Symmetries and Conversation Laws

[ophase]

According to Noether Theorem, Every symmetry corresponds to a conversation Law in the nature.
For example: If rotational symmetry exits, Angular momentum is conserved.

How can I be sure that which symmetry corresponds to which conversation Law?
Can you tell me the conversation Laws under these symmetries:
Charge symmetry,
Parity symmetry,
Time symmetry,
Flavor symmetry,
Helical symmetry,
Scale symmetry.

 

[malawi_glenn]

Time symmetry is conservation of Energy

 

[Vanadium 50]

According to Noether Theorem, Every symmetry corresponds to a conversation Law in the nature.

That's not what it says. It makes statements about specific kinds of symmetries in the Lagrangian.

Before going too far down this path, do you understand Noether's Theorem? Could you prove it if you had to? If the answer is no, probably the best we can do is to say some symmetries have this property, and to leave "some" vague.

 

[Meir Achuz]

I believe Noether's theorem applies to continuous symmetries.
Most of the symmetries you list are discrete, internal symmetries.

 

[blechman]

According to Noether Theorem, Every symmetry corresponds to a conversation Law in the nature.
For example: If rotational symmetry exits, Angular momentum is conserved.

How can I be sure that which symmetry corresponds to which conversation Law?
Can you tell me the conversation Laws under these symmetries:
Charge symmetry,
Parity symmetry,
Time symmetry,
Flavor symmetry,
Helical symmetry,
Scale symmetry.

so as the others have said, you should definitely go through and be sure you understand Noether's theorem. You also seem to apply the word "symmetry" and "conservation law" interchangeably in your question, thus jumping the gun! For example, I don't really know what "charge symmetry" is when read literally, but "charge" is CONSERVED due to the gauge symmetry of E&M!

So let me answer your question by brute force:

Charge conservation is a result of the gauge symmetry (to be technically correct, charge conservation actually follows from the GLOBAL part of this symmetry, for those who are in the know).

Parity is a discrete spacetime symmetries and Noether's theorem does not apply in the naive way, and mentioned above.

Flavor symmetry gives conservation of flavor quantum number. For example, things like "baryon number" and "lepton number".

Helical symmetry is angular momentum, and is a subset of the rotational symmetry already mentioned.

Scale symmetry is not actually a symmetry in most cases! But when it is a symmetry, it generates conserved charges called "Virasoro charges", but this is very advanced stuff and is not relevant for most phenomenology. The keyword to understand these charges is "Conformal Field Theory".

OK, there is a brute-force answer. The reason I know which charge goes with which symmetry is because I use Noether's theorem to CALCULATE it! That answers the other part of you question.

Hope that helps!

 

[ophase]

@malawi_glenn, @Vanadium 50, @clem

I should have said that Noether Theorem concerns continuous transformations on the scalar fields.
But doesn't it mean that for the discrete symmetries Noether theorem leaves us alone??

Before going too far down this path, do you understand Noether's Theorem? Could you prove it if you had to?

After infinitesimal translations to Lagrangian or generalized coordinates, i can show that Noether current or momentum conserved.
Is there any other proof of the theorem i should know?? If there is, it maybe helpful for me.

@blechman

Thank you for detailed answer.
I have to say that interchangeability between "symmetry" and "conservation law" confuses me a lot.
Am i right if i would say that Gauge transformations are invariant because of charge symmetry ??
or
Charge is conserved under Gauge symmetry ??
Don't you think there's a conceptual confussion??

Scale symmetry is not actually a symmetry in most cases! But when it is a symmetry, it generates conserved charges called "Virasoro charges"

Scale symmetry was the most important part of my question. Because i wonder why it is not included in the elementary field theory since it is an appearant symmetry in the universe.

Can we derivate other fields (including supersymmetric and unknown fields too) after some kind of scale transformations??
We know the supersymmetry transformation between boson and fermion fields. Can it be defined as a scale transformation??
Is it what they called supersymmetric conformal field theory??

 

[hamster143]

When proving Noether's theorem, the way you calculate the conserved quantity (constant of the motion) that corresponds to the continuous symmetry is by differentiating the action wrt a variable that parametrizes the symmetry. With discrete symmetries, there's no variable, Noether's theorem can't be applied directly.

Scale symmetry is only a symmetry when your fields are massless. If your fields are massive, you can have a symmetry that preserves the lagrangian by rescaling spacetime, all particle masses, and all dimensional coupling constants.

 

[humanino]

The way I remember it, scale symmetry is badly broken in the real world. Beginning with chiral symmetry breaking breaking in QCD and the special scale at which the coupling constant grows big (even with massless quarks), but with QED as well, chemical bonds set spatial scales, and a simply scaled structured, like say a chair, would collapse under its own weight, if for instance you would scale the planet and and the chair a factor (say) $10^{6}$

 

[malawi_glenn]

Am i right if i would say that Gauge transformations are invariant because of charge symmetry ??
or
Charge is conserved under Gauge symmetry ??
Don't you think there's a conceptual confussion??

No, the Lagrangian is invariant under a Local U(1)-gauge transformation of the fields (your generalized coordinates), and what is conserved, the Noether charge, is the Electric Charge.

 

[ophase]

@malawi_glenn

So I must say;

"Lagrangian is invariant under a Local U(1)-gauge transformations because of charge symmetry"

Do you mean that there's no concept called gauge symmetry?? or they mean another thing when they are talking about gauge symmetry??

Is it wrong to say "Charge is conserved under Gauge symmetry" ??

 

[malawi_glenn]

NO NO NO

You have to say:

The Lagrangian is invariant under a Local U(1)-Gauge transformation, the Noethers Theorem then gives you that Electric Charge is conserved.

Why is it so hard to read what we write?? "charge symmetry" is a sloppy usage of words.

What is symmetric is the Lagrangian under a Local U(1)-gauge transformation. This symmetry will give you charge conservation. Not vice versa or something in between.

 

[genneth]

I don't really want to confuse the issues even more, but can we please be accurate and say that charge conservation is due to the global U(1) symmetry? Apart from the global subgroup, gauge symmetries are not real symmetries; gauge degrees of freedom do not relate different physical states.

 

[malawi_glenn]

I don't really want to confuse the issues even more, but can we please be accurate and say that charge conservation is due to the global U(1) symmetry? Apart from the global subgroup, gauge symmetries are not real symmetries; gauge degrees of freedom do not relate different physical states.

yes, of course, we can even be that accurate :-)

 

[blechman]

malawi_glenn and genneth did a good job explaining the gauge symmetry part. That's a subtle issue about the global vs gauge symmetry - it has lead me to make more than one mistake here, on a QFT exam solution set (yikes!), and worst of all, in my own research! So don't feel too bad if you don't understand it.

Moving on to "the most important part of [ophase's] question":

WARNING: this might get complicated. If it confuses you, I'm sorry. Don't worry too much about it.

Scale symmetry $(x\rightarrow\lambda x)$ is a very tricky subject. CLASSICALLY, this is a symmetry of any theory of massless fields, but the problem is that QUANTUM MECHANICALLY, this symmetry is broken! These kinds of symmetries are called "anomalous symmetries".

The way this materializes in the real world is that if you start with a "scale-invariant" theory that has no masses in it, the problem is that the couplings themselves depend on the energy! Example: if you measure the electromagnetic charge of the electron (e) at rest, you get:

$$\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}=\frac{1}{137}$$

However, if you measure the electromagnetic charge of the electron moving with an energy of 100 GeV, you get:

$$\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}=\frac{1}{128}$$

So we find that the coupling gets LARGER as the energy increases! This breaks the scale invariance and is due to this "quantum anomaly" I mentioned.

So even massless theories typically do not have scale invariance, except for some VERY special examples, but they are rare.

I do not understand your question about SUSY. Supersymmetry is another spacetime symmetry, different from rotations, translations and scale transformations. Maybe you can clarify?

Anyway, I hope this helps and doesn't confuse. But this is why scale symmetry is treated a little differently than ordinary translations/rotations.

Note Added: Hey, while proofreading this post, I got a sudden rush of deja vu! ophase: did we have this conversation before?!

 

[blechman]

The way I remember it, scale symmetry is badly broken in the real world. Beginning with chiral symmetry breaking breaking in QCD and the special scale at which the coupling constant grows big (even with massless quarks), but with QED as well, chemical bonds set spatial scales, and a simply scaled structured, like say a chair, would collapse under its own weight, if for instance you would scale the planet and and the chair a factor (say) $10^{6}$

I just wanted to point out that what you say is quite right. However, in all of these cases, you are talking about MASSIVE field theories (except for your coupling growing big example). In MASSLESS QED, for example, there are no bound states. You can see this by two arguments:

1. Brute force: after a hard-work calculation, you find the bond lengths are inversely proportional to the mass of the electron, so as that mass goes to zero, the bond lengths go to infinity and there are no bound states.

2. By dimensional analysis: a massless theory has no scale, and therefore you cannot write down anything with dimensions of length. Therefore there cannot be a bound state.

So your chemistry argument fails in the case of massless electrons for example (not a "real world" scenario, of course, but I'm a theorist and that's never stopped me before! ). However, even massless QED has the running coupling, as I said above.

Anyway, just to clarify our positions.

 

[humanino]

Anyway, just to clarify our positions.

Indeed we agree, and as a matter of fact, I appreciate the clarification.

 

[ophase]

@hamster143

With discrete symmetries, there's no variable, Noether's theorem can't be applied directly.

If one have a discrete symmetry and Noether theorem can't be applicable how can one find out the conserved quantity?

@genneth

In the classification of symmetries, I never heard of a branch called real symmetries. Can you give us a more detailed definition of Real Symmetries?

@blechman

That's great to find some scale symmetric relations in QED. But I don't understand how did you calculate the constant while electron moving at 100 GeV.

I do not understand your question about SUSY. Supersymmetry is another spacetime symmetry, different from rotations, translations and scale transformations. Maybe you can clarify?

My question about SUSY: Why should it be a spacetime symmetry?? (I know the additional fields like W,D,F to satisfy the algebra and it disturbs me a lot)
Isn't it possible to construct (let's say) a scale symmetry between the fields??

Note Added: Hey, while proofreading this post, I got a sudden rush of deja vu! ophase: did we have this conversation before?!

Do you mean I'm the only one here asking silly questions after our past thread and you'r saying like "i recognize you wherever you go regardless of time" :)

https://www.physicsforums.com/showthread.php?t=205735 ??

I'll have more questions soon
:)

 

[Vanadium 50]

With discrete symmetries, there's no variable, Noether's theorem can't be applied directly.

Do you know how to apply Noether Theorem to discrete symmetries? It would be quite helpful if you can show.

He just told you - it doesn't apply in this case.

 

[ophase]

Ok, i edited it.. I think it's more clear now... Since english is not my native language, these happens. Sorry

 

[blechman]

@blechman

That's great to find some scale symmetric relations in QED. But I don't understand how did you calculate the constant while electron moving at 100 GeV.

Calculate the scattering cross section of electrons fusing to make a Z-boson, which then decays to muons (for example), when the CoM energy of the electrons is at the Z mass (91 GeV, which is close enough to 100). This answer is proportional to the fine structure constant (squared, actually).

Then measure said cross section at LEP. This gives you your value for the coupling.

I'm oversimplifying, but that's the idea.

My question about SUSY: Why should it be a spacetime symmetry?? (I know the additional fields like W,D,F to satisfy the algebra and it disturbs me a lot)
Isn't it possible to construct (let's say) a scale symmetry ??

SUSY is a spacetime symmetry because it has nontrivial relations with the other spacetime symmetry generators such as

$$[Q,\bar{Q}]_+\sim P$$

So it is a spacetime symmetry.

I'm not understanding what's disturbing you. What's "W"? Presumably, "D,F" are the auxiliary fields you introduce in SUSY - they're only introduced to make the MATH easier, you don't really need to introduce them, they're not new "physical" fields.

The scale symmetry is also a spacetime symmetry, but as I said in my last post, it is violated in all but the rarest of quantum field theories. If you try to write a "scale-invariant" theory down, you will find that quantum corrections break the scale symmetry. There's nothing to be done with that! SUSY does not have this problem. Nor does the usual Poincare invariance (translations/rotations).

Do you mean I'm the only one here asking silly questions after our past thread and you'r saying like "i recognize you wherever you go regardless of time" :)

https://www.physicsforums.com/showthread.php?t=205735 ??

I'll have more questions soon
:)

I seriously thought I wrote these words down before, perhaps with some other post that wasn't you. Maybe I'm just on drugs...

Anyway, ask away!

 

[samalkhaiat]

Noether theorem applies to (and ONLY TO) continueous transformations. Such transformations form a Lie-type group. That is, they can be Tylor expanded around the identity; we can consider an infinitesimal part of them

$$T( \omega ) = 1 + \omega , \ \ |\omega | \ll 1$$

There are two types of contineous symmetry transformations:

1) Space-time

these transformations change the Lagrangian by a total divergence:

$$\mathcal{L} \rightarrow \mathcal{L} + \partial_{a} \Lambda^{a} \ \ (1)$$

Examples are: spacetime (super)translations, (super)rotations, (super)scale and (super)conformal transformations. For mathematical details see

www.physicsforums.com/showthread.php?t=172461

2) Internal

under these transformations, the Lagrangian remains invariant;

$$\mathcal{L} \rightarrow \mathcal{L} \ \ (2)$$

Notice the difference between eq(1) & eq(2).

Examples are flavour and gauge symmetry transformations.

Discrete transformations (like parity , time reversal and charge conjugation) have no infinitesimal versions. Their symmetry considerations is based on the fact that their corresponding operators commute with the Hamiltonian.

regards

sam

 

[genneth]

Discrete symmetries do not have conserved quantities associated with them.

The term "gauge symmetry" is a historical accident, and misleading, as it does not refer to a symmetry.

 

[samalkhaiat]

Discrete symmetries do not have conserved quantities associated with them.

At least in em-interactions, the eigenvalues of the parity operator are very much conserved. An observable is a constant of motion (conserved), if and only if, its operator commutes with the generator of time-translation group (Hamiltonian).

The term "gauge symmetry" is a historical accident, and misleading, as it does not refer to a symmetry.

Symmetry is a mathematical operation that leaves the action integral and observables of your theory unchanged. As such, local and global "gauge" transformations are very much in the domain of definition of symmetry.

regards

sam

 

[hamster143]

The way this materializes in the real world is that if you start with a "scale-invariant" theory that has no masses in it, the problem is that the couplings themselves depend on the energy! Example: if you measure the electromagnetic charge of the electron (e) at rest, you get:

$$\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}=\frac{1}{137}$$

However, if you measure the electromagnetic charge of the electron moving with an energy of 100 GeV, you get:

$$\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}=\frac{1}{128}$$

Basically, we're having trouble defining a spherical horse in a vacuum which is massless QED. A complete massless theory obviously would not have any dimensional parameters or dependency on energy, but QED (massless or not) is not complete and we can't study it in isolation.

But you could say that massless QED has a hidden dimensional parameter (call it "Landau pole") and the renormalized coupling $$\alpha$$ is a function of this dimensional parameter. And then if you could look at the symmetry of the Lagrangian that corresponds to rescaling spacetime and Landau pole at the same time, Noether's theorem would apply and hopefully you'd get something interesting in return.

 

[genneth]

At least in em-interactions, the eigenvalues of the parity operator are very much conserved. An observable is a constant of motion (conserved), if and only if, its operator commutes with the generator of time-translation group (Hamiltonian).

Symmetry is a mathematical operation that leaves the action integral and observables of your theory unchanged. As such, local and global "gauge" transformations are very much in the domain of definition of symmetry.

regards

sam

In a quantum theory, yes, every operator that commutes with the Hamiltonian presents a quantity that does not change with time. This is more general and not quite as structured as the symmetry that one gets from the divergence-free current derived via Noether's Theorem. I'm happy to accept it as a perfectly well-defined conserved quantity. (I guess also classically, functions which have zero Poisson bracket counts...)

"Gauge symmetry" is definitely not a symmetry however. Gauge transformations do not change the state of the system, but only enact a relabelling. The basic tension is that certain objects are impossible to coordinatise in a smooth, non-singular way. This is especially clear in the quantum theory, where the distinction between operators and states is clearly defined. When formally constructing the theory of e.g. Yang-Mills, we have to be very careful to define the Hilbert space as the naive one modulo gauge transforms, otherwise it does not have a proper inner product. Failure to treat this correctly will result in nonsense such as negative probabilities.

An open question is whether there is some privileged role played by gauge theories in the wider space of theories which over-label states. Notice that when we say particles are "boson" or "fermion" we are really saying that we have over-coordinatised the system; interchanging two particles does not produce a different state with the same physical properties, but rather produces the exact same state. But it seems that we can only treat over-labelling via calling them bosons, fermions or gauge bosons; not even anyons are analytically treatable. I wonder if there is some deep reason for this?

 

[Bobhawke]

Im not understanding why gauge symmetries arent "real symmetries". Like Samn said, a symmetry to me means that you apply a certain transformation to the action and it is invariant, which is true of gauge symmetries. Are you guys using a more restricted definition of symmetry which excludes this? Could someone explain to me why gauge symmetries are not "real symmetries" in an understandable way, or point me towards something that does explain it.

 

[humanino]

Because gauge symmetries are just a convenient language to deal with constrained system. For instance, you want to describe the photon as a massless vector, so you must enforce the constraint that the longitudinal polarization vanishes. U(1) gauge invariance does that elegantly for you.

 

[jensa]

Im not understanding why gauge symmetries arent "real symmetries". Like Samn said, a symmetry to me means that you apply a certain transformation to the action and it is invariant, which is true of gauge symmetries. Are you guys using a more restricted definition of symmetry which excludes this? Could someone explain to me why gauge symmetries are not "real symmetries" in an understandable way, or point me towards something that does explain it.

Hi,
I am probably not the right person to try to answer this since I don't feel like I have a full understanding of this subject myself, but I will post a reply anyway in hopes to gain some more understanding from possible replies to my post.

I think what you call "real symmetries" is a matter of definition but there definitely is a difference between two types of "symmetry transformations". I think a good resource discussing this is Xiao-Gang Wen's book Quantum Field Theory of Many-Body Systems. As an illustration of the difference he begins with the example of a particle in an extended 1D periodic potential, say with period a, vs a particle constrained to live on a circle with circumference a. Clearly both particles are subject to a Hamiltonian which is invariant under spatial translations of multiples of a. The difference is that in the former case a translation takes you to a different point x+a while the in the latter case a translation takes you back to the same point you started with.

In other words if you act with the translation operator (which is a symmetry of the Hamiltonian) on a state $|x\rangle$ you end up with a state $|x+a\rangle$. In the first case the new state $|x+a\rangle$ really corresponds to a different state than $|x\rangle$ while in the latter case $|x+a\rangle$ is only a different "label" for the original state $|x\rangle$ so you really have not performed any operation at all.

In terms of global gauge invariance most people say that the two states $|\psi\rangle$ and $e^{i\varphi}|\psi\rangle$ are just different labels of the same "physical state". The space of physical states is said to be ray-representation of the Hilbert space. (for reference see Weinbergs Volume I of The quantum theory of fields)

How local gauge invariance fits into all of this I am not quite sure and would be very interested if someone has comments on this...

 

[humanino]

How local gauge invariance fits into all of this I am not quite sure and would be very interested if someone has comments on this...

As you said

constrained

 

[jensa]

As you said

Hi humanino, I am glad you answered. I have read some of your posts and see that you know your stuff. I hope I am not hijacking this thread by asking you this but I have been meaning to ask this question on the forum but did not feel like I knew quite how to phrase the question.

Local gauge invariance puzzles me a bit. Maybe I am looking for more of an explanation than exists but basically I am wondering where the "constraint" comes from. The act of "gauging" a symmetry seems to me to be somewhat random - like let's try to promote this symmetry to a local one and see if the resulting gauge theory works. Is this an accurate picture of how one thinks of local gauge theories or is there some more rigorous way to motivate gauging symmetries? And how can we think of the imposed constraint?

Is it possible to understand the origin of local gauge theories from the notion of ray representations directly. I understand gauge fields can be understood through connections on fiber bundles, but I am not sure how this works precisely. Do you have a good resource for this stuff?

Sorry if this post seems like rambling...I should probably think this over and start a new thread once I am able to formulate a better question.

Edit: I seem to remember reading somewhere that promoting a global gauge symmetry, let's say U(1) symmetry, to a local one is related to the problem of different observers agreeing on the same phase convention... Is this the right direction I should look into? Any good references?

 

[humanino]

Maybe more qualified people will contribute, but in the meantime, since you ask

let's try to promote this symmetry to a local one and see if the resulting gauge theory works. Is this an accurate picture of how one thinks of local gauge theories or is there some more rigorous way to motivate gauging symmetries?

It's not the way I think about it. Let's get back to U(1) again. You have wave functions, they are (sets of) complex numbers. The phase difference matters between them, but not the absolute phase. You can multiply them by a unitary complex number. This is a global gauge transformation. It amounts to a choice for the "origin" of the phase. Then you easily realize that you could choose this origin at every point in space-time. But by doing so, you monkey around with the derivative, which "connects" two neighboring space-time points. So you need in order to fix the derivative to introduce an additional field, and luck or magic happens, this is the photon (potential vector) field. It seems to me from this point of view rather well motivated that the absolute phase should be arbitrary at every point in space-time. This construction unravel something deeper, which can be geometrically described in terms of what mathematician call connections and fiber bundles. It produces interactions from the constraints that derivative should transform accordingly to the arbitrariness we want in our physics.

Do you have a good resource for this stuff?

There are many. Dirac Yeshiva's lecture in 1964, "The geometrical setting of gauge theories of the Yang-Mills type" by Daniel and Viallet (RMP 1980), or "the geometry of physics" by Frankel (Cambridge 2004) are some I like.

 

[jensa]

Thank you very much for your reply.

Then you easily realize that you could choose this origin at every point in space-time.

I think this is the step I have difficulty understanding... It seems to me that requiring that you can choose this origin differently at different space-time points is to impose an additional constraint, on top of having a global choice of phase-convention (i.e. origin of phase). The justification for imposing this stronger constraint seems to be that, if we do so, we end up with gauge fields which correspond to the EM field.

But by doing so, you monkey around with the derivative, which "connects" two neighboring space-time points. So you need in order to fix the derivative to introduce an additional field, and luck or magic happens, this is the photon (potential vector) field. It seems to me from this point of view rather well motivated that the absolute phase should be arbitrary at every point in space-time. This construction unravel something deeper, which can be geometrically described in terms of what mathematician call connections and fiber bundles. It produces interactions from the constraints that derivative should transform accordingly to the arbitrariness we want in our physics.

I think I understand these steps/arguments, given that we have the freedom to choose a different phase-convention at each space-time point. Question is just if there is some more fundamental motivation for imposing this freedom/arbitrariness other than "because it gives us the right physics".

There are many. Dirac Yeshiva's lecture in 1964, "The geometrical setting of gauge theories of the Yang-Mills type" by Daniel and Viallet (RMP 1980), or "the geometry of physics" by Frankel (Cambridge 2004) are some I like.

Thanks again! I will definitely try to get a hold of these books and check them out. Titles sound promising.

 

[humanino]

I think this is the step I have difficulty understanding... It seems to me that requiring that you can choose this origin differently at different space-time points is to impose an additional constraint, on top of having a global choice of phase-convention (i.e. origin of phase). The justification for imposing this stronger constraint seems to be that, if we do so, we end up with gauge fields which correspond to the EM field.

Well, I thought the difficult step was to understand how gauge invariance relates to the arbitrariness of the phase. Once you have imposed the global gauge transformation, will you accept that your choice in Washington today should automatically be valid on Saturn yesterday !?

 

[jensa]

Well, I thought the difficult step was to understand how gauge invariance relates to the arbitrariness of the phase. Once you have imposed the global gauge transformation, will you accept that your choice in Washington today should automatically be valid on Saturn yesterday !?

Thanks again humanino. I really appreciate you taking time to explain this.
So it is a natural consequence of relativity? That different observers cannot agree on a common choice for the origin of phase and therefore it must be arbitrary at every space-time point...or something like that? That sounds like reasonable justification.

Just a note to defend my stupidity: I deal only with non-relativistic QFT (in condensed matter) so such ideas usually pass me by.

 

[genneth]

jensa: don't feel stupid --- these are very deep questions you're trying to ask. As the end of Wen's book points out, the conventional picture of gauge theories are as systems where the configuration is defined by some connection on a fibre bundle. As for why: the historical path is Weyl's attempt to take the idea of general relativity and apply it further. In GR, we say that we cannot compare directions at different spacetime points without something to tell us how to; we call the object that informs us how neighbouring points' tangent spaces relate to each other the connection. Weyl thought that maybe the size of objects should also be local --- so there would need to be something that compares sizes at neighbouring points. He called this a gauge theory, because it's a theory that changes the size-measuring apparatus, i.e. a literal gauge. This produced some very neat maths, and actually causes something very much like EM to drop out. However, it also predicts things which are entirely contradictory to experimental evidence (the original paper was published with a footnote by Einstein explaining why it could not be correct). With the advent of quantum mechanics, the idea to replace size with phase was not too much of a stretch, with the advantage that it actually produces the right predictions.

This view that gauge fields are the result of needing to compare quantities at different spacetime points is fairly accepted, and used in both high-energy and condensed matter. However, it certainly raises the question of whether the phase is then supposed to be a real physical thing --- there are very good arguments from quantum information theory that the wavefunction (phase and amplitude) should only be thought of as book-keeping by some observer, and not actually some physical object. Nevertheless, even ignoring meta-physical pondering, it's certainly somewhat mysterious why phase should be incomparable, but size is.

Wen's angle is that the gauge theories we've looked at are only the tip of the iceberg of quantum systems with many-to-one labelling of states. For instance, he shows that you can get U(1) (and also non-abelian) gauge theories with massless fermions as the low-energy theory of some bosonic system. In this case, these things are all just book-keeping, and we choose to work with them because we know how too. In fact, the only "gauge structures" (Wen's term for theories with a many-to-one labelling) we can deal with are: identical particles of bosons or fermions (but not anyons), and certain gauge field theories (the word gauge now used in the traditional sense). So far, we can only solve problems that can be decomposed into combinations of these. As an example of a system we don't yet know how to perform the decomposition: non-linear sigma models.

Is Wen right? I don't know. But it's certainly an interesting point of view which deserves more wide-spread knowledge, especially outside of condensed matter.