Gauge Theory: Simplified Explanation & Examples for Physics

[vikram_n]

Could someone explain what a gauge theory is, both in general and how it applies to physics? Please try to keep definitions relatively simple, even though the topic is exceedingly complicated. Examples are also greatly appreciated. Thanks!

 

[bcrowell]

http://en.wikipedia.org/wiki/Introduction_to_gauge_theory (may be easier to understand than the main WP article)

 

[vikram_n]

To a novice, this is still just words. I need a definition that puts it more simply.

 

[atyy]

A gauge theory is a theory in which there is more than one mathematical label denoting a physical state. An example is the electric potential V in electrical circuits. If the potentials at the respective ends of a resistor R are V1 and V2, the current through the resistor is (V2-V1)/R. If we change all the potentials by a constant A, the current remains the same since (V2+A-V1-A)/R = (V2-V1)/R. Since A can be any number, there are infinite number of potentials that correspond to the same physical situation of a particular current. Only the potential *difference* is "physical", while the potentials are "gauge".

 

[Muphrid]

Have you ever thought about why you choose the x-axis to be in one direction and the y-axis in another?

In physical problems, you often have the ability to choose a coordinate system based on what is convenient. Why? Because we take as given that any meaningful result does not depend on our coordinate system. The overall physical situation and any information we extract from it must be the same regardless of whether we use cartesian, cylindrical, spherical, or any other type of coordinate system. It means you have the freedom to choose a coordinate system and get the same result regardless of what your choice was. Often, we use this to our advantage, choosing the most convenient coordinate system for a given problem.

Gauge theory deals with a different kind of freedom. In it, there must be a field (or maybe several fields) that we can adjust, tweak, or alter based on certain rules, but no matter how we adjust this field, the physical predictions we make based on it must be the same. This means that, like our choice of coordinate system, we have some freedom to choose what this field should be, and we often do choose it based on what is most convenient for the problem at hand.

 

[tom.stoer]

Think about a grid of clocks in your country to fix a reference system for time. All clocks shall be synchronized, and the time zone (+ daylight saving time) shall be identical for all clocks. Now we have constructed an absolute time for your country.

For physical processes this absolute time is irrelevant; all what matters are time intervals, i.e. how long does it take to get from your home to the campus.

Now we introduce a local gauge degree of freedom. That means that for each clock we can chose the time zone independently and we can vary the time zone at each time we like. If we do that we have no chance to calculate any time difference; comparing the clock at your home with the clock at your campus when arriving doesn't tell you anything.

Therefore we introduce a gauge field. It tells you at each place and each time how the setting of the time zone varies compared to other clocks at other places. This gauge field is implemented by messengers riding from clock A at place A to any other clock and being ready to report regarding the time zone information of A (when they started at A). Having enough messengers in place you are able to subtract the gauge effect, i.e. when arriving at the campus you can tell the messenger at which time you started at you home, he will be able to readjust the clock at the campus and therefore you are able to calculate the time difference.

Up to now this seems rather strange b/c all what we do is to introduce a gauhe degree of freedom which is subtract afterwards, so there is no imprint at all. All what I have said seems to be a waste of time. This would be the case if all gauge field configurations would be "pure gauge". In electrodynamics pure gauge means that we only deal with field configurations which are equivalent to vacuum, i.e. vanishing of all electromagnetic fields.

But - if you like - we can discuss the next step and implement dynamics for the gauge field. That means that the clocks will be affected by the messengers, and the messengers will be affected by the clocks. Then the messengers do not only carry information regarding time zone settings but the carry "physical information" (in electrodynamics this could be electromagnetic waves).

 

[the_pulp]

This is one of the cases that perhaps my ignorance is most useful than all the wisdom spreaded througout this thread. I will give you my view, it is very "naively understandable", but perhaps is wrong or not so precise. In that case, please teachers, correct me:
1) In the universe there are charges going around everywhere. In some place there is some negative charge, in some other some positive charge. In some place there is some green colour charge and so on.
2) These charges are conserved globally. That is to say that if in some moment there are in the whole universe 10000 negative charges, one minute later there will be the same amount of negative charges.
3) These charges are also conserved locally. That is to say that if in some moment, in some place, there are 10 red colour charges, then, one instant later, these 10 red colour charges will be in the same place or not too far.
4) There are some theorems, Noether, Ward Takanashi, etc, that says that if you want to make a theory about systems with local charges, you can make them rather simply building an object called "Lagrangian" which can be viewed as some sort of probability density (or probability amplitud density) which have to have some sort of symmetry (and that is the only way you can make them). The theories that are constructed under this procedure are called Gauge Theories (ie pick a symmetry and find the lagrangian asociated with it)
5) The amount of different charges that you observe in experiments and the way they interact should define the sort of symmetry that the Lagrangian should have (and, in that way, its precise mathematical formulation). (This is more or less what happened with the model that describes the strong interaction which is defined as a Lagrangian with SU(3) as a gauge symmetry)
6) viceversa, we can postulate that some not yet explained phenomena should be explained through a Lagrangian with some symmetry and, as a consequence, you will have as a result, a theory of some local charges which interact in some way. (That is what happened with the model that describes the electroweak interaction which is defined as a Lagrangian with SU(2)xU(1) as a gauge symmetry).

The symbols I used (SU(2) and so on) are things used to represent symmetries. They are groups. So, in some way you can say that point 4 is a way to unifiy Group Theory with Local Charge Physics (in some way, not taking into account that there can be broken symmetries or that the same group symmetry can be used with a theory of interactions of 2 type of particles or 20 or 2000 or so on). That is to say that for every group symmetry there is (up to what I stated in the las bracket) one physics model and viceversa.

Please, take into account that what I stated is not tooooo precise and that it may contain mistakes. More or less it represents my view on the subject.

 

[dydxforsn]

Here is a really good (and accessible) article on gauge theory I was shown by a friend some time ago:

http://www.vttoth.com/CMS/physics-notes/124-the-principle-of-gauge-invariance

Hope it helps :)