[Le facteur deux]
Examples are given of physical problems where factor two played an important role. First, we consider the irregularities of the Moon trajectory that Newton correctly, attributed to the differences in the attraction by the Sun of Earth and Moon because of the small differences in their location. Newton’s estimate of those (small) irregularities was off the measures by a factor close to two, which was restored mater by Clairaut and seen as a major success for Newton’s dynamical theory. Another factor two appears in the Sun’s estimate of the deviation of light. The deviation computed by using special relativity is half the one measured. As Einstein showed, the factor is restored by considering general relativity which considers the changes in the metric due to gravity. Lastly, we consider a more recent problem, the one of the apparent doubling of electric charges in a superconductor as measured by the quantum of flux, which is half the one predicted by London, namely ℏc/2e instead of ℏc/e. This doubling is classically explained by assuming that, in superconductors current carrying fermions are in pairs so that the unit charge becomes 2e instead of e. We argue instead that if one takes into account that the fermions carry a spin 1/2 the quantization element of magnetic flux is divided by two as implied by looking at the second class of homotopy of the group of transformation SU(2) × U(1) of the ground state, as it follows from Toulouse theory of defects in ordered media.
Nous donnons plusieurs exemples de problèmes physiques où un facteur 2 joue un rôle important. Nous examinons d’abord la question des irrégularités du mouvement de la lune attribuées correctement par Newton à la différence de l’attraction du soleil sur la terre et sur la lune légèrement décalées l’une par rapport à l’autre vues du soleil. L’estimation par Newton de ces irrégularités différait d’un facteur 2 environ des observations. Ce facteur 2 a été trouvé par Clairaut en utilisant la théorie dynamique de Newton qui fut ainsi été pleinement confirmée. Un autre facteur 2 apparaît dans le calcul de la déviation d’un rayon lumineux par la masse du soleil. La relativité restreinte donne un résultat différent de l’observation aussi par un facteur 2. Ce facteur 2 a été rétabli par Einstein en utilisant la relativité générale qui tient compte de la déformation de la métrique de l’espace-temps près des masses, ce que ne fait pas la relativité restreinte. Finalement nous considérons une question de physique plus récente, celle du quantum de flux magnétique dans un supraconducteur. Les mesures donnent un quantum moitié de celui prédit par London, soit ℏc/2e au lieu de ℏc/e. L’explication habituelle de ce facteur 2 est que dans les supraconducteurs les électrons porteurs de charge sont appariés, ce qui doublerait la charge du quantum de London. Nous montrons qu’il existe une autre possibilité d’explication de cette division par 2 du quantum de London, basée sur la théorie de Toulouse des défauts dans les milieux continus. Dans le cas présent, le quantum de flux magnétique est lié au premier groupe d’homotopie du paramètre d’ordre qui, si l’on inclut la phase de la fonction d’onde globale et le spin 1/2 a le groupe de symétrie SU(2) × U(1) qui génère des défauts pour lesquels la circulation vaut la moitié de celle du groupe U(1), qui serait donc limité à la seule phase (le groupe U(1) sans les rotations de spin le groupe SU(2)). Cette division par 2 de la circulation de la phase remonte à une remarque de Cauchy sur l’échange des deux valeurs propres d’une matrice 2 × 2 le long d’un cercle entourant (dans le plan complexe) un zéro du déterminant.
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Mots-clés : Défauts topologiques SU(2) × U(1), Quantum de London, Supraconductivité, Superfluidité
Yves Pomeau 1 ; Sergio Rica 2
CC-BY 4.0
@article{CRPHYS_2025__26_G1_349_0,
author = {Yves Pomeau and Sergio Rica},
title = {The factor two},
journal = {Comptes Rendus. Physique},
pages = {349--368},
publisher = {Acad\'emie des sciences, Paris},
volume = {26},
year = {2025},
doi = {10.5802/crphys.248},
language = {en},
}
Yves Pomeau; Sergio Rica. The factor two. Comptes Rendus. Physique, Volume 26 (2025), pp. 349-368. doi : 10.5802/crphys.248. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.248/
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