Ramanujan summation and the Casimir effect

Authors

  • Wolfgang Bietenholz Instituto de Ciencias Nucleares, UNAM

DOI:

https://doi.org/10.31349/SuplRevMexFis.3.020705

Keywords:

Ramanujan summation, Casimir effect, renormalization, Dark Energy

Abstract

Srinivasa Ramanujan was a great self-taught Indian mathematician, who died a century ago, at the age of only 32, one year after returning from England. Among his numerous achievements is the assignment of sensible, finite values to divergent series, which correspond to Riemann's $\zeta$-function with negative integer arguments. He hardly left any explanation about it, but following the few hints that he gave, we construct a direct justification for the best known example, based on analytic continuation. As a physical application of Ramanujan summation we discuss the Casimir effect, where this way of removing a divergent term corresponds to the renormalization of the vacuum energy density, in particular of the photon field. This leads to the prediction of the Casimir force between conducting plates, which has now been accurately confirmed by experiments. Finally we review the discussion about the meaning and interpretation of the Casimir effect. This takes us to the mystery surrounding the magnitude of Dark Energy.

References

W. Bietenholz, From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results,

Rev. Mex. F´ıs. E 18 (2021) 020203, https://doi.org/10.31349/RevMexFisE.18.020203

S. Ramanujan, Second Notebook (unpublished), Chapter VI. {reproduced in Chapter 8 of Ref. [4]}

B. Riemann, Uber die Anzahl der Primzahlen unter einer gegebenen Große, in ¨ Monatsberichte der Berliner Akademie,

November 1859.

B. Candelpergher, Ramanujan summation of divergent series, Lectures Notes in Mathematics, 2185 [hal01150208v2], 2017, https://hal.univ-cotedazur.fr/hal-01150208v2/document

H. B. G. Casimir and D. Polder, The Influence of Retardation on the London-van der Waals Forces, Phys. Rev. 73 (1948) 360–

, https://doi.org/10.1103/PhysRev.73.360

H. B. G. Casimir, On the attraction between two perfectly conducting plates, Proc. Kon. Ned. Akad. Wetensch. B 51 (1948) 793-795.

K. A. Milton, The Casimir effect: Physical manifestations of zero-point energy, World Scientific, 2001.

M. Bordag, G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko, Advances in the Casimir Effect, Oxford Scholarship Online, 2009, https://DOI:10.1093/acprof:oso/9780199238743.001.0001.

J. C. Collins, Renormalization, Cambridge University Press, 1984.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, 1972 (10th edition). https://personal.math.ubc.ca/∼cbm/aands/abramowitz and stegun.pdf.

G. Bressi, G. Carugno, R. Onofrio and G. Ruoso, Measurement of the Casimir Force between Parallel Metallic Surfaces, Phys. Rev. Lett. 88 (2002) 041804, https://doi.org/10.1103/PhysRevLett.88.041804

S. K. Lamoreaux, Demonstration of the Casimir Force in the 0.6 to 6 µm Range, Phys. Rev. Lett. 78 (1997) 5, https://doi.org/10.1103/PhysRevLett.78.5 [Erratum: Phys. Rev. Lett. 81 (1998) 5475, https://doi.org/10.1103/PhysRevLett.81.5475]

U. Mohideen and A. Roy, Precision Measurement of the Casimir Force from 0.1 to 0.9 µm, Phys. Rev. Lett. 81 (1998) 4549, https://doi.org/10.1103/PhysRevLett.81.4549

V. Sahni and A. Starobinsky, The Case for a Positive Cosmological Λ-term, Int. J. Mod. Phys. D 9 (2000) 373, https://doi.org/10.1142/S0218271800000542.

W. Bietenholz, What are Elementary Particles? From Dark Energy to Quantum Field Excitations, Rev. Cub. F´ıs. 37 (2020) 146, http://www.revistacubanadefisica.org/RCFextradata/OldFiles/2020/v37n2/RCF2020v37p146.pdf

S. M. Carroll, The Cosmological Constant, Living Rev. Rel. 4:1 (2001) 1, https://doi.org/10.12942/lrr-2001-1.

R. L. Jaffe, The Casimir Effect and the Quantum Vacuum, Phys. Rev. D 72 (2005) 021301, https://doi.org/10.1103/PhysRevD.72.021301.

T. H. Boyer, Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a

Charged Particle, Phys. Rev. 174 (1968) 1764, https://doi.org/10.1103/PhysRev.174.1764.

S. G. Mamaev and N. N. Trunov, Dependence of the vacuum expectation values of the energy-momentum tensor on the geometry and topology of the manifold, Theor. Math. Phys. 38 (1979) 28, https://doi.org/10.1007/BF0101854010.1007/BF01018540. J. Ambjørn and S. Wolfram, Properties of the vacuum. I. Mechanical and thermodynamic, Annals Phys. 147 (1983) 1, https://doi.org/10.1016/0003-4916(83)90065-9. G. J. Maclay, Analysis of zero-point electromagnetic energy and Casimir forces in conducting rectangular cavities, Phys. Rev. A 61 (2000) 052110, https://doi.org/10.1103/PhysRevA.61.052110; M. Bordag, U. Mohideen and V. M. Mostepanenko, New Developments in the Casimir Effect, Phys. Rept. 353 (2001) 1. https://doi.org/10.1016/S0370-1573(01)00015-1.

R. M. Cavalcanti, Casimir force on a piston, Phys. Rev. D 69 (2004) 065015. https://doi.org/10.1103/PhysRevD.69.065015. M. P. Hertzberg, R. L. Jaffe, M. Kardar and A. Scardicchio, Attractive Casimir Forces in a Closed Geometry, Phys. Rev. Lett. 95 (2005)

, https://doi.org/10.1103/PhysRevLett.95.250402; Casimir Forces in a Piston Geometry at Zero and Finite Temperatures, Phys. Rev. D 76 (2007) 045016, https://doi.org/10.1103/PhysRevD.76.045016.

J. N. Munday, F. Capasso and V. A. Parsegian, Measured longrange repulsive Casimir-Lifshitz forces, Nature 457 (2009) 170,

https://doi.org/10.1038/nature07610.

I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii, General theory of van der Waals’ forces, Sov. Phys. Uspekhi 4 (1961) 153, https://doi.org/10.1070/PU1961v004n02ABEH003330.

J. Schwinger, Casimir effect in source theory, Lett. Math. Phys. 1 (1975) 43, https://doi.org/10.1007/BF00405585. J. Schwinger, J. DeRaad and K. A. Milton, Casimir Effect in Dielectrics, Ann. Phys. (N.Y.) 115 (1978) 1, https://doi.org/10.1016/0003-4916(78)

-0.

K. Scharnhorst, On propagation of light in the vacuum between plates, Phys. Lett. B 236 (1990) 354, https://doi.org/10.1016/0370-2693(90)90997-K. [Erratum: Phys. Lett. B 787 (2018) 204, https://doi.org/10.1016/j. physletb.2018.12.002.]. G. Barton, Faster-than-c light between parallel mirrors. The Scharnhorst effect rederived, Phys. Lett. B 237 (1990) 559, https://doi.org/10.1016/0370-2693(90)91224-Y.

S. G. de Clark, The Scharnhorst Effect: Superluminality and Causality in Effective Field Theories, Ph.D. thesis, University of Arizona, 2016, https://repository.arizona.edu/handle/10150/622964

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Published

2022-03-31

How to Cite

1.
Bietenholz W. Ramanujan summation and the Casimir effect. Supl. Rev. Mex. Fis. [Internet]. 2022 Mar. 31 [cited 2024 Mar. 28];3(2):020705 1-6. Available from: https://rmf.smf.mx/ojs/index.php/rmf-s/article/view/6095