Casimir effect for a massive scalar field under mixed boundary conditions

Pinto, A.C. Aguiar; Britto, T. M.; Bunchaft, R.; Pascoal, F.; Rosa, F. S. S.

doi:10.1590/s0103-97332003000400042

Cited by 15 publications

(15 citation statements)

References 21 publications

(28 reference statements)

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“…(3.25), holding in the case of Dirichlet conditions on both the planes π 0 and π a . This means that in the present Dirichlet-Neumann case, the forces on the boundary planes are repulsive ( 8 ).The results in Eq.s (3.30) (3.31) agree with those reported, e.g., in[10,28] and Santos et al…”

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confidence: 90%

“…In passing let us notice that, for d = 3 and Dirichlet boundary conditions (see subsection 3.6), the above configuration is the one most typically considered when dealing with the (scalar) Casimir effect [4,25,30]. The case with Dirichlet boundary conditions on one plane and Neumann conditions on the other (discussed in subsection 3.7) was originally considered in the electromagnetic case by Boyer [5], who derived the total energy; later, computations of the total energy and boundary forces for a (massless or massive) scalar field at both zero and non-zero temperature were performed by Pinto et al [10,28] and Santos et al [32] (see also [2,4]). Finally, let us also mention the monography by Fulling [22] where the stress-energy VEV for the model with periodic boundary conditions is given; see also [14,15] and, again, [4,25] for the derivation of the total energy in the same configuration.…”

Section: 15mentioning

confidence: 99%

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Local Zeta Regularization and the Scalar Casimir Effect

Fermi

Pizzocchero

2017

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All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.Printed in Singapore Local Zeta Regularization and the Scalar Casimir Effect Downloaded from www.worldscientific.com by 54.202.27.70 on 05/12/18. For personal use only. PrefaceZeta regularization is a method to treat the divergent quantities appearing in several areas of mathematics and physics; these are managed by introducing a complex parameter, with the role of a regulator, and defining their "renormalized versions" in terms of the analytic continuation with respect to the regulator. The standard textbook example of this procedure deals with the divergent series +∞ =1(1) which, in the zeta approach, is interpreted as the analytic continuation at s = −1 of the regularized series ζ(s) := +∞ =1 −s . The latter series only converges for s > 1 but the function s → ζ(s) defined in this way, the well-known Riemann zeta function, possesses a unique analytic extension to C \ {1} and ζ(−1) = −1/12; in this sense, the sum (1) "equals" −1/12. In a more pictorial language, one could say that −1/12 is the "renormalized" value of the series (1).The application of zeta regularization to the divergences of quantum field theory was first proposed by Dowker and Critchley [46], Hawking [84] and Wald [146] to renormalize local observables, especially the vacuum expectation value (VEV) of the stress-energy tensor. The ultimate purpose was the semiclassical treatment of quantum effects in general relativity, e.g. using the stress-energy VEV as a source term in Einstein's equations. Due to the attention to vacuum states, the zeta approach was connected from its very beginning to Casimir physics. However, the above mentioned pioneers focused their investigations on the conceptual validity of the method, rather than on its implementation for actual computations in specific configurations. For example, the elegant formula of Dowker-Critchley-Hawking, which gives the VEV of the stress-energy tensor as the functional derivative of the (renormalized) effective field action with respect to the space-time metric is not useful for actual computations in a given geometry, apart from very special cases. This is due to the fact that its use would require computing the effective action for all spacetime metrics close the one under consideration. For this reason, Birrel and Davies have described the application of this formula as "impossibly difficult" (see [18], page 190).Moreover, papers [46,84, 146] deal with a quantized field on the whole spacetime manifold and do not consider the possibility of confining the field to a given space ...

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confidence: 90%

Section: 15mentioning

confidence: 99%

Local Zeta Regularization and the Scalar Casimir Effect

Fermi

Pizzocchero

2017

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“…On the other hand, the Casimir effect for the massive scalar field also studied by some authors [24,25,26]. As is known that the Casimir effect disappears as the mass of the field goes to infinity since there are no more quantum fluctuations in the limit.…”

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confidence: 94%

“…The repulsive Casimir force has special importance in that it can be applied to microelectromechamical systems (MEMS) [21,22]. We discussed Casimir pistons for a massless scalar field with hybrid boundary conditions and obtained the repulsive Casimir force on the piston [23].On the other hand, the Casimir effect for the massive scalar field also studied by some authors [24,25,26]. As is known that the Casimir effect disappears as the mass of the field goes to infinity since there are no more quantum fluctuations in the limit.…”

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confidence: 97%

Casimir Pistons for Massive Scalar Fields

Zhai

Zhang

2009

Mod. Phys. Lett. A

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The Casimir force on two-dimensional pistons for massive scalar fields with both Dirichlet and hybrid boundary conditions is computed. The physical result is obtained by making use of generalized ζ-function regularization technique. The influence of the mass and the position of the piston in the force is studied graphically. The Casimir force for massive scalar field is compared to that for massless scalar field.Keywords: Casimir force; piston; generalized ζ-function. About 60 years ago Casimir gave the prediction that an attractive force should act between two plate-parallel uncharged perfectly conducting plates in vacuum [1]. Especially in recent 10 years, the effect has been paid more attention because of the development of precise measurements [2]. At the same time, Casimir energies and forces have been calculated theoretically in various different configurations. Different properties of the Casimir force (attractive or repulsive) can be obtained for different boundary conditions and different geometries [3,4]. For example, it has been claimed that the Casimir energy inside rectangular cavities can be either positive or negative depending on the ratio of the sides [5,6,7,8,9]. But the conclusion is worth suspecting because the calculations ignore the divergent term associated with the boundaries and the nontrivial contribution from the outside region of the box [10,11,12]. Recently, a modification of the rectangle−"Casimir piston" was introduced to avoid the above problems [13]. The Casimir force on the piston is a well-defined force because the position of the piston is independent of the divergent terms in the interval vacuum energy and external region. Successively, the study of this geometry attracted a lot of interests. The Casimir force on the piston was studied for different dimensions, different fields and different boundary conditions [14,15,16,17,18,19,20]. The results indicate that the Casimir force on the piston can be attractive or repulsive for different cases. The repulsive Casimir force has special importance in that it can be applied to microelectromechamical systems (MEMS) [21,22]. We discussed Casimir pistons for a massless scalar field with hybrid boundary conditions and obtained the repulsive Casimir force on the piston [23].On the other hand, the Casimir effect for the massive scalar field also studied by some authors [24,25,26]. As is known that the Casimir effect disappears as the mass of the field goes to infinity since there are no more quantum fluctuations in the limit. But the precise way the Casimir energy varies as the mass changes is worth studying [27]. In this paper,we consider the Casimir force on the piston for the massive scalar field with two types of boundary conditions, that is, Dirichlet and hybrid. We obtain our physical results using ζ−function regularization technique. We discuss the influence of the mass and the ratio of the sides graphically. The results tell us the expectable properties of the force on the piston as is for massless scalar field and also tell us t...

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“…4,85,155 As is known that the Casimir effect vanishes as the mass m of the field goes to infinity since there are no more quantum fluctuations in the limit. We review here the study of the precise way the Casimir energy varies as the mass changes.…”

Section: Massive Scalar Field With Dirichlet Bcsmentioning

confidence: 99%

Some developments of the Casimir effect in p-cavity of (D + 1)-dimensional space–time

Zhai

Lin

Feng

et al. 2014

Int. J. Mod. Phys. A

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The Casimir effect for rectangular boxes has been studied for several decades. But there are still some points unclear. Recently, there are new developments related to this topic, including the demonstration of the equivalence of the regularization methods and the clarification of the ambiguity in the regularization of the temperature-dependent free energy. Also, the interesting quantum spring was raised stemming from the topological Casimir effect of the helix boundary conditions. We review these developments together with the general derivation of the Casimir energy of the p-dimensional cavity in (D + 1)-dimensional spacetime, paying special attention to the sign of the Casimir force in a cavity with unequal edges. In addition, we also review the Casimir piston, which is a configuration related to rectangular cavity.

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Casimir effect for a massive scalar field under mixed boundary conditions

Cited by 15 publications

References 21 publications

Local Zeta Regularization and the Scalar Casimir Effect

Local Zeta Regularization and the Scalar Casimir Effect

Casimir Pistons for Massive Scalar Fields

Some developments of the Casimir effect in p-cavity of (D + 1)-dimensional space–time

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