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- Direct mode summation for the Casimir energy of spherical shell and compact ball
- Direct mode summation for the Casimir energy of spherical shell and compact ball | SpringerLink
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Direct mode summation for the Casimir energy of spherical shell and compact ball
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No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. These methods are distinguished by the approach used to carry out the calculations of the Casimir energy, and it is clear that the physical result must be independent from the regulators or the method employed for them.
But the literature has shown that the results found there exhibit a divergence among them.
Direct mode summation for the Casimir energy of spherical shell and compact ball | SpringerLink
On the other hand, in a global one, we start with an expression for the vacuum energy where there is no space-time variables present as they already were integrated. In the present work, we pretend detailing a global approach 55, 56 for the calculation of Casimir energy. Section 3 exhibits the calculations for the contributions of the inner and the outer regions of the spherical shell.
We analyze in Section 4 the results and compare them with those ones in the literature and make some considerations. Advances in High Energy Physics 3 2. This theorem gives the summation of zeros and poles of an analytic function as a contour integral. This contour is a curve that encompasses the interior region of the complex plane which contains the zeros and poles 65— In our case, we are interested in the root functions which match the conditions 2.
In the above equation, the argument for logarithm must involve the product of all root functions.
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The contour to be taken on the calculations is given by 68 according to Figure 1. The vacuum energy 2. This procedure is sensible, but it already has been made clear by Boyer 69, Now, group together these two developments, and 2. The other limits will be taken in an appropriate moment after the cancelation of possible remaining divergences. As it can be observed, the above contributions were written in such a way that the term for j 0 was detached from the summation on j.
Internal Mode Now, we proceed with the calculations of 3. The Debye expansion gives 6. The contribution 3. Advances in High Energy Physics 7 The contributions 3. Both divergences, the logarithm in 3. External Mode The contribution of the external modes comes by 3. We proceed with the calculations in an analogous way to that of the previous subsection. The term 3. Advances in High Energy Physics 9 After collecting the terms 3. Equation 4. Conclusions Our purpose in this work was to show the form and nature of each divergent term that appears in the calculation of Casimir energy and demonstrate that the method proposed is mathematically consistent and that it is in accordance with the results existing in literature.
As we can see, the second and fourth terms in 5. The result 5. Besides, this calculation presented at this work shows the desired agreement with the results existing in the literature. Acknowledgment The authors wish to thank Dr. Ludmila Oliveira H. References 1 H. D, vol. Inami and K. Kikkawa, T. Kubota, S.
Sawada, and M. Kikkawa and M. Advances in High Energy Physics 11 8 K.
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