Squeezed and Nonclassical Light pp 73-80 | Cite as

# Classical and Quantum Statistics in Partition of Highly Degenerate Light

Chapter

## Abstract

The statistical properties of photons have been found to be well accounted for by the methods of Quantum Electrodynamics. In spite of this satisfactory situation, recently the attention of some theoretical physicists has been attracted by an interesting paradox which shows, in the context of light beam-splitting, the existence of a surprising conflict between the expectations of QED and those dictated by the indistinguishability property of the photon ^{1,2.} Let us present the problem first.

## Keywords

Quantum Correlation Indistinguishability Principle Quantum Measurement Theory Indistinguishability Hypothesis Partition Probability
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## References

- 1).R.P Feynman, R.B.Leighton and M.Sands,
*The Feynman Lectures on Physics*, Vol.3, Ch.4, Addison Wesley, R.ading, 1965; S.Prasad, M.O.Scully and W.Martienssen, Opt. Comm. 62, 139, 1987; Z.Y.Ou, C.K.Hong and L.Mandel, Opt.Comm. 63, 118, 1987.Google Scholar - 2).R.Loudon,
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*The Principles of Statistical Mechanics*,Oxford U.Press, 1967, Ch.X; K.Huang,*Statistical Mechanics*,Wiley, N.Y.1963, Ch.9.Google Scholar - 5).M.Born and E.Wolf,
*Principles of Optics*,Macmillan, N.Y.1964, Ch.7, 8; L.Mandel, Opt.Soc.Am.51, 797, 1961, reports light-degeneracy parameters for various types of incoherent sources. Typically, n=0.005 for a strong high-pressure Hg-arc.Google Scholar - 6).W.Martienssen and E.Spiller, Am.J.Phys, 32, 919, 1964.Google Scholar
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*“Subtle is the Lord..”The Science and Life of Albert Einstein*,Oxford University Press, 1982, pag.430; and in Revs.Mod.Phys. 51, 4, 1979, pag.896. P.A.M.Dirac, Proc.R.Soc. A 112, 661, 1926, first related (B-E) statistics to symmetrization of particle microstates (A.Pais, Rev.Mod.Phys. op.cit. pag.894). Note that the formulation of the (B-E) partition law which is generally used has been introduced by A.Einstein, Sitzungsber.1925, op.cit. pag.3.Google Scholar - 9).T.S.Kuhn,
*The Structure of Scientific Revolutions*. The Chicago U. Press, 1962.Google Scholar - 10).The origin of the “intrinsic indistinguishability” concept belongs to the pre-quantal era as it may be traced down to the work by L.Boltzmann, “Weiter studien uber die Warmegleichgewicht unter Gasmolekulen”, Wiener Ber., II, 66, 1872. The first formulation of a partition law that agrees with Planck’s law is actually due to M.Planck: “Uber eine Verbesserung der Wien’schen Spektralgleichung” in Verh. d.D. Phys.Ges, 2, 202, 1900; and Ann.d. Phys.4, 553, 1901. According to T.S.Kuhn (op.cit. below), the origin of the first formal derivation of the partition law could be identified as a combinatorial theory (“Combinationslehre”) that Planck could have found in: E.Netto, “Combinatorik” in:
*“Encyclopadie der Mathematischen Wissenschaften*,Vol.1, Part 1, Leipzig, 1898 or in: L.Boltzmann, ”Uber die Beziehung zwischen dem Zweiten Hauptsatze der mechanischen Warmetheorie und der Warhrscheinlichkeitsrechnung respektive den Satzen uber das Warmegleichgewicht“, Wiener Ber., H, 76, 373, 1877. A full historical account of the origin of Planck’s theory is found in: T.S.Kuhn,*Black-Body Theory and Quantum Discontinuity*,Oxford, 1978, (Ch.4, text and note 20); and in works by M.J.Klein and M.J.Kangro, quoted therein.Google Scholar - 11).Consider
*the first*part of the following aphorism by Ludwig Wittgenstein: “We cannot think what we cannot think; so what we cannot think we cannot say either”. (L.Wittgenstein,*Tractatus Logicus Philosophicus*,5.61. The Humanities Press, New Jersey, 1974). In our case, if the hypothesis of intrinsic particle indistinguishability is adopted, this hypotesis*cannot*be violated in*any*step of the development of a valid formal theory. It appears evident to us that, if a*correct*theory (like the wavefunctionsymmetrization theory) cannot be formulated without making recourse to a side assumption that violates an hypothesis, that hypothesis itself is false. Then, in our case the “indistinguishability hypothesis” should be rejected, on logical grounds. Although this way of reasoning may appear somewhat artificial, it is nevertheless similar to the one used by N.Bohr in his debate with Einstein on the EPR argument. Cfr. N.Bohr, in: P.A.Schlipp ed.,*Albert Einstein Scientist Philosopher*,Cambridge University Press, 1982.Google Scholar - 12).N.Bohr: “Can quantum-mechanical description of physical reality be considered complete?”, Nature, 136, 65, 1935 and Phys.Rev. 48, 696, 1935, reprinted in: J.A. Wheeler and W.Zurek (eds.),
*Quantum Theory and Measurement*,Princeton University Press, 1983. In these papers, which open up the field of “Quantum Measurement Theory” and reply to the argument of Einstein, Podolsky and Rosen (EPR), Bohr anticipates the detection of a “Visibility loss” if a light and mobile screen is adopted in an IF. This effect, that according to Heisenberg Principle counterbalances the gain of infoicuation on the paths taken by photons within IF, is precisely the one shown by our B-E curve V(ii), in Fig.2.Google Scholar - 13).Highly-degenerate beams of particles may be obtained, in practice, only with photons and phonons. In general, other Bose-particles, like tt-mesons, are not created at a sufficient rate in the actual high-energy experiments.Google Scholar
- 14).The validity of this proposition is being tested experimentally in our Laboratory by reproducing “microscopic” photon-momentum transfer effects in a macroscopic IF, under controlled conditions. Preliminary results validate our proposition. (B-E) Hanbury Brown-Twiss effects in hadron-hadron scattering have been investigated by: G.Goldhaber, W.B.Fowler, S.Goldhaber, T.F.Hoang, T.E. Kalogropoulos and W.R.Powell, Phys.Rev.Lett. 3, 181, 1959; G.Goldhaber, W.Lee and A.Pais, Phys.Rev. 120, 300, 1960; A.Giovannini, G.C.Mantovani and S.P.Ratti: “Old and New Variables, Old and New Optical Concepts in High-Energy Hadron-Hadron Scattering”, Rivista del Nuovo Cimento, 2, 1, 1979.Google Scholar
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*is not*an “Hidden-Variable” theory, nor implies the exotic assumptions of “Stochastic-Electrodynamics” or the ones of the Bohm’s “Quantum-Potential Theory”. (Cfr. T.Marshall and E.Santos in:*New Techniques and Ideas in Quantum Measurement Theory*,ed.by D.Greenberger, The N.Y. Academy of Sciences, N.Y. 1986; D.Bohm, Phys.Rev.85, 166, 1952).Google Scholar - 16).H.B.Dwight,
*Tables of Integrals*, MacMillan, N.Y., 1961, pg. 209.Google Scholar - 17).J.Tersoff and D.Bayer, Phys.Rev.Lett. 50, 553, 1983.Google Scholar

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