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!
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].
...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.
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".
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.
where ◻· is the 4-divergence. Again, the equation is manifestly covariant and very elegant. There are reasons to believe that the electromagnetic potential is in a sense more fundamental than the electromagnetic field:
Nice to see this article available on the Internet: I read a PDF that was passed around by email at the time it was published (Frank Yang is a relative of one of my neighbors and I once tried to chat with him about Maxwell over dinner...).
It was pretty amazing and I was so such awe combined with fear of saying something stupid that I don't remember everything. He mostly talked about non-Physics subjects. He did talk about Fermi and working for Oppenheimer. I asked about Teller...do not now remember what he said to that. I had attempted to understand something about Yang-Mills Gauge Theory in preparation for dinner, but completely failed, so instead I figured as a Scot and card-carrying EE I'd ask him what he thought about the apparent quantum jump in progress made by Maxwell -- how was Maxwell able to come up with such modern looking physics in the age of steam, for example. At the time I did not know that the history of Maxwell was one of his subjects of interest. He talked about some of the themes that you can see in the article above (which was written 10 years later). He also mentioned, in a joking way, his prediction years earlier that "In the next ten years, the most important discovery in high-energy physics is that `the party's over'.".
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?
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.
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.
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.
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?
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.
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.
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.
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?
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.
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 page 8, footnote.
But again, I surely don't know!
These "non-physical redundancy" you are referring to can't be picked out totally at random. They do have some interdepencencies.
My understanding is that at each point in space you can pick a DIFFERENT gauge transformation, but there are some constraints. And if you do a sort of integral through a path of these points, the sum of that thing represents EXACTLY the electo-magnetic field along that path.
A very famous physicists made an economic analogy, using currency exchange rates between countries as gauge fixings:
https://arxiv.org/abs/1410.6753
That is also my understanding, you can of course not make absolutely arbitrary choices or you could trivially turn two distinct physical states into each other but as far as I know you still have huge freedoms.
I read the paper some time ago but I don't remember it as especially enlightening. Maybe I will take the opportunity to read it again over the weekend, maybe I can get something new out of it now that I missed the first time.
Okay I can't resist - this stuff is just too cool. Here is my take on some of this stuff; assume grains of salt throughout as I am an engineer of sorts.
(And sorry if this is old hat - and that the direction I go in strays pretty far, at times, from the original question about gauge theory actually having to do with physical things)
It's funny that charge conservation, noted below as a consequence of gauge symmetry, is not exact. See CP symmetry violation and "all that". Furthermore there are various notions of internal symmetries... in engineer talk, perhaps gauge symmetry is a symmetry of parameterization - in math talk the base space of a fibre bundle "parameterizes" (in a coordinate free way) a larger space. --Gauge symmetry is a freedom in how exactly this is done (shout out to your comment about equivalence classes - totally agree).
(For example the space time manifold showing how to put a vector space at every location in the manifold via tangent space constructions, or similar)
I get lost sometimes..
Anyway, looking at part of the Wikipedia charge conservation article we read that:
"...an electromagnetic field is unchanged when the scalar and vector potential are shifted by the gradient of an arbitrary scalar field."
In quantum mechanics the scalar field is equivalent to a phase shift in the wavefunction of the charged particle..."
This shift in the Lagrangian (to call it a different way) is gauge symmetry (U(1)) again. Interestingly, the phase shift can occur even if the space is "E&M flat" (no detectable field) and this shift IS observable:
-A purely topological effect due to the fact that loops (around the solenoid in the experiment) in the space cannot be shrunk to a point! (not without leaving "the space" that is)
Any phase shift is in fact a manifestation of Weyl's "path dependent length" which he'd speculated about to Einstein. (Einstein shot it down, but it was reborn as phase) But wait, path dependent lengths (generalized path length shift == phase shift) mean we have some, admittedly abstract, notion of curvature!
This is the connection (ahem) between E&M theories and General Relativity.
(and even closer connection with E&M-like, but non-abelian (Yang Mills) )
The potential, A, in electrodynamics plays a role analogous to that of the connection coefficient (Christoffel) in differential geometry. The tensor field strength in electrodynamics is analogous to curvature! (luckily for those who like their minds un-exploded E&M "happens" in flat space.)
Once you get to Yang Mills, the commutator of the potential has to be added to the derivative of the Faraday tensor to get the total field strength. This is just the same as in General Relativity, where the curvature is the derivative of the connection plus its commutator.
I guess what I am saying is, depending on how you look at it, the field theories are sort of shot-through with gauge symmetry, to both quantifiable effect and mathematical purpose, and it is neat.
I don't know gauge theory and haven't read the link, but the Lagrangian also needs to be relativistically invariant. This is a global symmetry instead of a gauge symmetry. I don't know if there's such a thing as, say, Euclidean gauge theory.
