Abstract
The two-point correlation functions of classical electromagnetic zero-point radiation fields are evaluated in four-vector notation. The manifestly Lorentz-covariant expressions are then shown to be invariant under scale transformations and under the conformal transformations of Bateman and Cunningham. As a preliminary to the electromagnetic work, analogous results are obtained for a scalar Gaussian random classical field with a Lorentz-invariant spectrum.
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References
E. Cunningham,Proc. London Math. Soc. 8, 77 (1910); H. Bateman,Proc. London Math. Soc. 8, 223 (1910).
See the reviews of stochastic electrodynamics by T. H. Boyer, inFoundations of Radiation Theory and Quantum Electrodynamics, A. O. Barut, ed. (Plenum, New York, 1980), and by L. de la Peña, inProceedings of the Latin American School of Physics, Cali, Colombia 1982, B. Gomezet al., eds. (World Scientific, Singapore, 1983). See also T. H. Boyer,Sci. Am. 253, 70 (Aug. 1985).
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T. H. Boyer,Phys. Rev. 182, 1374 (1969);186, 1304 (1969).
Marshall's work in Ref. 5 is one of the few articles to use 4-vector notation. Marshall gives in his Eq. (3.2) our Eq. (47), except for the evaluation of the scalar correlation function in closed form. However, Marshall does not consider the questions of scale or conformal invariance.
T. H. Boyer,Phys. Rev. D21, 2137 (1980).
See Ref. 8, p. 2140, Eqs. (15) and (16).
See, for example, J. D. Bjolken and S. D. Drell,Relativistic Quantum Fields (McGraw-Hill, New York, 1965), p. 27, Eq. (12.7).
N. D. Birrell and P. C. W. Davies,Quantum Fields in Curved Spaces (Cambridge University Press, Cambridge, 1982), p. 23, Eq. (2.79). In the present article unrationalized units are used, corresponding to the choice for the electromagnetic case. See also the discussion of the Δ, function in W. Heitler,The Quantum Theory of Radiation, 3rd edn. (Oxford University Press, Oxford, 1954), p. 72, Eq. (27b).
See the discussion in T. H. Boyer, to be published.
See the review by T. Fulton, F. Rohrlich, and L. Witten,Rev. Mod. Phys. 34, 442 (1962).
H. A. Kastrup,Ann. Phys. (Leipzig) 9, 388 (1962).
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See, for example, I. Bialynicki-Birula and Z. Bialynicka-Birula,Quantum Electrodynamics (Pergamon, New York, 1975), p. 134.
See, for example, J. D. Jackson,Classical Electrodynamics, 2nd edn. (Wiley, New York, 1975), p. 550, Eq. (11.138).
The structure of the result in Eq. (47) has similarities to expectation values found in quantum electrodynamics. See, for example, A. I. Akhiezer and V. B. Berestetskii,Elements of Quantum Electrodynamics (Oldbourne Press, London, 1959), p. 81. Also see J. D. Björken and S. D. Drell,Relativistic Quantum Fields (McGraw-Hill, New York, 1965), pp. 78–81. (Dr. D. C. Cole pointed out to me that the 4-vector form for the correlation function (47) was given by T. W. Marshall,Proc. Camb. Phil. Soc. 61, 537 (1965), Eq. (3.2).)
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Boyer, T.H. Conformal symmetry of classical electromagnetic zero-point radiation. Found Phys 19, 349–365 (1989). https://doi.org/10.1007/BF00731830
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DOI: https://doi.org/10.1007/BF00731830