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Classical physics of thermal scalar radiation in two spacetime dimensions
1.See, for example, P. M. Morse, Thermal Physics, 2nd ed. (Benjamin/Cummins, Reading, 1981).
2.See, for example, R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, New York, 1985).
3.See, for example, Ref. 1, pp. 78–79.
4.See, for example, B. H. Lavenda, Statistical Physics: A Probabilistic Approach (Wiley, New York, 1991), pp. 67–70.
5.See, for example, B. F. Schutz, A First Course in General Relativity (Cambridge U. P., Cambridge, 1986), p. 150.
6.See, for example, W. Rindler, Essential Relativity: Special, General, and Cosmological, 2nd ed. (Springer-Verlag, New York, 1977), pp. 49–51.
7.In two spacetime dimensions, the analysis involves only trigonometric and exponential functions. In four spacetime dimensions, the Rindler frame normal modes involve modified Bessel functions of imaginary argument.
8.See, for example, Ref. 4, p. 63.
9.M. J. Sparnaay, “Measurement of the attractive forces between flat plates,” Physica (Amsterdam) 24, 751–764 (1958).http://dx.doi.org/10.1016/S0031-8914(58)80090-7
10.S. K. Lamoreaux, “Demonstration of the Casimir force in the 0.6 to range,” Phys. Rev. Lett. 78, 5–8 (1997);http://dx.doi.org/10.1103/PhysRevLett.78.5
10.S. K. Lamoreaux, Phys. Rev. Lett. “Demonstration of the Casimir Force in the 0.6 to range,” 81, 5475–5476(E) (1998).http://dx.doi.org/10.1103/PhysRevLett.81.5475
11.U. Mohideen, “Precision measurement of the Casimir force from 0.1 to ,” Phys. Rev. Lett. 81, 4549–4552 (1998);http://dx.doi.org/10.1103/PhysRevLett.81.4549
11.H. B. Chan, V. A. Aksyuk, R. N. Kleinman, D. J. Bishop, and F. Capasso, “Quantum mechanical actuation of microelectromechanical systems by the Casimir force,” Science 291, 1941–1944 (2001);http://dx.doi.org/10.1126/science.1057984
11.G. Bressi, G. Caarugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir force between parallel metallic surfaces,” Phys. Rev. Lett. 88, 041804 (2002).http://dx.doi.org/10.1103/PhysRevLett.88.041804
12.The original calculation was made in terms of the zero-point energy of quantum field theory by H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proc. Ned. Akad. Wetenschap. 51, 793–795 (1948);
12.The same results appear in classical electrodynamics which includes classical electromagnetic zero-point radiation. See, for example, T. H. Boyer, “Random electrodynamics: The theory of classical electrodynamics with classical electromagnetic zero-point radiation,” Phys. Rev. D 11, 790–808 (1975).http://dx.doi.org/10.1103/PhysRevD.11.790
13.Boltzmann’s derivation is discussed in, for example, R. Resnick and D. Halliday, Physics (Wiley, New York, 1967), Supplementary Topic IV, p. 11;
13.R. Becker and G. Leifried, Theory of Heat, 2nd ed. (Springer, New York, 1967), p. 94.
14.Some aspects of the scalar field case in four spacetime dimensions are given by T. H. Boyer, “Derivation of the Planck spectrum for relativistic classical scalar radiation from thermal equilibrium in an accelerating frame,” Phys. Rev. D 81, 105024 (2010).http://dx.doi.org/10.1103/PhysRevD.81.105024
15.H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, 1981), pp. 575–580.
16.For an extensive discussion of scaling, see T. H. Boyer, “Scaling symmetry and thermodynamic equilibrium for classical electromagnetic radiation,” Found. Phys. 19, 1371–1383 (1989).http://dx.doi.org/10.1007/BF00732758
17.Additional comments on scaling symmetry can be found in T. H. Boyer, “Scaling symmetries of scatterers of classical zero-point radiation,” J. Phys. A: Math. Theor. 40, 9635–9642 (2007).http://dx.doi.org/10.1088/1751-8113/40/31/031
18.See, for example, Ref. 15, p. 137.
19.Discussion of random radiation in terms of random phases can be found in S. O. Rice, “Mathematical analysis of random noise,” in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954), p. 138.
20.See, for example, Ref. 1, p. 339.
21.See, for example, T. H. Boyer, “The classical vacuum,” Sci. Am. 253 (2), 70–78 (1985).http://dx.doi.org/10.1038/scientificamerican0885-70
22.This divergence is termed the “infrared divergence” in two-dimensional massless scalar fields in Minkowski spacetime. See S. R. Coleman, “There are no Goldstone bosons in two dimensions,” Commun. Math. Phys. 31, 259–264 (1973).http://dx.doi.org/10.1007/BF01646487
23.S. A. Fulling and P. C. W. Davies, “Radiation from a moving mirror in two dimensional space-time: Conformal anomaly,” Proc. R. Soc. London, Ser. A 348, 393–414 (1976), p. 407.http://dx.doi.org/10.1098/rspa.1976.0045
24.R. C. Tolman, Thermodynamics and Cosmology (Dover, New York, 1987), p. 318;
24.R. C. Tolman and P. Ehrenfest, “Temperature equilibrium in a static gravitational field,” Phys. Rev. 36, 1791–1798 (1930).http://dx.doi.org/10.1103/PhysRev.36.1791
25.See, for example, M. D. Greenberg, Advanced Engineering Mathematics, 2nd ed. (Prentice Hall, Upper Saddle River, 1998), Sec. 17.7;
25.J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (Benjamin/Cummins, Reading, 1970), pp. 264 and 338.
26.I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 494. The desired integral is .
27.In classical physics, the blackbody radiation spectrum follows from the structure of relativistic spacetime. It appears that thermal radiation in a static spacetime coordinate system is the one-parameter spectrum of isotropic random classical radiation obtained from the scale-invariant spectrum of zero-point radiation by the time-dilating conformal transformation that preserves the wave equation and carries radiation normal modes into normal modes.
28.See, for example, Ref. 2, p. 12.
29.T. H. Boyer, “Equilibrium of random classical electromagnetic radiation in the presence of a nonrelativistic nonlinear electric dipole oscillator,” Phys. Rev. D 13, 2832–2845 (1976);http://dx.doi.org/10.1103/PhysRevD.13.2832
29.T. H. Boyer,“Statistical equilibrium of nonrelativistic multiply periodic classical systems and random classical electromagnetic radiation,” Phys. Rev. A18, 1228–1237 (1978).http://dx.doi.org/10.1103/PhysRevA.18.1228
30.T. H. Boyer, “Blackbody radiation and the scaling symmetry of relativistic classical electron theory with classical electromagnetic zero-point radiation,” Found. Phys. 40, 1102–1116 (2010).http://dx.doi.org/10.1007/s10701-010-9436-0
31.P. C. W. Davies, “Scalar particle production in Schwarzschild and Rindler metrics,” J. Phys. A 8, 609–616 (1975).http://dx.doi.org/10.1088/0305-4470/8/4/022
32.See the recent review by L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, “The Unruh effect and its applications,” Rev. Mod. Phys. 80, 787–838 (2008).http://dx.doi.org/10.1103/RevModPhys.80.787
33.Connections between the classical and quantum treatments are described by T. H. Boyer, “Classical and quantum interpretations regarding thermal behavior in a coordinate frame accelerating through zero-point radiation,” e-print arXiv:1011.1426.
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