What is the global and local symmetry in gauge theory?

[TimeRip496]

For details and to try to explain the above, the complex scalar field is a useful example. The complex scalar field has an action
S=∫d4x(∂μϕ∗)(∂μϕ)−V(|ϕ|).
There is a global, continuous symmetry to this action -- an overall phase. That is, if one replaces ϕ→eiαϕ, then the action does not change. This type of change is called a gauge transformation. The symmetry group of this transformation is the Lie group U(1).

http://www.quora.com/What-is-Gauge-Theory-%28intuitively%29

Can anyone explain this to me as simple as possible? I don't really need to know it but I want to know what is all the symmetry about. I roughly know what does symmetry mean in physics terms but what is up with the local symmetry or global symmetry? And what is symmetry group?

From my understanding, symmetry of a physical system is a physical or mathematical feature of the system that is preserved or remains unchanged under some transformation.

I know I am way too early to even touch this but I am really curious about it.

 

[DEvens]

Unless you work through the maths, you are unlikely to get more than a motivational understanding. But here goes.

Motivational example:

Think about the set of functions f(x) on the real numbers. Now think about two classes of those functions.

Class 1: f(x) = f(-x) even functions
Class 2: g(x) = - g(-x) odd functions

Suppose you had a physical process that exchanged f(x) for f(-x). For example, reflection off a mirror at x=0. You now have a situation where even and odd functions could be naturally split apart. The even functions could wind up having one behaviour on reflection, the odd a completely different behaviour. The symmetry of the situation naturally divides into two collections of functions. At this point you would start studying group theory, in particular representations of groups, action on basis vectors, and so on.

So if you have any old function F(x) you can project out the odd part and the even part.

f(x) = 1/2 (F(x) + F(-x))
g(x) = 1/2(F(x) - F(-x))

It should be obvious that f(x) = f(-x) and g(x) = -g(-x). So we can pull out the odd and even part. So if some physical process gave zero on he odd part and non-zero on the even, we could project out the even part and only use that. If it was helpful we could use that.

Motivational example:

Think about a molecule made of four identical atoms arranged in a square. I have no idea what molecule that might be, or even if it is in fact possible. But go with it.

This square has symmetry. If you rotate it 90 degrees around the axis perpendicular to the square it looks the same. Also, if you mirror reflect it in some planes you wind up getting something that looks exactly the same. Now think about starting with the square and giving one atom a little tug out of place. The whole molecule will want to vibrate. But how will it do that? It turns out the symmetry can tell you a lot about that. You can construct (with a lot of that group theory I mentioned) an operator that projects out the normal modes. Just like the odd and even parts of a function, we can project out the normal vibration modes of a symmetric object. Insert more math than I am willing to type into the forum. But if a physical action only interacted with one mode, we could possibly simplify by starting with that one mode. For example, we could project out the rotation part and the translation part for the molecule, and ignore them.

For example, one vibration mode is the "breathing mode." This is where the square gets bigger and smaller, each atom moving in unison straight in or out. Every molecule has this mode.

Motivational example:

Think about a big pile of pins. If the pins are all oriented parallel to each other the pile can be denser. Now suppose the pins are piled on a table. The pins will tend to be parallel to the table, and with each other. So it will pick out a direction. If you dumped all the pins on the table then kept jiggling and pushing them into a pile, they would tend to line up into parallel clumps. But the specific direction of these clumps is not important. It could have been a different angle. Level with the table but pointing north-south. Or 5 degrees off north-south. Or 10. Or 20. It does not make any difference as long as they are the same angle. This is a rotational symmetry.

But suppose one clump was at 37 degrees, and the next clump at 45 degrees. There is an interaction between these two clumps. If you rotated the table (slowly and carefully so as not to move anything) the 37-45 could become 137-145, and everything would effectively be the same. It is the difference that matters.

So there is a rotational symmetry, but the difference between the clumps produces an interaction. The absolute value does not matter, but differences between neighbours does.

Now think of your $\phi$ field. It can be rotated in phase. That's a little circle. If you rotated every point by exactly the same amount, you get no difference. There is a symmetry. But if you rotate one part by one amount, and a neighbouring part by a different amount, it creates clumps. And those clumps have an interaction. Just like the needles.

It turns out that interaction can be made to look just like electromagnetism. The interaction is carried by the photon. The phase is experienced by particles with charge. Absorbing or emitting a photon changes the phase, like rotating the needles. The specific "vibration" mode it sits in tells you how a charged particle interacts.

 

[TimeRip496]

Unless you work through the maths, you are unlikely to get more than a motivational understanding. But here goes.

Motivational example:

Think about the set of functions f(x) on the real numbers. Now think about two classes of those functions.

Class 1: f(x) = f(-x) even functions
Class 2: g(x) = - g(-x) odd functions

Suppose you had a physical process that exchanged f(x) for f(-x). For example, reflection off a mirror at x=0. You now have a situation where even and odd functions could be naturally split apart. The even functions could wind up having one behaviour on reflection, the odd a completely different behaviour. The symmetry of the situation naturally divides into two collections of functions. At this point you would start studying group theory, in particular representations of groups, action on basis vectors, and so on.

So if you have any old function F(x) you can project out the odd part and the even part.

f(x) = 1/2 (F(x) + F(-x))
g(x) = 1/2(F(x) - F(-x))

It should be obvious that f(x) = f(-x) and g(x) = -g(-x). So we can pull out the odd and even part. So if some physical process gave zero on he odd part and non-zero on the even, we could project out the even part and only use that. If it was helpful we could use that.

Motivational example:

Think about a molecule made of four identical atoms arranged in a square. I have no idea what molecule that might be, or even if it is in fact possible. But go with it.

This square has symmetry. If you rotate it 90 degrees around the axis perpendicular to the square it looks the same. Also, if you mirror reflect it in some planes you wind up getting something that looks exactly the same. Now think about starting with the square and giving one atom a little tug out of place. The whole molecule will want to vibrate. But how will it do that? It turns out the symmetry can tell you a lot about that. You can construct (with a lot of that group theory I mentioned) an operator that projects out the normal modes. Just like the odd and even parts of a function, we can project out the normal vibration modes of a symmetric object. Insert more math than I am willing to type into the forum. But if a physical action only interacted with one mode, we could possibly simplify by starting with that one mode. For example, we could project out the rotation part and the translation part for the molecule, and ignore them.

For example, one vibration mode is the "breathing mode." This is where the square gets bigger and smaller, each atom moving in unison straight in or out. Every molecule has this mode.

Motivational example:

Think about a big pile of pins. If the pins are all oriented parallel to each other the pile can be denser. Now suppose the pins are piled on a table. The pins will tend to be parallel to the table, and with each other. So it will pick out a direction. If you dumped all the pins on the table then kept jiggling and pushing them into a pile, they would tend to line up into parallel clumps. But the specific direction of these clumps is not important. It could have been a different angle. Level with the table but pointing north-south. Or 5 degrees off north-south. Or 10. Or 20. It does not make any difference as long as they are the same angle. This is a rotational symmetry.

But suppose one clump was at 37 degrees, and the next clump at 45 degrees. There is an interaction between these two clumps. If you rotated the table (slowly and carefully so as not to move anything) the 37-45 could become 137-145, and everything would effectively be the same. It is the difference that matters.

So there is a rotational symmetry, but the difference between the clumps produces an interaction. The absolute value does not matter, but differences between neighbours does.

Now think of your $\phi$ field. It can be rotated in phase. That's a little circle. If you rotated every point by exactly the same amount, you get no difference. There is a symmetry. But if you rotate one part by one amount, and a neighbouring part by a different amount, it creates clumps. And those clumps have an interaction. Just like the needles.

It turns out that interaction can be made to look just like electromagnetism. The interaction is carried by the photon. The phase is experienced by particles with charge. Absorbing or emitting a photon changes the phase, like rotating the needles. The specific "vibration" mode it sits in tells you how a charged particle interacts.

Thanks a lot! Thats very informative. But I have some doubt here. When you talk about the electromagnetic interaction, what do you mean by the phase of the charge particle? Is it related to the wave nature of the charged particle?

For the square molecule, what do you mean by constructing (with a lot of that group theory I mentioned) an operator that projects out the normal modes?

 

[ddd123]

Thanks a lot! Thats very informative. But I have some doubt here. When you talk about the electromagnetic interaction, what do you mean by the phase of the charge particle? Is it related to the wave nature of the charged particle?

There is a phase change in the wavefunction.

For the square molecule, what do you mean by constructing (with a lot of that group theory I mentioned) an operator that projects out the normal modes?

Above, concerning odd and even functions, he took it as obvious and didn't specify that F(x) = f(x) + g(x) . Try substituting and see that it's true. It's similar to vector addition from base vectors, and, conversely, projecting the overall vector into one of the base directions. This projection tells you the specific behavior filtering out the other unrelated (for your purposes) ones. It's similar to classical mechanics with projectile motion: if you project the velocity vector of a cannon ball to the y (gravity) axis, you only get the gravitational (accelerated motion) behavior, if you project it on the x-axis you get the inertial (constant velocity) behavior.

With vibrational modes, or with molecules where there's rotation and vibrations etc... it's the same, you can filter out specific aspects using projection. This projection is generally performed using an operator (in the simplest of cases it's a matrix acting on vectors).

 

[TimeRip496]

There is a phase change in the wavefunction.
Above, concerning odd and even functions, he took it as obvious and didn't specify that F(x) = f(x) + g(x) . Try substituting and see that it's true. It's similar to vector addition from base vectors, and, conversely, projecting the overall vector into one of the base directions. This projection tells you the specific behavior filtering out the other unrelated (for your purposes) ones. It's similar to classical mechanics with projectile motion: if you project the velocity vector of a cannon ball to the y (gravity) axis, you only get the gravitational (accelerated motion) behavior, if you project it on the x-axis you get the inertial (constant velocity) behavior.

With vibrational modes, or with molecules where there's rotation and vibrations etc... it's the same, you can filter out specific aspects using projection. This projection is generally performed using an operator (in the simplest of cases it's a matrix acting on vectors).

Thanks a lot! Really help a novice like me