
The conceptual origins of Maxwell's equations and gauge theory (2014) [pdf] - espeed
http://www.physics.umd.edu/grt/taj/675e/OriginsofMaxwellandGauge.pdf
======
zaph0d_
I once took an advanced seminar course on the mathematical foundations of
electrodynamics in parallel to my theoretical electrodynamics course during my
third bachelor semester. I did not have any clue about differential geometry
and did not understand the advanced formalism the lecturer introduced in the
seminar. But I was quite shocked how easily Maxwells equations can be derived
and how compact the formula was. The article suggests that Gauge theory and
fiber bundels are subjects, where math and theoretical physics seem to help
each other, which is absolutely facinating!

~~~
espeed
Note the article is by Nobel Laureate C.N. Yang [1] who also worked with James
Simons and co-authored what has become known as the "Wu-Yang Dictionary" [2].

[1] Yang-Mills Theory:
[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory](https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory)

[2] "Wu-Yang Dictionary"
[http://www.indiana.edu/~jpac/QCDRef/1970s/Concept%20of%20non...](http://www.indiana.edu/~jpac/QCDRef/1970s/Concept%20of%20nonintegrable%20phase%20factor%20and%20global%20formulation%20of%20gauge%20fields%20-%20Wu,%20Yang%20-%201975.pdf)

 _...The mathematics of these results is in fact well known to the
mathematicians in fiber bundle theory. An identification table of
terminologies is given in Sec. V. We should emphasize that our interest in
this paper does not lie in the beautiful, deep, and general mathematical
development in fiber bundle theory. Rather we are concerned with the necessary
concepts to describe the physics of gauge theories. It is remarkable that
these concepts have already been intensively studied as mathematical
constructs._

"Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying
Rosetta Stone for Geometry in Application"
[https://www.youtube.com/watch?v=h5gnATQMtPg](https://www.youtube.com/watch?v=h5gnATQMtPg)

~~~
zaph0d_
Thank you for the interessting material! I have heard a lecture on the
geometric and topological applications to solid-state physics. This is one of
the things that excites me a lot that whole areas of physics can be
"geometrized".

~~~
Retra
Part of me believes that geometry is really just a subset of human thinking
that comes very naturally/quickly to us, and thus we are more successful
studying physics through geometry than through less 'algebraic' frameworks. So
we made the most advances there just because we are biased toward doing so.

But then I think that we may be so strongly biased toward doing so because
there's something fundamentally easy about evolving a brain that comprehends
geometry. That information with geometric representations are fundamentally
easier to evolve good mental models for than other kinds of information.

------
21
So I've read that it's enough to know that "there is a U(1) gauge symmetry in
the Lagrangian" to derive the Maxwell equations from. Basically U(1) =>
Maxwell equations.

Is that true? Are there any other assumptions required?

~~~
Y_Y
It's a great question, but slightly subtle. If you already know the Lagrangian
(-1/4 F^{\mu\nu} F_{\mu\nu}) all you need is Noether's theorem. If you want to
come up with the Lagrangian you can do so by trying to find an expression with
local gauge symmetry (the A field is invariant under addition of the four-grad
of a scalar field). I'm afraid I don't have a good reference off the top of my
head, but it's not difficult.

~~~
danbruc
Is that why those symmetry groups are so often mentioned, because they single
out unique theories? As far as I know, but I am not a physicist, gauge
symmetries are more of a defect than a feature of field theories. Because we
lack theories without redundancies we just declare all states in the
mathematical model that are related by a gauge transformation to be equivalent
to each other and to correspond to just one physical state. In consequence I
always wondered why one would then stress those symmetries but it would of
course make sense if they were some kind of unique labels.

~~~
Y_Y
Yes! The symmetries dictate the physics, gauge and otherwise. Unfortunately
there isn't a nice way to see that without a ton of field theory background.

~~~
danbruc
Which seems quite strange to me. I assume we would be really happy to find
theories equivalent to existing field theories but without the gauge
redundancies so that we have a one-to-one mapping between physical states and
mathematical states. Why would a part we want to get rid of determine the part
we are really interested in?

~~~
Retra
The 'mathematical states' here are descriptions of nature. A symmetry in the
description doesn't correspond to identified physical states, it corresponds
to a constraint on the description. For instance, if you want to prove that
something is unique, you have to show how all the different ways of describing
it lead to the same conclusions.

~~~
danbruc
_A symmetry in the description doesn 't correspond to identified physical
states [...]_

As I said before, I am not a physicist, but I am pretty sure that you are
wrong here. Unless you are a physicist and you are sure about that. My
understanding is that gauge symmetries are a misnomer and should better be
called gauge redundancies because they essentially establish equivalence
classes of mathematical states corresponding to identical physical states.
This is not the same kind of symmetry as for example the Poincaré symmetry of
special relativity.

Let me illustrate my understanding with a silly example that would almost
certainly not actually work out mathematically but it should get the point
accross. Say all you have in your mathematical toolbox are complex numbers and
you come up with a theory of thermodynamics where temperatures are described
by complex numbers. Everything works out nicely besides that there is no
discernable physical difference between all the temperatures differing only by
their imaginary parts. So you declare that there is a gauge symmetry in your
theory of thermodynamics and all temperatures only differing by their
imaginary parts are actually the same physical temperature. This theory has
translations along the imaginary axis as a gauge symmetry.

~~~
tobmlt
Ah, you're looking for charge I think.

[https://en.wikipedia.org/wiki/Charge_conservation](https://en.wikipedia.org/wiki/Charge_conservation)

The whole thing spins my head around differently every time I try to learn
more but I hope I am helping by pointing this out.

"The symmetry that is associated with charge conservation is the global gauge
invariance of the electromagnetic field"

~~~
danbruc
No, I really meant what I said, gauge symmetries are non-physical redundancies
in gauge theories. Gauge fixing is the act of choosing one of the equivalent
alternatives for carrying out calculations, gauge transformations allow
switching between those alternatives. The first paragraph of the Wikipedia
article on gauge fixing [1] actually says pretty much exactly this,
unfortunately I just looked at it seconds ago.

 _In the physics of gauge theories, gauge fixing (also called choosing a
gauge) denotes a mathematical procedure for coping with redundant degrees of
freedom in field variables. By definition, a gauge theory represents each
physically distinct configuration of the system as an equivalence class of
detailed local field configurations. Any two detailed configurations in the
same equivalence class are related by a gauge transformation, equivalent to a
shear along unphysical axes in configuration space. Most of the quantitative
physical predictions of a gauge theory can only be obtained under a coherent
prescription for suppressing or ignoring these unphysical degrees of freedom._

And because you mentioned the Aharonov–Bohm effect in your other comment,
further down at the end of the gauge freedom section you can find a statement
that the Aharonov-Bohm effect does not enable experimentally distinguishing
different gauges either.

 _Not until the advent of quantum field theory could it be said that the
potentials themselves are part of the physical configuration of a system. The
earliest consequence to be accurately predicted and experimentally verified
was the Aharonov–Bohm effect, which has no classical counterpart.
Nevertheless, gauge freedom is still true in these theories. For example, the
Aharonov–Bohm effect depends on a line integral of A around a closed loop, and
this integral is not changed by A → A + ∇ψ._

I also think you are wrong when you state in your other comment that charge
conservation is not exact. I am pretty sure it is exact and you are mixing up
CPT-symmetry and the violations of its components with the gauge symmetry of
QED.

[1]
[https://en.wikipedia.org/wiki/Gauge_fixing](https://en.wikipedia.org/wiki/Gauge_fixing)

~~~
tobmlt
Good point about charge conjugation symmetry vs "charge conservation"\- I was
totally mixing up one for the other! Thanks (a lot) for driving that home. For
anyone who might follow this - C in CPT - is charge conjugation - the swap of
a particle for its anti-particle. Certainly not the same thing as charge
conservation itself. Thanks for that correction. I tossed around a lot of
stuff I shouldn't have.

Okay, so I think I get you now. I was basically working from the point of view
that "gauge matters" because the current 4 vector is conserved only if the E&M
Lagrangian is gauge invariant. Hence charge conservation is inextricably
linked with gauge symmetry - it "matters" to the derivation of the physical
dynamics.

But, yes, to your mathematical description of gauge fixing above... And to not
being able to experimentally distinguish gauges. Phase differences, after all,
are not gauge differences. Whatever the worth of the Aharonov–Bohm effect, it
is not going to matter here. Good point. I get excited about this stuff.

To your question: "Why would a part we want to get rid of determine the part
we are really interested in"

I'd answer: Why do we "want to get rid of" it?

We cannot detect which gauge we are in, and we can twice confirm that it
doesn't matter which one we are in precisely because current four vector is
conserved. The redundant descriptions of the Lagrangian are the way to ensure
physical descriptions of reality.

I think this is a matter of what we like or don't like, not physics or math.
But hey, sometimes those are clues!

Gauge symmetries are non-physical redundancies... which are essential to our
mathematical description of real physics. Maybe they are a clue that there is
a better, undiscovered, way to do business?

Maybe.

~~~
danbruc
_I 'd answer: Why do we "want to get rid of" it?_

I'd counter with why would you want additional degrees of freedom in your
theory that don't correspond to something real? To pick up my silly example,
why would you use complex temperatures to do the math if the physical
temperature is real and only has one and not two degrees of freedom? I can
only think of two sensible scenarios.

The first one is that you have a theory with real temperatures but you can
make working with that theory easier if you switch to complex temperatures for
the calculations in witch case the redundancies are purely a mathematical
tool. The second one is that you suspect that temperatures are actually
complex and you just haven't figure out how to detect the imaginary component
in experiments.

Neither option seems to be true for gauge theories - as far as I know there
are no alternative equivalent theories that don't have gauge symmetries and
are just harder to work with mathematically and I also think nobody or at
least not many believe that gauge redundancies correspond to yet undiscovered
physical things.

~~~
tobmlt
Okay cool.

I take your point. Maybe you are right, but I'd argue that the first option
actually applies to gauge theory in the following way:

First, I have to assert that Lagrangian formulations of standard physics are
generally simpler than say, Newtonian form. This is because the equations of
motion all come from one principle - least action.

Second, I assert that when working with a Lagrangian, it is impossible to
derive all the needed physics without introducing gauge freedom somewhere. For
a simple example (that is within range of my brain+sources (see footnote1))
Try it with the magnetic field, B.

There is a constraint on B that it must be everywhere divergence free. The
easiest way to ensure this is to write the magnetic field as the curl of
something. This introduced redundancy in the description but ensures we do not
have to worry about the fact that B is constrained to be divergence free (-
now there is no alternative!) Furthermore, there is no way to derive Lorentz's
force law from a Lagrangian without adding a gauge freedom term. -end example

This example is from Susskind's "The Thoeretical Minimum," page 211 (for the
case of the Lorenz force law) This kind of thing generalizes well to
differential forms and comes in handy deriving, e.g. Yang Mills, or making
sure some particular theory is "manifestly co-variant" \- that sort of thing.
It makes the thing easier, not harder to properly formulate.

One could build the theory without using a Lagrangian (or without anything
"math-mechanically equivalent" like a Hamiltonian) but then you loose the
simplicity of the dynamics falling out of principle of least action. This use
for least action is universal across everything we currently know about
fundamental physical theories.

In sum, I assert it makes sense to go with the first "sensible scenario."
...And that it leads directly to gauge redundant theories just like we have.
But of course this assertion of what is easier is subjective (see footnote2).

To get even more subjective for a minute, there are these amazing
coincidences, such as the fact that GR looks like E&M, and even more like Yang
Mills when they are viewed in this "action oriented" way. -A way that
basically requires the use of gauge freedom to get to the correct dynamics.
Why turn away from that? Well you would if you found something else simpler
and more beautiful. To me, it looks like with gauge theory we have something
too amazing, and too pretty, to be a coincidence or to ignore. But hey, if you
have something up your sleeve to beat it, don't leave us all out!

Anyway, if you see it differently, all well and good. If you think the above
stuff is ...just that... then by all means, teach me better!

This has been fun.

Cheers.

(footnote1) Further investigation reveals that there appear to be theorems and
classification results proved by Cartan, Weyl, and others implying that second
order quasi-linear field equations for the metric tensor possessing symmetries
and conservation laws of the Einstein equations necessarily arise from a
variational principle. This claim taken from: Gauge invariance, charge
conservation,and variational principles, by Manno, Pohjanpelto, and Vitolo -I
wonder what else is out there in this regard? -I am betting there is more but
I really do not know.

(footnote2) Okay, to try and get past the subjectivity, how about comparing
two non-equivalent theories to see how "different they are"? Could there be
some use for gauge redundancy here? Skimming around when I could during a
meeting at work I find hints of this idea in "General covariance from the
perspective of Noether's theorems" from Brown and Brading. [http://philsci-
archive.pitt.edu/821/1/TorrettiB&B.pdf](http://philsci-
archive.pitt.edu/821/1/TorrettiB&B.pdf) page 8, footnote. But again, I surely
don't know!

~~~
danbruc
I'll have to think about that for a day or two.

