Self-consistent theory for a plane wave in a moving medium and light-momentum criterion

What is it about?

A covariant theory of light propagation in a medium is set up based on the symmetries of Maxwell electromagnetic theory, momentum-energy conservation law, and Fermat’s principle, required by the principle of relativity. Macroscopic Maxwell equations are the foundation of classical electrodynamics, while Einstein’s light-quantum hypothesis is the basis of account of atomic transitions in quantum mechanics. Macroscopic Maxwell equations combined with Einstein’s light-quantum hypothesis lead to a self-consistent Minkowski photon four-momentum for a plane light wave, and thus this four-momentum macroscopically describes the light-matter interactions through atomic absorption and re-emission. Thus, Minkowski photon carries the unique correct light momentum in a medium.

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Why is it important?

The theory developed in my work takes into account not only (i) the invariance of Maxwell equations argued by Minkowski, but also (ii) the invariance of physical definitions, and (iii) the four-vector covariance of momentum and energy of light. All conditions (i), (ii), and (iii) are required by the principle of relativity, and without (iii), the definition of light momentum in a medium is indefinite. Usually, there is little attention paid to the principle of relativity in the analysis of Abraham-Minkowski debate on the momentum of light in a medium. For example, the invariance of Maxwell equations in Minkowski’s electrodynamics of moving media is realized based on the condition that all EM fields constitute two field-strength tensors. From this it follows that the wave phase must be a Lorentz invariant, and the frequency and the wave vector must constitute a Lorentz four-vector. Unfortunately, this fundamental physical result from the Minkowski’s electrodynamics is widely ignored in the analysis of Abraham-Minkowski debate, although Einstein has already made a pioneer demonstration in his 1905 paper for a plane wave in free space. Planck constant as a Lorentz invariant is an implicit postulate in Dirac’s quantum mechanics [P. A. M. Dirac, The principles of quantum mechanics, (Oxford, London, 1958), §66, p. 254]. Invariance of Planck constant is the sufficient and necessary condition for the wave four-vector multiplied by the Planck constant to be a four-vector. The paper (10.1139/cjp-2015-0167) provides a rigorous proof of the invariance of Planck constant. The procedure of proof is as follows: (1) firstly, Planck constant is used to quantize classical electromagnetic momentum and energy, and then (2) the quantized momentum and energy are proved to constitute a Lorentz four-vector. It is significant to turn Dirac’s postulate to a reality, especially for relativistic quantum mechanics and quantum field theory. That is because if Planck constant were not invariant, the uncertainty principle, the De Broglie relations, and the commutation relations would all break down.

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