IKBFU's Vestnik

2021 Issue №1

Back to the list Download an article

The closed piecewise uniform string revisited

Pages
51-72

Abstract

We reconsider the composite string model introduced 30 years ago to study the vacuum energy. The model consists of a scalar field, describing the transversal vibrations of a string consisting of piecewise constant sections with different tensions and mass densities, keeping the speed of light constant across the section. We consider the spectrum using transfer matrices and Chebyshev polynomials to get a closed formula for the eigenfrequencies. We calculate vacuum and free energy as well as the entropy of this system in two approaches, one using contour integration and another one using a Hurwitz zeta function. The latter results in a representation in terms of finite sums over polynomials. Several limiting cases are considered as well, for instance, the high-temperature expansion, which is expressed in terms of the heat kernel coefficients. The vacuum energy has no ultraviolet divergences, and the corresponding heat kernel coefficient  is zero due to the constancy of the speed of light. This is in parallel to a similar situation in macroscopic elect­ro­dynamics.

Reference

1. Brevik I., Nielsen H. B. Casimir Energy for a Piecewise Uniform String // Phys. Rev. D. 1990. Vol. 41 (4). P. 1185—1192.

2. Brevik I., Bytsenko A. A., Gonçalves A. E. Mass and decay spectra of the piecewise uniform string // Physics Letters B. 1999. Vol. 453 (3). P. 217—221.

3. Li Xz., Shi X., Zhang Jz. Generalized Riemann Zeta-Function Regularization and Casimir Energy for a Piecewise Uniform String // Phys. Rev. D. 1991. Vol. 44 (2). P. 560—562.

4. Brevik I. H., Nielsen H. B., Odintsov S. D. Casimir energy for a three piece relativistic string // Phys. Rev. D. 1996. Vol. 53. P. 3224—3229.

5. Hadasz L., Lambiase G., Nesterenko V. V. Casimir energy of a nonuniform string // Phys. Rev. D. 2000. Vol. 62. 025011.

6. Brevik I., Bytsenko A. A., Sollie R. Thermodynamic properties of the 2N-piece relativistic string // J. of Mathematical Physics. 2003. Vol. 44 (3). P. 1044—1055.

7. Bayin S. Ећ., Krisch J. P., Oezcan M. The casimir energy of the twisted string loop: Uniform and two segment loops // J. of Mathematical Physics. 1996. Vol. 37 (8). P. 3662—3674.

8. Berntsen M. H., Brevik I., Odintsov S. D. Casimir theory for the piecewise uniform relativistic string // Ann. Phys. 1997. Vol. 257 (1). P. 84—108.

9. Brevik I., Elizalde E., Sollie R., Aarseth J. B. A new scaling property of the Casimir energy for a piecewise uniform string // J. Math. Phys. 1999. Vol. 40 (3). P. 1127—1135.

10. Schwinger J., DeRaad L. L. (Jr.), Milton K. A. Casimir Effect in Dielectrics // Ann. Phys. 1978. Vol. 115. P. 1—23.

11. Bordag M., Kirsten K., Vassilevich D. V. On the ground state energy for a penetrable sphere and for a dielectric ball // Phys. Rev. D. 1999. Vol. 59. 085011.

12. Kronig R. De L., Penney W. G. Quantum Mechanics of Electrons in Crystal Lattices // Proc. R. Soc. A. 1931. Vol. 130. P. 499.

13. Bordag M., Pirozhenko I. G. Surface plasmons for doped graphene // Phys. Rev. D. 2015. Vol. 91. 085038.

14. Asorey M., Alvarez D. Garcia, Munoz-Castaneda J. M. Casimir effect and global theory of boundary conditions // J. Phys. A: Math. Gen. 2006. Vol. 39. P. 6127—6136.

15. Bordag M., Castañeda Muñoz J. M., Santamaría-Sanz L. Vacuum energy for ge­ne­ra­lised Dirac combs at . 2018. ArXiv:1812.09022.

16. Bordag M. On Bose-Einstein condensation in one-dimensional lattices of delta functions // Mod. Phys. Lett. 2020. Vol. A35 (03). 2040005.

17. Shajesh K. V., Brevik I., Cavero-Peláez I., Parashar P. Casimir energies of self-similar plate configurations // Phys. Rev. D. 2016. Vol. 94. 065003.

18. Bordag M., Klimchitskaya G. L., Mohideen U., Mostepanenko V. M. Advances in the Casimir Effect. Oxford University Press, 2009.

19. Geyer B., Klimchitskaya G. L., Mostepanenko V. M. Thermal corrections in the Casimir interaction between a metal and dielectric // Phys. Rev. A. 2005. Vol. 72. 022111.

20. Milton K. A., Kalauni P., Parashar P., Li Y. Casimir self-entropy of a spherical electromagnetic -function shell // Phys. Rev. D. 2017. Vol. 96. 085007.

21. Bordag M., Kirsten K. On the entropy of a spherical plasma shell // J. Phys. A: Math. Gen. 2018. Vol. 51. 455001.

22. Bordag M. Entropy in some simple one-dimensional configurations. 2018. ArXiv: 1807.10354quant-ph.

23. Bordag M., Munoz-Castaneda J. M., Santamaría-Sanz L. Free energy and entropy for finite temperature quantum field theory under the influence of periodic backgrounds // Eur. Phys. J. C. 2020. Vol. 80 (3).

24. Brevik I., Sollie R. On the Casimir energy for a 2N-piece relativistic string // J. of Mathematical Physics. 1997. Vol. 38 (6). P. 2774—2785.

25. Griffiths D. J., Steinke C. A. Waves in locally periodic media // American J. of Physics. 2001. Vol. 69 (2). P. 137—154.

26. Elizalde E. Ten Physical applications of Spectral Zeta Functions. Springer, 1995.

27. Bordag M., Pirozhenko I. G., Nesterenko V. V. Spectral analysis of a flat plasma sheet model // J. Phys. 2005. Vol. A38. 11027.

28. Bordag M., Khusnutdinov N. On the vacuum energy of a spherical plasma shell // Phys. Rev. D. 2008. Vol. 77. 085026.

29. Milton K. A., Kalauni P., Parashar P., Li Y. Remarks on the Casimir self-entropy of a spherical electromagnetic -function shell // Phys. Rev. 2019. Vol. D99 (4). 045013.

30. Brevik I., Nielsen H. B. Casimir Theory for the Piecewise Uniform String: Division into 2N Pieces // Phys. Rev. D. 1995. Vol. 51 (4). P. 1869—1874.

31. Brevik I., Bytsenko A. A., Nielsen H. B. Thermodynamic properties of the piecewise uniform string // Classical and Quantum Gravity. 1998. Vol. 15 (11). P. 3383—3395.