# Physical significance of gauge invariance

1. Dec 26, 2012

### ShayanJ

I've read that gauge invariance leads to a fundamental phenomenon.What is that?
Thanks

2. Dec 27, 2012

### algebrat

First guess: gauge invariance I associate with the idea of symmetries and conservation laws. So if some aspect of our universe cannot tell the difference for some transformation, or symmetry, we can call that a change in gauge.

For instance, we can't tell if the universe shifts back one second, the laws of physics are invariant under a change in time gauge. Mathematically, we expect a conserved quantity, which has already been named actually, energy.

So also for spatial translation, another change in gauge. The associated conserved quantity is momentum.

It's sort of like having a differential equation on the plane, for instance say it was rotationally invariant. Then the solution sets might be circles on the plane. The conserved quantity is the radius. You can't change the radius with out breaking the differential equation enforced.

So the differential equation is like the (local) law of physics, and the radius is like the conserved quantity, and you are constrained to an equipotential.

I think we call it gauge invariance when we interact with modern experimental physicists (random guess). So the fundamental phenomenon depends on which gauge invariance we are looking at. But I would guess by fundamental phenomenon, they are referring to the associated conserved quantity, whether it be spin, etc.

The other issue you may run into, countless times I've read in textbooks the phrase something like "we'll discuss that later, or explain it later in the text". Often, it seems they've forgotten to bring it back up, or failed to succeed at explaining it in a form anyone other than a powerful graiate student psyche could penetrate. I'd chalk it up to the author is only human.

:).

3. Dec 27, 2012

### ShayanJ

I know about Noether's theorem and the relationship between symmetries and conserved quantities.
By gauge invariance I mean transformations like $\vec{A} \rightarrow \vec{A}+\vec{\nabla}\psi$ and $\phi \rightarrow \phi + \frac{\partial \psi}{\partial t}$ which leave the fields associated to the potentials,invariant.
I ask this because this kind of symmetry is treated differently than the others and also by Noether's theorem,I know there is a conserved quantity but I don't know what's that.
Another thing is that sometimes gauge invariance is said to have some physical significant which is a little mysterious to me.
Also force carriers are sometimes called gauge bosons which again brings up the question of this topic.

Last edited: Dec 27, 2012
4. Dec 27, 2012

### andrien

5. Dec 27, 2012

### VantagePoint72

algebrat, that answer is totally wrong. Gauge symmetries are fundamentally different from global symmetries, not just a name we use for symmetries when associated with particle physics. Spatial translation is definitely not a gauge transformation.

Shyan, generally speaking gauge symmetries don't have a conserved current associated with them because Noether's theorem applies to global symmetries. A Gauge symmetry is sort of the opposite of physically significant: it means part of our description is not physically significant and hence can be changed (even locally) while describing the same physical situation. That said, a gauge symmetry will generally have a global symmetry as a special case of it, in which you are actually looking at a different physical configuration. For instance, the electromagnetic field has U(1) gauge symmetry. This means that, in general, you can do a different U(1) transformation at every spacetime point and still get the same physics. The EM gauge potential transforms under U(1) in the way you state above, and the way you become familiar in introductory EM classes when you first encounter gauge freedom. If you do the same U(1) transformation at every point, you're actually doing a global transformation—which, by Noether's theorem, corresponds to conservation of charge.

The physical significance of gauge invariance is that two configurations that are related by a gauge transformations are the exact same configuration—they don't just behave the same way as with global symmetries, they are physically the same.

6. Dec 27, 2012

### alemsalem

according to Nima Arkani-Hamed (who calls it a gauge redundancy) its not physically significant its only in your description (describing the same physical situation many ways).
he also says that you can start with a Lagrangian that is not gauge invariant and make it gauge invariant by introducing redundancies.

Sorry I don't remember which lecture this is.

7. Dec 28, 2012

### ShayanJ

So you say two configurations A and B are the same if they are related by a gauge transformation which is a symmetry of both of them and this has,as an special case,a global symmetry.
But the statement that A and B treat the same way with respect to some aspects has as an special case the statement that A and B are the same.
Now if we apply your interpretation of gauge and global symmetries,we reach to the conclusion that gauge symmetry is a special case of global symmetry and also vice versa!!!

8. Dec 28, 2012

### algebrat

I'll let someone else cover this, but before they do, I'd like to point out that at least I'm confused on a few points. My uninformed guess, others will have a hard time responding to your last post.

On the following points:

How is a gauge translation relating A and B, AND symmetries of each.

"this has, as a special case, a global symmetry." Not sure what's going on there, but again, I haven't seem the details.

"A and B are same in some aspects, specializes to them being the sale globally".

Let me start to try to rephrase what you are saying, correct me if I'm off. Say A and B are two parametrizations of the same system. There is a transformation from one to the other.

Hmm, I'm struggling here, but at any rate, I think you are trying to say you don't see much of a difference between local and global symmetries, but I'm really not qualified to catch what's going on.

I guess I just wanted to claim that you may want to make your question more clear.

9. Dec 28, 2012

### element4

Dear Shyan,

what you write is not really correct. There are extremely important differences between "gauge symmetries" and true "global symmetries", the first kind is NOT a real symmetry but a redundancy as mentioned above.

Say a Hamiltonian (which correspond to energy) has a certain symmetry, this means that two physically distinct configurations A and B related by this symmetry have the same energy! If you have translational symmetry, you can do an experiment at point A or at point B but the physics is the same although A and B correspond to physically different places in space.

But gauge "symmetries" correspond to giving the same physical state, many redundant labels. Like naming a person "Peter" and "Ed", gauge transformations map "Peter" to "Ed" and vice-versa. But its still the same person! Therefore the true physical states in gauge theories correspond to so-called gauge-orbits, meaning that you identify all states related by a gauge transformation as the SAME physical configuration.

Gauge "symmetries" are important because they kill unphysical degrees of freedom, for example they are responsible for removing one of the polarizations of photons.

One can formulate the difference more technically. For example classically, gauge symmetries correspond to so-called "first-class constrants" (see http://en.wikipedia.org/wiki/First_class_constraint).

10. Dec 28, 2012

### atyy

I've seen this statement, eg. in David Tong's QFT notes http://www.damtp.cam.ac.uk/user/tong/qft.html (p138): "This is because among the infinite number of gauge symmetries parameterized by a function λ(x), there is also a single global symmetry: that with λ(x) = constant. This is a true symmetry of the system, meaning that it takes us to another physical state."

However, I've also seen the opposite statement, eg. in Greiter's exposition http://arxiv.org/abs/cond-mat/0503400. He distinguishes between a global symmetry (Eq 98) and a gauge symmetry (Eq 27, 99). He comments (p 14) "This "unphysical" symmetry, however, seems to contain the physical symmetry as the special case ... The formal equivalence of the transformation (98) and (27) with (99) is at the root of the widely established but incorrect interpretation of (98) as a gauge transformation, and in particular of the spontaneous violation of (98) as a spontaneous violation of a gauge symmetry. ... The problem here is that the equivalence is only formal. ... In the literature, (98) is often referred to as a global gauge transformation, and the conservation of charge attributed to gauge invariance. This view, however, is not consistent. If one speaks of a global gauge symmetry, this symmetry has to be a proper subgroup of the local gauge symmetry group. ... The difference between the global phase rotation (98) and a global gauge rotation (99) is even more at evident at the level of quantum states. The BCS ground state (1) is, for example, not invariant under (98), while it is fully gauge invariant"

I think Greiter is correct for the situation he is talking about. But then is Tong wrong, or is the situation he's talking about different?

Last edited: Dec 28, 2012
11. Dec 28, 2012

### VantagePoint72

I'm sorry, but I have no idea what your second sentence is supposed to be saying. I'll try to explain a bit more though. In a true global spacetime symmetry, if something has rotational symmetry about an axis it means that if you rotate every point around that axis, the physics remains unchanged. The consequence of this is that the component of angular momentum around that axis is conserved. However, the important point is that you do the exact same thing to every point. You can't rotate one point by 5 degrees and another point by 30 degrees and expect to get the same physics, that doesn't make any sense. Suppose somehow you could, though. Suppose you were free to rotate things however much you wanted at every point in spacetime and still get the same physics. Then the conclusion would be that the coordinate being rotated isn't actually physical—it's just a redundant part of the description of the physical state. Obviously this doesn't work for a spacetime symmetry, but suppose there were some other sense in which you could 'rotate' things.

For example, imagine we were looking at the dynamics of a single complex-valued field. A global 'rotation' symmetry would be if the physics were unchanged if you rotated the complex argument of the field by the same the amount at every point: i.e. $\phi(\vec{x},t) = R(\vec{x},t)e^{i\theta (\vec{x},t)}$ and you substitute $\theta (\vec{x},t) \rightarrow \theta (\vec{x},t) + \beta$ for some fixed β. As it happens, such a global symmetry generally gives rise to conservation of some kind of charge. But suppose the physics were unchanged even if β were allowed to be an arbitrary function of spacetime position. Then, as before, it would follow that θ can't be understood as a truly meaningful physical parameter, since it can be arbitrarily changed without consequences to the physics. This is a gauge invariance (in fact, very similar to one found in electromagnetism) and it implies we have some redundant information in our description of the system—in this case, the complex argument of the field. Since it's not a physical symmetry, it doesn't correspond to any additional conservation laws beyond the one corresponding to the global symmetry of doing the same (local) gauge transformation at every spacetime point.

It's a fairly complicated subject and is not easy to explain without delving into the mathematics of group theory and representations. This is about the best I can do informally.

Last edited: Dec 28, 2012
12. Dec 28, 2012

### alemsalem

so a local symmetry MUST be nonphysical?
why can't we have two physical configurations related by a local transformation (they are fields after all) and have that be a symmetry?

13. Dec 28, 2012

### VantagePoint72

A local symmetry that is allowed to freely vary at different points. The point isn't that you have two configurations that are related by a local transformation at one place, the point is that you have two configurations that yield the same physics after doing arbitrary local transformations everywhere. That is by definition not a physical symmetry. Just think about geometry: when we say something is reflection symmetric, that means you get the same thing back when you reflect every point around the same axis. Likewise for rotational, etc., symmetry.

Here's an example: imagine you had a round bowl with a marble rolling around in it. You notice that if you rotate the bowl, the motion of the marble released in the same initial conditions is unchanged. From this, you conclude that as far as the physics is concerned, the problem is rotational symmetric. Now suppose the bowl is also brightly coloured with many paints. You notice that these colours don't affect the marble's motion: you can repaint the bowl and still get the same marble motion. Is this the same as the rotational symmetry? No. You can paint the bowl arbitrarily: whereas for a rotation, you had to do the same thing to every point, you can repaint each point of the bowl how you like. As far as the physics is concerned, the bowl's colour is an arbitrary local symmetry: a gauge symmetry. Whereas rotating the bowl gave us a new physical configuration that just reproduced the same physics, repainting the bowl actually leaves it the exact same—because the paint job is redundant information for describing the system, as far as the physics is concerned.

14. Dec 28, 2012

### ShayanJ

Now I see it.
Actually,when I was writing my last post,I was viewing the subject from a wrong direction but now I get it.
Thanks all guys

15. Dec 28, 2012

### samalkhaiat

Yes, it is called interaction between matter fields and the gauge field.

Sam

16. Dec 29, 2012

### VantagePoint72

I think maybe that's putting the cart before the horse a bit. Yes, imposing gauge invariance on a non-interacting theory turns it into an interacting theory but it's perfectly possible (in theory) to have an ungauged interacting theory to begin with. I think a better way of putting it is that imposing gauge invariance severely limits which interactions are possible. Maybe it's splitting hairs, but I wouldn't say, "We have gauge theories, therefore interactions," even if it's a formally valid logical statement. I would say, "We have interacting theories, and they are gauge invariant." But then, maybe I'm just looking at it from the historical perspective of how these theories were developed, rather the more natural perspective of what's fundamental to nature...hmm...

17. Dec 29, 2012

### andrien

Sam is talking about local gauge invariance which leads to interaction between matter field and gauge field.it is the basis of modern treatment of gauge theory.Electromagnetism for example arises directly due to local gauge invariance.it comes from the work of yang and mills

18. Dec 29, 2012

### VantagePoint72

I am well aware of how Yang-Mills theory works, which I think was pretty clear from my post since I described that exact process.

19. Dec 29, 2012

### samalkhaiat

Then you should also aware of the fact that the gauge bosons (and their interactions) arise naturally from the local gauge principle with no extra input. For theoreticians, this is a BIG DEAL and fundamental.

20. Dec 29, 2012

### VantagePoint72

Did you even read my earlier comment I was referring to? It certainly doesn't appear that you did.

21. Dec 30, 2012

### the_pulp

Hi, I asked something like this in another thread and the idea I got from that was that if you want a Conservation Law you need a Global Symmetry in the Lagrangian. If you want that Conservation Law to be Local you need that Symmetry to be a Gauge Symmetry. Since this idea has not been precisely state in this thread, Id like to know if you agree with this or if you think that Im wrong.

Right now, this is my understanding about the Physical Interpretation of Gauge Invariance.

22. Dec 30, 2012

### dextercioby

Not excluding general relativity*, all interacting field theories are obtained by starting with free gauge theories and free gauge-less theories. So <we have gauge theories, therefore interactions> is the only valid judgement.

*As a field theory, GR can be derived from gauging the global Lorentz symmetry of a flat-spacetime (see post 24) or by consistent self-couplings of a spin 2 field again on flat space-time (see Bill's post 23 and my comment in post 25 to Bill's post).

Last edited: Dec 30, 2012
23. Dec 30, 2012

### Bill_K

Actually, what is gauged in the case of General Relativity is not the Lorentz group but the translation group. An infinitesimal coordinate transformation is a position-dependent translation

xμ → xμ + ξμ(x)

under which the gravitational field undergoes the gauge transformation

hμν → hμν + ξμ,ν + ξν,μ

24. Dec 30, 2012

### dextercioby

Hi Bill,

GR comes from gauging the Lorentz group antisymmetric infinitesimal generators as shown by Utyiama in 1956 (Invariant Theoretical Interpretation of Interaction, Phys.Rev, Vol.101, No.5, page 1597).

It's true that some of his arguments were a little <by hand>, as advocated by Kibble on first page of his article in 1961 (Loreritz Invariance and the Gravitational Field, J. Math. Phys., Vol.2, No.2, page 212), but nonetheless, under reasonable assumptions one can reach GR as a field theory by gauging the Lorentz group generators.

25. Dec 30, 2012

### Bill_K

dextercioby, I'm not familiar with Utiyama's work - is it really about the same thing? I did find this review article on arxiv, which mentions it, and has this to say: