Multidimensional electrostatic energy and classical renormalization

Author(s)
Sami M. ALJaber

ABSTRACT

Recent interest in problems in higher space di mensions is becoming increasingly important and attracted the attention of many investigators in variety of fields in physics. In this paper, the electrostatic energy of two geometries (a charged spherical shell and a nonconducting sphere) is calculated in higher space dimension, N. It is shown that as the space dimension increases, up to N = 9, the electrostatic energy of the two geometries decreases and beyond N = 9 it increases. Furthermore, we discuss a simple example which illustrates classical renormalization in electrostatics in higher dimensions.

Recent interest in problems in higher space di mensions is becoming increasingly important and attracted the attention of many investigators in variety of fields in physics. In this paper, the electrostatic energy of two geometries (a charged spherical shell and a nonconducting sphere) is calculated in higher space dimension, N. It is shown that as the space dimension increases, up to N = 9, the electrostatic energy of the two geometries decreases and beyond N = 9 it increases. Furthermore, we discuss a simple example which illustrates classical renormalization in electrostatics in higher dimensions.

Cite this paper

ALJaber, S. (2010) Multidimensional electrostatic energy and classical renormalization.*Natural Science*, **2**, 760-763. doi: 10.4236/ns.2010.27095.

ALJaber, S. (2010) Multidimensional electrostatic energy and classical renormalization.

References

[1] Kalnins, E.G., Miller, W. and Pogosyan, G.S. (2002) The Coulomb oscillator relation on ndimensional spheres and hyperboloids. Physics of Atomic Nuclei, 65(6), 10861094.

[2] ALJaber, S.M. (1998) Hydrogen atom in N dimensions. International Journal of Theoretical Physics, 37(4), 12891298.

[3] Halberg, A.S. (2001) The central symmetric screened Coulomb potential in N dimensions. Hadronic Journal, 24(5), 519530.

[4] Brack, M. and Murthy, M.V. (2003) Harmonically trapp ed fermion gases: Exact and asymptotic results in arbitrary dimensions. Journal of Physics A: Mathematical and General, 36(4), 11111133. Weldon, H.A. (2003) Quantization of higherderivative field theories. Annals of Physics, 305(2), 137150.

[5] Griffiths, J.B. and Podolsky, J. (2010) The LinetTian solution with a positive cosmological constant in four and higher dimensions. Physical Review D, 81(6), 064015064020.

[6] Wetterich, C. (2009) Dilation symmetry in higher dimensions and the vanishing of the cosmological constant. Physical Review Letters, 102(14), 141303141306.

[7] Li, W. and Yang, F. (2003) Ndimensional spacetime unit spheres and Lorentz transformation. Advances in Applied Clifford Algebras, 13(1), 5764.

[8] Plyukhin, A.V. (2010) Stochastic process leading to wave equations in dimensions higher than one. Physical Review E, 81(2 Pt 1), 021113021117.

[9] Kajimoto, N., Manaka, T. and Iwamoto, M. (2006) Electrostatic energies stored in dipolar films and analysis of decaying process of a large surface potential of ALq3 films. Chemical Physics Letters, 430(46), 340344.

[10] Ferreira, G.F. (2000) The electrostatic energy of thin charged straight threads and coils and the work to bend straight threads into coils. Journal of Electrostatics, 49(12), 2330.

[11] Pask, J.E. and Sterne, A. (2005) Realspace formulation of the electrostatic potential and total energy of solids. Physical Review B, 71(11), 113101113104.

[12] Ryder, L.H. (1996) Quantum filed theory. Cambridge University Press, Cambridge.

[13] Perskin, M.E. and Schroeder, D.V. (1995) An introduction to quantum field theory. AddisonWesley, Reading, Massachusetts.

[14] Alexandre, J. (2005) Concepts of renormalization in phy sics. Science Progress, 88(1), 116.

[15] Corbò, G. (2010) Renormalization in classical field theory. European Journal of Physics, 31(1), L5L8.

[16] Tort, A.C. (2010) Another example of classical renormalization in electrostatics. European Journal of Physics, 31(2), L49L50.

[17] ALJaber, S.M. (1999) Fermi gas in Ddimensional space. International Journal of Theoretical Physics, 38(3), 919923.

[18] Griffiths, D.J. (1999) Introduction to electrodynamics. AddisonWesley, Reading, Massachusetts.

[19] Boas, M.L. (2006) Mathematical methods in the physical sciences. John Wiley, New York.

[20] Connes, A. and Kreimer, D. (1999) Renormalization in quantum field theory and the RiemannHilbert problem. Journal of High Energy Physics, 9, 024.

[21] Falk, S., H?ubling, R. and Scheck, F. (2010) Renormalization in quantum field theory: An improved rigorous method. Journal of Physics A: Mathematical Theoretical, 43(3), 035401.

[22] Borsányi, S. and Reinosa, U. (2009) Renormalized nonequilibrium quantum field theory: Scalar fields. Physical Review D, 80(12), 125029125046.

[1] Kalnins, E.G., Miller, W. and Pogosyan, G.S. (2002) The Coulomb oscillator relation on ndimensional spheres and hyperboloids. Physics of Atomic Nuclei, 65(6), 10861094.

[2] ALJaber, S.M. (1998) Hydrogen atom in N dimensions. International Journal of Theoretical Physics, 37(4), 12891298.

[3] Halberg, A.S. (2001) The central symmetric screened Coulomb potential in N dimensions. Hadronic Journal, 24(5), 519530.

[4] Brack, M. and Murthy, M.V. (2003) Harmonically trapp ed fermion gases: Exact and asymptotic results in arbitrary dimensions. Journal of Physics A: Mathematical and General, 36(4), 11111133. Weldon, H.A. (2003) Quantization of higherderivative field theories. Annals of Physics, 305(2), 137150.

[5] Griffiths, J.B. and Podolsky, J. (2010) The LinetTian solution with a positive cosmological constant in four and higher dimensions. Physical Review D, 81(6), 064015064020.

[6] Wetterich, C. (2009) Dilation symmetry in higher dimensions and the vanishing of the cosmological constant. Physical Review Letters, 102(14), 141303141306.

[7] Li, W. and Yang, F. (2003) Ndimensional spacetime unit spheres and Lorentz transformation. Advances in Applied Clifford Algebras, 13(1), 5764.

[8] Plyukhin, A.V. (2010) Stochastic process leading to wave equations in dimensions higher than one. Physical Review E, 81(2 Pt 1), 021113021117.

[9] Kajimoto, N., Manaka, T. and Iwamoto, M. (2006) Electrostatic energies stored in dipolar films and analysis of decaying process of a large surface potential of ALq3 films. Chemical Physics Letters, 430(46), 340344.

[10] Ferreira, G.F. (2000) The electrostatic energy of thin charged straight threads and coils and the work to bend straight threads into coils. Journal of Electrostatics, 49(12), 2330.

[11] Pask, J.E. and Sterne, A. (2005) Realspace formulation of the electrostatic potential and total energy of solids. Physical Review B, 71(11), 113101113104.

[12] Ryder, L.H. (1996) Quantum filed theory. Cambridge University Press, Cambridge.

[13] Perskin, M.E. and Schroeder, D.V. (1995) An introduction to quantum field theory. AddisonWesley, Reading, Massachusetts.

[14] Alexandre, J. (2005) Concepts of renormalization in phy sics. Science Progress, 88(1), 116.

[15] Corbò, G. (2010) Renormalization in classical field theory. European Journal of Physics, 31(1), L5L8.

[16] Tort, A.C. (2010) Another example of classical renormalization in electrostatics. European Journal of Physics, 31(2), L49L50.

[17] ALJaber, S.M. (1999) Fermi gas in Ddimensional space. International Journal of Theoretical Physics, 38(3), 919923.

[18] Griffiths, D.J. (1999) Introduction to electrodynamics. AddisonWesley, Reading, Massachusetts.

[19] Boas, M.L. (2006) Mathematical methods in the physical sciences. John Wiley, New York.

[20] Connes, A. and Kreimer, D. (1999) Renormalization in quantum field theory and the RiemannHilbert problem. Journal of High Energy Physics, 9, 024.

[21] Falk, S., H?ubling, R. and Scheck, F. (2010) Renormalization in quantum field theory: An improved rigorous method. Journal of Physics A: Mathematical Theoretical, 43(3), 035401.

[22] Borsányi, S. and Reinosa, U. (2009) Renormalized nonequilibrium quantum field theory: Scalar fields. Physical Review D, 80(12), 125029125046.