On the group notation of the 1975 Wu-Yang paper

[pines-demon]

TL;DR Summary

What means the group subindex in the notation of the Wu–Yang paper

In the 1975 Wu–Yang paper on electromagnetism=fiber bundle theory Table 1: https://journals.aps.org/prd/pdf/10.1103/PhysRevD.12.3845

Wu & Yang use the notation ##\mathrm{U}_1(1)## for the bundle of electromagnetism and ##\mathrm{SU}_2## for the isospin gauge field.
I am unfamiliar with this groups, unless we take the subindices, for example what the subindex 1 in ##\mathrm{U}_1(1)## means? Same for ##\mathrm{SU}_2## ?

 

[vanhees71]

They simply extend the discussion to non-Abelian gauge groups. They write ##\mathrm{U}_1## instead of ##\mathrm{U}(1)##. It's just their notation. ##\mathrm{U}(1)## is the symmetry of wave functions under multiplying the wave function with a phase factor. Electrodynamics occurs when gauging this global symmetry to make it a "local symmetry", while one must be somewhat careful when calling such a "gauge symmetry" a symmetry, but that's another story. The same holds for ##\mathrm{SU}_2## vs. ##\mathrm{SU}(2)## here the gauged symmetry is isospin symmetry. That's only a toy model. In the Standard Model you rather have ##\mathrm{SU}(3)## as the gauge group of Quantum Chromodynamics describing the Strong Interactions and ##\mathrm{SU}(2) \times \mathrm{U}(1)##, describing the strong and electromagnetic interactions (gauging the weak-isospin and weak-hypercharge symmetry and then "Higgsing" it to the electromagnetic ##\mathrm{U}(1)##, i.e., getting the three W- and Z-gauge bosons massive and keeping the photon massless, without distroying the local gauge symmetry underlying the electroweak sector of the Standard Model.

 

[Orodruin]

TL;DR Summary: What means the group subindex in the notation of the Wu–Yang paper

In the 1975 Wu–Yang paper on electromagnetism=fiber bundle theory Table 1: https://journals.aps.org/prd/pdf/10.1103/PhysRevD.12.3845

TIL: My university library has cancelled the subscription to APS jounals …

 

[pines-demon]

They simply extend the discussion to non-Abelian gauge groups. They write U1 instead of U(1).

But that is not what they do, the do not write ##\mathrm U_1## or ##\mathrm U(1)## they write ##\mathrm U_1(1)## does having two 1s imply something else?

 

[Orodruin]

TIL: My university library has cancelled the subscription to APS jounals …

Ok, I was wrong. Just not connected to OpenAthens, had to access via the library homepage.

Generally, they seem to be using ##U_1## and ##SU_2## as described by @vanhees71 but then there is this table:

I did not find the notation in other places but the look I gave was very cursory.

 

[pines-demon]

Ok, I was wrong. Just not connected to OpenAthens, had to access via the library homepage.

Generally, they seem to be using ##U_1## and ##SU_2## as described by @vanhees71 but then there is this table:

I did not find the notation in other places but the look I gave was very cursory.

I did not find it elsewhere either they also write ##\mathrm{SO}_3##. In a precedent paper by Yang, cited in the above he uses ##\mathrm U(1)## without subindex :https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.33.445

 

[martinbn]

May be it is a misprint.

 

[pines-demon]

May be it is a misprint.

You are probably right, I was thinking that it was some special group or notation for bundles. Thanks!

 

[ohwilleke]

When considering whether this might be a misprint, it is worth noting that scientific journal typesetting was pretty primitive in 1975.

It had to be done analog style, because advanced word processing computers weren't widely available. Lots of physicists at that time didn't know how to type at all, and relied upon their department secretaries or a university typing pool for that, and those (invariable female) administrative aides, who had no formal training in graduate level physics or mathematics, typed the final version of the article from the physicists' handwritten manuscripts, which were written using pens or mechanical pencils.

A significant subset of the charts and illustrations were also handwritten (either free hand, or drawn with straight edges and compasses and protractors) in that era, and were literally cut and pasted into the final draft. And, it was also around the time that the transition from slide rules to calculators was just reaching the tipping point. Many university computers in 1975 still had punchcard data entry, although keyboards at dumb terminals connected to mainframe computers were starting to become more common then.

My father was an engineering professor at Georgia Tech in 1975. He didn't have access to a mainframe with a dumb terminal with a keyboard until 1977 (at a new university with more resources to spend on computers), and he didn't have a computer of his own to use in his office or at home until 1983. He learned to type on his own at that point (at the same time I did as a middle school aged kid), because men pursuing careers in science and engineering weren't taught to type in his generation.

 

[Vanadium 50]

In 1975 you would have your secretary type out your paper, equations and all, on an IBM Selectric, and then the journal would use a Monotype machine to do the final formatting. I wouldn't say "primitive" so much as "specialized". And "expensive" - it took a minimum of three different people.

Xerox would be happy to sell - actually, lease, as they sold very little - a graphics printer in the 1970s, although the Dover in the early 80s was a huge step forward.

 

[pines-demon]

Aside of misprinting problems the (1) could be a reference to either a equation or reference but seems unlikely.