Casimir energy in the space–time with one extra compactified dimension

2018 ◽  
Vol 33 (20) ◽  
pp. 1850122 ◽  
Author(s):  
L. C. N. Santos ◽  
C. C. Barros
Keyword(s):  
Scalar Field ◽  
Angular Velocity ◽  
Physical Region ◽  
Casimir Energy ◽  
Space Time ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Rotating Frame ◽  
Rotating System ◽  
Noninertial Effects

In this paper we study the quantization of a massless scalar field in a rotating frame. In particular, we obtain the Casimir energy in a space–time with one extra compactified dimension for a rotating observer. We consider a uniformly rotating system on the circle S1 and present an equation for spin-0 bosons where noninertial effects can be taken into account. It is shown that the spectrum of the scalar field depends on the angular velocity of the rotating system and in this way, positive and negative modes can be defined through an appropriate choice of the angular velocity. We show that noninertial effects restrict the physical region of the space–time where particles can be placed, and furthermore that the Casimir energy in the space–time with one extra compactified dimension is shifted by these effects. In addition, we pointed out that rotating effects modify the length of the extra dimension for a co-rotating observer in this kind of space–time.

scholarly journals Remarks on Some Results Related to the Thermal Casimir Effect in Einstein and Closed Friedmann Universes with a Cosmic String

Universe ◽  
2021 ◽  
Vol 7 (7) ◽  
pp. 232
Author(s):  
Valdir Barbosa Bezerra ◽  
Herondy Francisco Santana Mota ◽  
Celio Rodrigues Muniz ◽  
Carlos Augusto Romero Filho
Keyword(s):  
Free Energy ◽  
Scalar Field ◽  
Internal Energy ◽  
Cosmic String ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Thermodynamic Quantities ◽  
Spinor Fields ◽  
Massless Spinor

In this paper, we present a review of some recent results concerning the thermal corrections to the Casimir energy of massless scalar, electromagnetic, and massless spinor fields in the Einstein and closed Friedmann universes with a cosmic string. In the case of a massless scalar field, it is shown that the Casimir energy can be written as a simple sum of two terms; the first one corresponds to the Casimir energy for the massless scalar field in the Einstein and Friedmann universes without a cosmic string, whereas the second one is simply the Casimir energy of the electromagnetic in this background, multiplied by a parameter λ=(1/α)−1, where α is a constant that codifies the presence of the cosmic string, and is related to its linear mass density, μ, by the expression α=1−Gμ. The Casimir free energy and the internal energy at a temperature different from zero, as well as the Casimir entropy, are given by similar sums. In the cases of the electromagnetic and massless spinor fields, the Casimir energy, free energy, internal energy, and Casimir entropy are also given by the sum of two terms, similarly to the previous cases, but now with both terms related to the same field. Using the results obtained concerning the mentioned thermodynamic quantities, their behavior at high and low temperatures limits are studied. All these results are particularized to the scenario in which the cosmic string is absent. Some discussions concerning the validity of the Nernst heat theorem are included as well.

scholarly journals DIRICHLET CASIMIR ENERGY FOR A SCALAR FIELD IN A SPHERE: AN ALTERNATIVE METHOD

2010 ◽  
Vol 25 (06) ◽  
pp. 1165-1183 ◽  
Author(s):  
M. A. VALUYAN ◽  
S. S. GOUSHEH
Keyword(s):  
Scalar Field ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Leading Order ◽  
Spatial Dimensions ◽  
Complicated Pattern

In this paper we compute the leading order of the Casimir energy for a free massless scalar field confined in a sphere in three spatial dimensions, with the Dirichlet boundary condition. When one tabulates all of the reported values of the Casimir energies for two closed geometries, cubical and spherical, in different space–time dimensions and with different boundary conditions, one observes a complicated pattern of signs. This pattern shows that the Casimir energy depends crucially on the details of the geometry, the number of the spatial dimensions, and the boundary conditions. The dependence of the sign of the Casimir energy on the details of the geometry, for a fixed spatial dimensions and boundary conditions has been a surprise to us and this is our main motivation for doing the calculations presented in this paper. Moreover, all of the calculations for spherical geometries include the use of numerical methods combined with intricate analytic continuations to handle many different sorts of divergences which naturally appear in this category of problems. The presence of divergences is always a source of concern about the accuracy of the numerical results. Our approach also includes numerical methods, and is based on Boyer's method for calculating the electromagnetic Casimir energy in a perfectly conducting sphere. This method, however, requires the least amount of analytic continuations. The value that we obtain confirms the previously established result.

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