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American Journal of Physics

The photon picture of radiation transport

J. Rosato

Published Jul 14, 2010• 5 citations
Publisher

Abstract

The photon picture of radiation transport is examined. Because the photon has no mass and spin one, a position operator for a photon particle cannot be defined properly. This lack leads to ambiguity if the radiative transfer equation is associated with a quantum transport equation for photons. The issue of photon nonlocalizability is discussed and illustrated through a wavefunction-based approach. It is next shown that the phase space formalism due to Wigner provides a suitable framework for a proper derivation of a photon transport equation from first principles.

DOI
10.1119/1.3377132

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The photon picture of radiation transport is examined. Because the photon has no mass and spin one, a position operator for a photon particle cannot be defined properly. This lack leads to ambiguity if the radiative transfer equation is associated with a quantum transport equation for photons. The issue of photon nonlocalizability is discussed and illustrated through a wavefunction-based approach. It is next shown that the phase space formalism due to Wigner provides a suitable framework for a proper derivation of a photon transport equation from first principles.
You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) œeld. However, Heisenberg’s uncertainty principle tells us that a state of deœnite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indeœnite in space and time. It suggests the diµculty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, a§er a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with œnite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic fallo˜ of energy density exist, and that a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999).
Light occupies a special position in our attempts to understand nature both classically and quantum mechanically. We recall that Newton, who made so many fundamental contributions to optics, championed a particle description of light and was not favorably disposed to the wave picture of light. However, the beautiful unification of electricity and magnetism achieved by Maxwell clearly showed that light was properly understood as the wave-like undulations of electric and magnetic fields propagating through space. The central role of light in marking the frontiers of physics continues on into the twentieth century with the ultraviolet catastrophe associated with black-body radiation on the one hand and the photoelectric effect on the other. Indeed, it was here that the era of quantum mechanics was initiated with Planck's introduction of the quantum of action that was necessary to explain the black-body radiation spectrum. The extension of these ideas led Einstein to explain the photoelectric effect, and to introduce the photon concept. It was, however, left to Dirac to combine the wave-and particlelike aspects of light so that the radiation field is capable of explaining all interference phenomena and yet shows the excitation of a specific atom located along a wave front absorbing one photon of energy. In this chapter, following Dirac, we associate each mode of the radiation field with a quantized simple harmonic oscillator, this is the essence of the quantum theory of radiation.
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