# What is a gauge theory?

1. May 15, 2014

### ShayanJ

From the first moments that I read about gauge theories, till now, after years, They are still a mystery to me.Maybe that's because I never had someone explaining them to me or never actually seen any real calculation regarding them, but I think I should be able to understand them now.
Anyway, this time while I was reading about flux quantization, this paragraph reminded me my ignorance about GTs.

Source

I know its better to have explicit sharp questions but when I think about that I realize that my questions are so hazy that persuades me I don't know enough about it to ask questions about it. I always tried reading about it but never have been able to understand it. So...can somebody explain about it?
Thanks

2. May 15, 2014

### WannabeNewton

Explain what?

3. May 15, 2014

### DrDu

I am not a specialist on that, but here is my understanding.
Gauge degrees of freedom are in some way artifacts from working with non-observable quantities. If you try to eliminate these from theory, you have to deal with non-local observables instead, namely so-called Wilson loops.
In EM, these loops correspond to the magnetic flux enclosed by the loop.
Apparently, it is easier to work with local operators which aren't observables than to work with nonlocal observables.

4. May 15, 2014

### ShayanJ

What is a gauge theory?
What does it mean that gauge symmetries define force laws?
What is the role of symmetry in physics?
Is it that because laws should have a special kind of symmetry, they become as they are?
And other questions regarding Gauge symmetries and Gauge Theories.
As I said, my questions are hazy...and many...because the whole thing isn't clear to me!

5. May 15, 2014

### WannabeNewton

Your questions are far too broad to be answered in a single post. Do you have a more specific question in mind? Have you seen minimal coupling through gauge covariant derivatives? Have you seen how to enforce the coupling of the Dirac field to the electromagnetic 4-potential from local $U(1)$ gauge invariance of the Dirac Lagrangian? If not you should probably first see these things before asking such broad questions. Almost every QFT book I have ever come across explains these things in stark detail. See, for example, chapter 3 of Maggiore.

6. May 15, 2014

### Simon Bridge

Yike: what education level is this to be at?
All these questions basically make a very big topic - the sort of thing that would be covered in several University papers. Can you narrow it down a bit?
i.e. most of the questions can be googled to find answers and courses. This forum is better for discussing the material you can find elsewhere, where you may be having trouble understanding it.
Short answers are likely to get circular. i.e.

A theory of physics involving gauge mathematics.
http://en.wikipedia.org/wiki/Introduction_to_gauge_theory
http://arxiv.org/abs/math-ph/9902027

Exactly that.
The observed force laws can be described as emergent properties of the Universe. Key to understanding how this happens is to observe overall symmetries.

It's pretty.
We see a lot of it in Nature and physics is the study of Nature so we need to study symmetries.
(My emph.) It's the opposite. The gauge theory describes force laws as emergent properties of the underlying symmetries we find in Nature. It is not a case of laws should or should not have a certain symmetry but what symmetries they actually have is thought to provide clues to the underlying Rules.

7. May 15, 2014

### abitslow

Wikipedia article on gauge theory is a good start. But if you haven't got a good understanding of what symmetry means and why symmetry is important in physics, you aren't anywhere close to where you need to be to get a firm grasp of gauge theories.
Critical in that is a grasp of LaGrangian Mechanics and Noether's Theorem.

8. May 15, 2014

### Jilang

Even Wiki does a terrible job on this! It is actually pretty straightforward. A gauge theory is one that is gauge invariant. Gauge invariance just means it doesn't matter what you define as zero. For example you might define a field everywhere by V(x). Adding a constant to it so it becomes V(x) + C will not change the physics. Wiki sort of states this, but it's not very clear.

9. May 15, 2014

### WannabeNewton

That's certainly grandly over-simplifying things.

10. May 15, 2014

### Jilang

Last edited: May 15, 2014
11. May 15, 2014

### HomogenousCow

This may be interesting to the OP
http://www.problemsinelectrodynamics.com/relativistic-electrodynamics/gauge-invariance-action-charge-conservation#show-solution [Broken]

Last edited by a moderator: May 6, 2017
12. May 15, 2014

### dextercioby

Gauge symmetry is a symmetry of the Lagrangian action derived from the property of the system's constraints to be 1st class.

13. May 15, 2014

### Matterwave

A gauge theory is any field theory (quantum or classical) wherein the fields exhibit so-called "gauge symmetries". This means that the field strengths (usually tensors) are derivable from some potential (usually a vector potential) which is not uniquely determined by physical systems. In other words, given a physical configuration of e.g. particles that source this field, the potential functions (often vector potentials) are determined only up to "gauge transformations".

The simplest example of this is the scalar potential theory Jilang quoted. Given a potential energy function (e.g. for gravity, or for a spring, or some such) V(x), V(x)+C, where C is a constant, describes the exact same physics. From physical descriptions, and physical laws, I cannot determine V(x) absolutely. This is such a simple version of the idea, though, that usually one does not even consider the "gauge symmetries" at play here. One might consider this a "trivial" example of a gauge freedom.

The simplest, non-trivial, example of a gauge theory is the classical theory of electrodynamics. In relativistic language, given the four-vector potential $A_\mu=(\phi,\vec{A})$ one can add a total divergence to this term without changing the physics. In other words given: $A'_\mu=A_\mu+\partial_\mu\psi(x)$, $A'_\mu$ and $A_\mu$ give the same physics (in geometric language, this transformation would be given as $\bf{A}'=\bf{A}+\bf{d}\psi$). This is because the "real" physics is encoded in the field strength tensor (the Faraday tensor) $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$ (in geometric language simply: $\bf{F}=\bf{d}\bf{A}$) and this field strength tensor is unchanged by the above gauge transformation (check this yourself, essentially this is true simply because the partial derivatives commute, or in geometric language, because the exterior derivative is closed). Therefore, one is said to have "gauge freedom". And this is an example of a classical gauge theory.

Finally, we get to the idea of how "gauge theories" explain "forces". Well, as it turns out three (the strong, weak, and EM) of the four fundamental forces of nature are described by such gauge theories (gravitation as well has "gauge symmetries" but this is a little bit different since the "gauge" freedom in general relativity is really the freedom to choose coordinate systems - sometimes called diffeomorphism invariance). The quantum version of the EM theory, for example, must exhibit the same freedom of gauge, and this actually leads to some trouble in the usual canonical quantization system of the 4-vector potential since the actual dynamical degrees of freedom have not been properly isolated (in the naive approach). The strong force and the weak force (combined with E&M to become the electroweak force in the Weinberg Salam model) are similarly defined by gauge theories. However, in those cases, the gauge symmetry is more complicated because the gauge group (the group for which the vector potentials form the Lie Algebra) is so-called non-abelian (SU(3) for the strong force, and SU(2)xU(1) for the weak force, rather than U(1) for E&M). To go into this in more detail would take quite a bit longer. I hope I have provided at least a little background to the problem for you.

Last edited: May 16, 2014
14. May 15, 2014

### Simon Bridge

... and so on. All of which is why you usually learn about guage theories from the bottom up - building on more common subjects. Try to go from the top down and you get a lot of "what does this stand on" type questions before you get to something you can understand ... and you still have to work your way back up again.

15. May 16, 2014

### ShayanJ

Thanks to all
I think I need much more study for attacking this one, still your posts were useful.

16. May 16, 2014

### the_pulp

I think I have the question that you are trying to find (at least, it is a very usual question). Why every lagrangian that is used in fundamental particle physics is gauge invariant? Why does nature choose to work with gauge invariant lagrangians? My conclusion, after giving this subject a lot of thought and -amateur- investigation, is that theorists still dont know why. There are good papers that give some insight about it (there is one very good of Rovelli called "Why gauge?") but I think that they havent found any final resolution yet. This is my view about what is your question and which is its answer.