An investigation of the convergence of Casimir stress in one-dimensional inhomogeneous media
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摘要: 文章基于Lifshitz理论，在现有计算Casimir stress的规范化方法基础上，对之做进一步的数学演绎来研究一维情况下Casimir stress的收敛性. 通过分析电磁场格林函数的Galerkin变分方程，文章严格地证明并得出结论：在现有的Lifshitz公式理论体系及常规的规范化方法下，只要介质中某处的介电常数ε或磁导率μ的一阶导数不为零，Casimir stress在这些导数非零处就是发散的. 该研究成果证明了现有的常规规范化方法不适用于非均匀介质，为改进现有物理模型和寻找新的适用于非均匀介质的规范化方法提供了理论参考.
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关键词:
- 发散
Abstract: This paper, based on the Lifshitz theory and the existing regularization employed to calculate Casimir stress, does further mathematics deduction on them to explore the convergence of Casimir stress in one dimension. By analyzing the Galerkin variational equation of Green's function of electromagnetic field, this paper strictly proves and concludes that under the existing Lifshitz formulism and standard regularization, as long as either the derivative of permittivity ε or the derivative of permeability μ is nonzero somewhere in the media, Casimir stress is divergent in this place with nonzero derivative. This investigation proves that existing standard regularization are not applicable to inhomogeneous media, which provides theoretical reference for the improvement of the current physical model and the exploration of new regularization applicable to inhomogeneous media.-
Keywords:
- Divergence
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[1]B.G. Casimir. .On the attraction between two perfectly conducting plates[J].Proceedings of the Royal Netherlands Academy of Arts and Sciences, 1948, (51):793-795 [2]U.Mohideen,Anushree RoyPrecision Measurement of the Casimir Force from 01 to 0.9 μm[J].Physical Review Letters, 1998, 81(21):4549-4552 [3]AttaallahAlmasi, Philippe Brax, DavideIannuzzi, René I.P. SedmikForce sensor for chameleon and Casimir force experiments with parallel-plate configuration[J].Physical Review D, 2009, 91(10):102002-1 [4]J.N. Munday,FCapasso,V. A. Parsegian. .Measured long-range repulsive Casimir–Lifshitz forces[J].NATURE, 2009, 457(7226):170-173 [5]S.I. Goto,ACHale,R. W. Tucker,T. J. Walton. Numerical regularization of electromagnetic quantum fluctuations in inhomogeneous dielectric media[J].Physical Review A, 2012, 85(3):034103-1 [6]Fanglin Bao, Bin Luo, Sailing He.First-order correction to the Casimir force within an inhomogeneous medium[J].Physical Review A, 2015, 91(6):063810-1 [7]C.Genet,ALambrecht,SReynaud. Casimir force and the quantum theory of lossy optical cavities[J].Physical Review A, 2003, 67(4):043811-1 [8]S.A. EllingsenCasimir force on real materials—the slab and cavity geometry[J].Journal of Physics A: Mathematical and Theoretical, 2007, 40(13):3643-3664 [9]William M.R. Simpson,SAR. Horsley,U. Leonhardt. Divergence of Casimir stress in inhomogeneous media[J].Physical Review A, 2013, 87(4):043806-1 [10]E.M. LifshitzThe Theory of Molecular Attractive Forces between Solids[J].Soviet Physics, 1956, 2(1):73-83 [11]I.E. Dzyaloshinskii,EMLifshitz,L. P. Pitaevskii. The general theory of van der Waals forces[J].Advances in Physics, 1961, 10(38):165-209 [12]T.G. Philbin,CXiong,U. Leonhardt..Casimir stress in an inhomogeneous medium[J].Annals of Physics, 2010, 325(3):579-595 [13]C.Xiong,TW.Kelsey,S.A. Linton,U. Leonhardt.Casimir forces for inhomogeneous planar media.[J].Journal of Physics: Conference Series, 2013, 410(1):012165-1 [14] U.Leonhardt. Essential Quantum Optics: From Quantum Measurements to Black Holes [M]. Cambridge: Cambridge University Press, 2010: 231-249. [1]B.G. Casimir. .On the attraction between two perfectly conducting plates[J].Proceedings of the Royal Netherlands Academy of Arts and Sciences, 1948, (51):793-795 [2]U.Mohideen,Anushree RoyPrecision Measurement of the Casimir Force from 01 to 0.9 μm[J].Physical Review Letters, 1998, 81(21):4549-4552 [3]AttaallahAlmasi, Philippe Brax, DavideIannuzzi, René I.P. SedmikForce sensor for chameleon and Casimir force experiments with parallel-plate configuration[J].Physical Review D, 2009, 91(10):102002-1 [4]J.N. Munday,FCapasso,V. A. Parsegian. .Measured long-range repulsive Casimir–Lifshitz forces[J].NATURE, 2009, 457(7226):170-173 [5]S.I. Goto,ACHale,R. W. Tucker,T. J. Walton. Numerical regularization of electromagnetic quantum fluctuations in inhomogeneous dielectric media[J].Physical Review A, 2012, 85(3):034103-1 [6]Fanglin Bao, Bin Luo, Sailing He.First-order correction to the Casimir force within an inhomogeneous medium[J].Physical Review A, 2015, 91(6):063810-1 [7]C.Genet,ALambrecht,SReynaud. Casimir force and the quantum theory of lossy optical cavities[J].Physical Review A, 2003, 67(4):043811-1 [8]S.A. EllingsenCasimir force on real materials—the slab and cavity geometry[J].Journal of Physics A: Mathematical and Theoretical, 2007, 40(13):3643-3664 [9]William M.R. Simpson,SAR. Horsley,U. Leonhardt. Divergence of Casimir stress in inhomogeneous media[J].Physical Review A, 2013, 87(4):043806-1 [10]E.M. LifshitzThe Theory of Molecular Attractive Forces between Solids[J].Soviet Physics, 1956, 2(1):73-83 [11]I.E. Dzyaloshinskii,EMLifshitz,L. P. Pitaevskii. The general theory of van der Waals forces[J].Advances in Physics, 1961, 10(38):165-209 [12]T.G. Philbin,CXiong,U. Leonhardt..Casimir stress in an inhomogeneous medium[J].Annals of Physics, 2010, 325(3):579-595 [13]C.Xiong,TW.Kelsey,S.A. Linton,U. Leonhardt.Casimir forces for inhomogeneous planar media.[J].Journal of Physics: Conference Series, 2013, 410(1):012165-1 [14] U.Leonhardt. Essential Quantum Optics: From Quantum Measurements to Black Holes [M]. Cambridge: Cambridge University Press, 2010: 231-249.
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