# Why are gauge theories called so?

Why are gauge theories called so? I guessed it was because gauge also means to estimate, so when one is trying to find the gauge theory for such and such interactions one has to estimate what might be the best gauge group for that interaction. Does this make sense?

• In a book I have written by Leonard Susskind he says: It is called a gauge transformation. Why "gauge"? Its a historical glitch. At one time it was wrongly thought to reflect ambiguities in gauging lengths at different locations. – SaudiBombsYemen Oct 11 '16 at 18:31
• Would History of Science and Mathematics be a better home for this question? – Qmechanic Oct 11 '16 at 18:37
• @Qmechanic That's the same question I asked myself. I'm not sure, though. – Patrick Oct 11 '16 at 18:41
• I'm voting to close this question as off-topic because it is about history, not science, and should be on history of science and mathematics. – heather Oct 11 '16 at 21:28

## 1 Answer

Because Weyl's original gauge theory (1918-1920), which was also the very first unified field theory of electromagnetism and gravity (Kaluza only published in 1921), had a field of scales/gauges needed to completely localize Riemannian metric, see When and how was the geometric understanding of gauge theories developed? Weyl's prototype was general relativity, but as it is Riemannian geometry is not completely local. Lengths of vectors at different points are numbers, and can be compared in the absolute sense. Weyl's idea was that all measurements must be relativized to local scales only.

So he replaced Riemannian with conformal metric, and added a field of scales instead of absolute scalars, the gauge field. This gauge field specifies how the scales are transported, but not minimally, different fields may specify the same transport, in which case they are related by a gauge transformation. Weyl's theory did not work out (Einstein quickly noticed that it was unphysical), but the gauge idea had a bright future. Weyl's principle of gauge invariance stated that the form of physical laws must be invariant under local gauge transformations. Selecting a particular version of a gauge field among many equivalent ones is now called "fixing the gauge", the Lorentz gauge for the vector potential is an example.

Thus, the first gauge group was $\mathbb{R}^+$ of positive reals under multiplication, in late 1920s Weyl went from a field of gauges to a field of phases, replacing $\mathbb{R}^+$ with $U(1)$, but the name stuck. In this new theory Weyl was able to give a gauge theoretic explanation of the conservation of electric charge. Dirac replaced the $U(1)$ fields with the sections of associated complex line bundles in 1931, and noticed the possibility of magnetic monopoles when the bundles were globally non-trivial. Non-Abelian gauge theories did not appear until Yang and Mills in 1950s.

Varadarajan's paper Vector Bundles and Connections in Physics and Mathematics is a good historical survey of early gauge theory, with technical details.

• Did not Weyl explicitly call his invariance "Eichinvarianz" and the English word is simply a calque of this one? – Selene Routley Oct 13 '16 at 0:03
• @WetSavannaAnimalakaRodVance Yes, and Eich is calibration/gauge/scale in German. But modern gauge theories are closer in spirit to the $U(1)$ version, so it should really be phase theory, phase fixing and phase invariance :) – Conifold Oct 14 '16 at 0:49
• @Conifold Just a note in passing : Weyl did not adapt his theory to U(1) group until London and Schrödinger proved it might give a nice playground for the newly discovered quantum mechanics. See the book The Dawning of Gauge Theory by O'Raifertaigh, and a review by the same author on Review of Modern Physics. – FraSchelle Oct 24 '16 at 11:45