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I can not identify clearly who was the first one to realize that the electromagnetic tensor is the curvature 2-form of a U(1)-connection. Looking at Weyl's work, it seems that he came pretty close to it in 1929 but I can not find any clear indication that he realized that $A$ was a connection and $F$ its curvature. I can not see any clear statement about this until Wu and Yang's paper in 1975. Could anyone help me?

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  • $\begingroup$ Crossposted from physics.stackexchange.com/q/796428/2451 $\endgroup$
    – Qmechanic
    Jan 8 at 1:15
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    $\begingroup$ Weyl could not realize it in 1929 because connections on general bundles were only defined by Koszul and Ehresmann in 1950. However, what he did already in 1918-20, in terms of Christoffel symbols and parallel transport, is considered essentially equivalent. Varadarajan's survey has a section titled "The work of Weyl and his discovery that the electromagnetic vector potential is a connection on a suitable bundle on spacetime and the electromagnetic field is the curvature of this connection." $\endgroup$
    – Conifold
    Jan 8 at 2:24
  • $\begingroup$ Thank you very much! It seems however that even after Ehresmann definition in 1950, it took a long while (~1970) before physicists and mathematicians realized they were using the same objects. $\endgroup$ Jan 8 at 8:40
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    $\begingroup$ Andrzej Trautman identified $F$ as a curvature of a U(1) connection before Wu and Yang. In his 1967 lectures he talks about it (published here doi.org/10.1016/0034-4877(70)90003-0 ) $\endgroup$
    – Mauricio
    Jan 8 at 13:43
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    $\begingroup$ Might worth checking Lubkin's work (1963): doi.org/10.1016/0003-4916(63)90194-5 $\endgroup$
    – Mauricio
    Jan 8 at 13:50

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I still have only a partial answer to the question. Looking at Trautman's lecture notes published in 1970 (https://doi.org/10.1016/0034-4877(70)90003-0), it is more than clear that he knew for sure that $A$ was a U(1) connection and $F$ its curvature. Lubin's work in 1963 (https://doi.org/10.1016/0003-4916(63)90194-5) is rather technical and seems to already point towards that direction. However, going back as far as Utiyama's work of 1955 (https://doi.org/10.1103/PhysRev.101.1597), right after the work of Yang and Mills, it seems that he generalizes the work of Weyl (1929) to non-abelian group in a very technical way, comparing the affine connexions and the gauge fields without truly identifying them as a the same mathematical/geometrical structure. Any complement to this would of course be welcome!

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