17

Can anyone give a physical reason that $U(1)$ is the gauge group for classical electromagnetism?

I am familiar with the principal bundle formalism for Yang-Mills theory and see that since the Lie algebra of $U(1)$ is $\mathbb{R}$, the Yang-Mills equations reduce to Maxwell's equations. However, I am looking for a plausible physical reason that $U(1)$ is related to classical electromagnetism.

This seems to be the same question: Classical electrodynamics as an $\mathrm{U}(1)$ gauge theory , but does not have a satisfactory answer.

Qmechanic
  • 201,751
Bendy
  • 273
  • 3
    it depends on what you mean by "physical". To me, rigorous group theoretical arguments are as physical as it comes... (and, ultimately, the fact that the model works very well is the best argument I can think of) – AccidentalFourierTransform May 27 '16 at 10:58
  • Okay, so suppose I have some electromagnetic field configuration on the base space of the principal $U(1)$-bundle. Take a path in the base space and move along it; the electromagnetic field varies and, lifting the path to the total space, so does the ' $U(1)$ phase'. Is there any physical meaning to this $U(1)$ phase? – Bendy May 27 '16 at 12:58
  • @Bendy Gauge degrees of freedom in general do not affect the physics of the theory. A field that satisfies Maxwell's equations still does so even after you've applied a U(1) transformation to it, but an effect of such a transformation may be that the electric field turns into magnetic and vice versa, or electromagnetic in general. – auxsvr May 27 '16 at 13:53
  • 2
    Sorry, @auxsvr - but gauge transformations cannot map electric fields to magnetic ones or vice versa. You must have confused gauge transformations with (S-) dualities. Otherwise, the answer to the OP is that $U(1)$ is the only connected Abelian simple Lie group. Electromagnetism has to be Abelian because the light doesn't interact with itself (linearity, which implies the absence of higher-order interactions from F mu nu F mu nu), and it is a long-range force which bans breaking. But in particle physics, U(1) does arise as a leftover of a larger group. – Luboš Motl May 27 '16 at 14:07
  • @LubošMotl Well, I meant applying a U(1) transformation to $F + i\star F$. If this is called S-duality, then you're right. – auxsvr May 27 '16 at 14:32
  • 1
    Sorry but $F+i*F$ (just like $F$ itself) is gauge-invariant, so the gauge transformation doesn't change a damn thing about it. – Luboš Motl May 27 '16 at 14:41
  • @LubošMotl Perhaps I'm using wrong terminology here or I'm missing something fundamental: isn't the effect of U(1) on $F$: $\cos a F+i \sin a\star F$? – auxsvr May 27 '16 at 14:59
  • 4
    I'm not sure what you are looking for. The four-potential transforms as $A\mapsto A+\mathrm{d}\chi$ under a gauge transformation, so the Lie algebra of your gauge group is $\mathfrak{u}(1)=\mathbb{R}$, and so the gauge group can be either $\mathrm{U}(1)$ or $\mathbb{R}$. Since classical electromagnetism doesn't have any other fields transforming in a representation of the gauge group, but just the current $j^\mu$ with $\mathrm{d}{\star}j = 0$ that transforms trivially, you can't decide, classically, which one to choose. We take $\mathrm{U}(1)$ because this is what get in the quantum theory. – ACuriousMind May 27 '16 at 15:04
  • 1
    Thanks for these comments, guys. A $U(1)$ transformation turning an electric field into a magnetic field is the kind of answer I would like, but doesn't seem to be correct. Must be Abelian because light doesn't interact with itself is also fairly satisfying. @auxsvr, in general $F$ transforms as $F \mapsto gFg^{-1}$ for $g \in G$ (the gauge group) which reduces to $F \mapsto F$ for $G$ Abelian. – Bendy May 27 '16 at 15:12
  • @Bendy That is the conjugation action; I'm writing about a left/right action on $F + i \star F$. – auxsvr May 27 '16 at 16:12
  • @auxsvr - the field strength $F$ transforms and has to transform in the adjoint representation, so the only correct or physically meaningful rule for its transformation is the conjugation. Transforming it by a left action only means to forget about the second color index. – Luboš Motl May 27 '16 at 16:40
  • 1
    @auxsvr - In response to your question: Isn't the effect of $U(1)$ on $F$: $\cos a F + i \sin a \ast F$? The answer is absolutely not. The effect of $U(1)$ on the gauge field $A \to A + d \chi$ and since $F = d A$, the effect of $U(1)$ on $F$ is simply $F \to F$. What you are thinking of there is known as the electric-magnetic duality (or more generally $S$-duality). This has nothing whatsoever to do with the $U(1)$ gauge symmetry. – Prahar May 27 '16 at 16:53
  • @LubosMotl :The Einstein-Maxwell-Hilbert lagrangian is invariant under $\mathrm{SU}(2,1)\times \mathrm{U}(2)$, with gauge subgroup U(1), which may be observed, for example, as left action on the coset representative; this is expressed in more simple terms in my previous answers. This action may not be valid in the context of the quantum theory, but it is in the classical theory, which is the subject of the question by Bendy, and it has the effect of turning e.g. magnetic field into electric. Perhaps there's a disagreement in terminology, or I'm still missing something fundamental. – auxsvr May 27 '16 at 21:47
  • @Prahar Please have a look at my previous answer. – auxsvr May 27 '16 at 21:47
  • 1
    aux: no overlap between your comments and reality. No Lagrangian is called "Einstein-Maxwell-Hilbert" in the West, no important Lagrangian has SU(2,1) x U(2) symmetry, no symmetry could change the fact that the gauge field has to transform in the adjoint, there's no difference between the classical and quantum theory when it comes to the transformation rules under the gauge group, no Yang-Mills symmetry allows to mix electric and magnetic. Can we please stop this fruitless exchange? – Luboš Motl May 28 '16 at 04:31

2 Answers2

5

When Maxwell formulated his equations he did so using quaternions, which BTW is a more elegant formalism, and Heaviside formulated them as we normally read them. Our standard vector forms of the Maxwell equations are more convenient for electrical engineering. These equations linearly add the electric and magnetic fields. This linear property is a signature of the abelian nature of the Lie group $U(1)$ for electrodynamics.

Let me argue this in somewhat more modern language with quantum mechanics. Suppose we have a quantum field (or wave) that transforms as $\psi(\vec r) \rightarrow e^{i\theta(\vec r)}\psi(\vec r)$. Now act on this with the differential operator $\hat p = -i\hbar\nabla$ and we find $$ \hat p\psi(\vec r) = -i\hbar\nabla\psi(\vec r) = -i\hbar\left(\nabla\psi(\vec r) + i\psi(\vec r)\nabla\theta\right) $$ We then see this does not transform in a homogeneous fashion, so we change the operator in to a covariant one by $\nabla \rightarrow \nabla - ie\vec A$ where $\vec A$ is the vector potential that subtracts out the $\nabla\theta$. We can now perform quantum mechanical calculations that include the electromagnetic field in a consistent manner.

We now consider the covariant differential one-form ${\bf D} = {\bf d} - ie{\bf A}$ such that ${\bf d} = dx\cdot\nabla$ and ${\bf A} = {\vec A}\cdot dx$. We can now look as the action of ${\bf D}\wedge {\bf D}$ on a unit or constant test function $$ {\bf D}\wedge {\bf D}\odot\mathbb I = {\bf d}\wedge{\bf d} \odot\mathbb I - e^2{\bf A}\wedge{\bf A} \odot\mathbb I - ie{\bf d}\wedge{\bf A} \odot\mathbb I - ie{\bf A}\wedge{\bf d} \odot\mathbb I , $$ where the wedge product of a p-form with itself is zero and elementary manipulations gives $$ {\bf D}\wedge {\bf D}\odot\mathbb I = ie\left({\bf d}\wedge{\bf A}\right) \odot\mathbb I . $$ Breaking out the differential form on the vector potential gives the magnetic field by $B = -\nabla\times A$ that the vector potential one-form wedged with itself is zero is a signature of the abelian or $U(1)$ symmetry of the electromagnetic field. For other gauge fields there is a color index, where there are different sorts of charges, and so ${\bf A}\wedge{\bf A}$ is nonzero and this is a signature of the nonabelian nature of other gauge fields, in particular the weak and strong nuclear forces.

This is done in a three dimensional or nonrelativistic manner, and of course this must be generalized to relativistic QM or the Dirac equation for the quantum dynamics of a fermion. So this is a bit of an elementary introduction. The main upshot though is the reason electromagnetism is abelian or $U(1)$ is there is only one electric charge $e$, with positive and negative values, while other gauge fields have an array of color-charges.

Lawrence B. Crowell
  • 12,735
  • Thanks for this. Are you basically saying that, if the gauge group were nonabelian, there would be terms corresponding to different charges in the Lagrangian? – Bendy May 27 '16 at 15:36
  • 2
    In effect that is the case. If there were several charges there would be different vector potentials, or what in more advanced language are called gauge connections. These would form 2-form products and physically it would means the photon would carry these charges. In QCD this happens and is why the field is confining. The gluons attract each other and instead of forming a dipole field similar to the B field of a magnet they tend for form a sort of flux tube that is self binding or attracting. – Lawrence B. Crowell May 27 '16 at 15:45
  • 1
    Yes, but this question is asking about classical E&M, not quantum (QED). – The_Sympathizer Jun 23 '20 at 01:24
1

A speculative classical reason is that the existence of classical magnetic monopoles requires a $U(1)$ gauge group. It is well-known that generic electromagnetic fields in non-trivial spacetimes only admit an electromagnetic potential $A$ that is defined locally, meaning that you have one expression for $A$ on one patch of spacetime and another in another. On the overlaps between these patches, the expressions must be related to one another by a gauge transformation. In the case of a magnetic monopole, the gauge transformation required is not single valued but rather takes values on $U(1)$. This is explained on the first few pages of professor Tong's notes https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf.

It would be interesting to see other classical configurations of the electromagnetic field where this happens and without monopoles. I feel like there should be some out there (maybe in condensed matter physics books). Let me know if someone finds them!

Ivan Burbano
  • 3,875