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Physical significance of gauge invariance 
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#1
Dec2612, 10:45 PM

P: 928

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


#2
Dec2712, 12:29 AM

P: 428

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
Dec2712, 12:46 AM

P: 928

I know about Noether's theorem and the relationship between symmetries and conserved quantities.
By gauge invariance I mean transformations like [itex] \vec{A} \rightarrow \vec{A}+\vec{\nabla}\psi [/itex] and [itex] \phi \rightarrow \phi + \frac{\partial \psi}{\partial t} [/itex] 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. 


#4
Dec2712, 03:34 AM

P: 1,020

Physical significance of gauge invariance



#5
Dec2712, 11:42 AM

P: 718

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
Dec2712, 06:19 PM

P: 159

according to Nima ArkaniHamed (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
Dec2812, 12:45 AM

P: 928

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
Dec2812, 01:57 AM

P: 428

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
Dec2812, 09:38 AM

P: 107

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 viceversa. But its still the same person! Therefore the true physical states in gauge theories correspond to socalled gaugeorbits, 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 socalled "firstclass constrants" (see http://en.wikipedia.org/wiki/First_class_constraint). 


#10
Dec2812, 12:06 PM

Sci Advisor
P: 8,788

However, I've also seen the opposite statement, eg. in Greiter's exposition http://arxiv.org/abs/condmat/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? 


#11
Dec2812, 12:36 PM

P: 718

For example, imagine we were looking at the dynamics of a single complexvalued 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. [itex]\phi(\vec{x},t) = R(\vec{x},t)e^{i\theta (\vec{x},t)}[/itex] and you substitute [itex]\theta (\vec{x},t) \rightarrow \theta (\vec{x},t) + \beta[/itex] 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. 


#12
Dec2812, 12:56 PM

P: 159

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
Dec2812, 01:24 PM

P: 718

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
Dec2812, 02:04 PM

P: 928

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
Dec2812, 07:54 PM

Sci Advisor
P: 924

Sam 


#16
Dec2912, 12:39 AM

P: 718




#17
Dec2912, 01:09 AM

P: 1,020




#18
Dec2912, 01:35 AM

P: 718

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



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