- #1
- 2,810
- 605
I've read that gauge invariance leads to a fundamental phenomenon.What is that?
Thanks
Thanks
it is a very broad field now.see hereShyan said:I've read that gauge invariance leads to a fundamental phenomenon.What is that?
Thanks
LastOneStanding said: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.
Shyan said: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!
Shyan said: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!
LastOneStanding said: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.
Shyan said: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!
Yes, it is called interaction between matter fields and the gauge field.Shyan said:I've read that gauge invariance leads to a fundamental phenomenon.What is that?
Thanks
samalkhaiat said:Yes, it is called interaction between matter fields and the gauge field.
Sam
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 millsLastOneStanding said: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...
LastOneStanding said: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.
LastOneStanding said:[...]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." [...]
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*As a field theory, GR can be derived from gauging the global Lorentz symmetry of a flat-spacetime
So are they talking about Einstein-Cartan theory instead of GR??He [Utiyama] gauged the Lorentz group SO(1, 3), inter alia. Using some ad hoc assumptions, like the postulate of the symmetry of the connection, he was able to recover GR. This procedure is not completely satisfactory, as is also obvious from the fact that the conserved current, linked to the Lorentz group, is the angular momentum current. And this current alone cannot represent the source of gravity. Accordingly, it was soon pointed out by Sciama and Kibble (1961) that it is really the Poincare' group R4 ⊃× SO(1, 3), the semi-direct product of the translation and the Lorentz group, which underlies gravity. They found a slight generalization of GR, the so-called Einstein-Cartan theory (EC), which relates – in an Einsteinian manner – the mass-energy of matter to the curvature and – in a novel way – the material spin to the torsion of spacetime.
Bill_K said:[...]So are they talking about Einstein-Cartan theory instead of GR??
the_pulp said: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 I am wrong.
I cannot see it.Can you be more specific.thanks.GR can be derived from gauging the global Lorentz symmetry of a flat-spacetime
andrien said:I cannot see it.Can you be more specific.thanks.
No, that's not correct. Gauge symmetries do not give conservation laws (subject to the caveats already discussed earlier in this thread). A global symmetry will gives a conservation law that is local, which implies global as well. This is true because local conservation of X means, "The change in X in some volume is equal to the amount of X flowing out of the boundary of that volume." So if you make the volume all of space, there is no where for X to flow out of (with a few mathematical caveats) and so X is also conserved globally. So, local conservation implies global conservation and so both follow from a global symmetry.
Gauge invariance is a fundamental principle in physics that states that the physical laws and equations describing a system should not change under a transformation of its underlying mathematical variables. This means that the choice of a specific reference point or frame of reference should not affect the physical predictions of the system. It is important because it allows us to simplify and generalize physical theories, making them more elegant and applicable to a wide range of situations.
Gauge invariance is closely related to the concept of symmetry in physics. Symmetry refers to the invariance of a system under certain transformations. Gauge invariance is a specific type of symmetry that involves transformations of the mathematical variables used to describe a physical system. This symmetry allows us to simplify and unify physical theories, making them more elegant and powerful.
One example of gauge invariance in action is in the theory of electromagnetism. Maxwell's equations, which describe the behavior of electric and magnetic fields, are invariant under a transformation of the electric and magnetic potentials. This means that the choice of a specific reference point or gauge does not affect the physical predictions of the equations. This allows us to describe electromagnetic phenomena in a more general and elegant way.
Gauge invariance plays a crucial role in our understanding of the fundamental forces in nature. The Standard Model of particle physics, which describes the electromagnetic, weak, and strong forces, is based on the principle of gauge invariance. By applying this principle, we are able to unify these forces and explain their fundamental interactions in a more elegant and comprehensive way.
While gauge invariance has been a powerful tool in advancing our understanding of the physical world, it also presents some challenges and limitations. One challenge is that it can be difficult to find the correct gauge or reference point for a given physical system. Additionally, there are some physical phenomena, such as the Higgs mechanism, that cannot be fully explained using gauge invariance alone. However, overall, gauge invariance has been a fundamental principle that has greatly advanced our understanding of the universe.