Classification of Gauge Theories

[MManuel Abad]

Hi there:

I was just searching about Gauge theories and stuff and I find it very confusing. My major complication is the classification. I'd like you to tell me some definitions and construct a "family tree". I guess it goes something like this:

Gauge Theory: A Field Theory in which the lagrangain is invariant under the action of a Lie Group.

Yang-Mills Theory: A Gauge Theory in which the gauge group is SU(N).

QED: A type of gaugue theory in which the gaugue group is U(1)

QCD cousin: without quarks: YM SU(N=3)

QCD: A type of Yang-Mills in which N=3 (the 3 stands for the number of colors, doesn't it?)- How is the existence of quarks expressed in the class. of QCD?

Please, I need this. I'd like you to complete this mini-tree (probably wrong) and include Electroweak theory, Super-Yang-Mills and stuff like that.

Thankyou very much.

 

[element4]

Actually, your definition of Gauge theory is not correct

Gauge Theory: A Field Theory in which the lagrangain is invariant under the action of a Lie Group.

For example any relativistic field theory is invariant under the Lorentz group, but not necessarily a gauge theory. Gauge theories are field theories which are invariant under local transformations, meaning that the transformation is a different element of the Lie group at each space-time point.**

There is also a correction to your first "branch", in the family tree. Yang-Mills theories are not restricted to $SU(N)$, but can be constructed from more general Lie groups (although historically they only considered $SU(2)$, I think). But there is an important detail in the definition: a Y-M theory is a gauge theory where the dynamics of the gauge field, $A_{\mu}$, is of the form $\mathcal L_{YM}\propto \text{tr}(F^{\mu\nu}F_{\mu\nu})$. You can consider QED and QCD as special cases.
I am not sure what you mean with >>How is the existence of quarks expressed in the class. of QCD?<<. Quarks are included, essentially, just by including some fermions, minimally coupled to the gauge field.

My last comment is that there exist gauge theories which are not part of the Yang-Mills branch. A very important one is for example Chern-Simons theory which the action is $S_{CS} = \frac k{4\pi}\int_{\mathcal M}\text{tr}\left(A\wedge dA + \frac 23 A\wedge A\wedge A\right)$, where $A = A_{\mu}dx^{\mu}$ is an Lie algebra valued one-form (I have used a more slick notation for simplicity). This action is actually not gauge invariant at the classical level, but it is at the quantum level if $k$ is an integer. It was originally introduced as an addition to the Yang-Mills action in order to make the gauge bosons massive i three-dimensions (without using the Higgs mechanism). But later it was discovered that pure Chern-Simons theory has many very deep and interesting mathematical features. Physically this gauge theory plays a central role in highly exotic systems in Condensed Matter Physics, such as the Fractional Quantum Hall effect.

But there are also other gauge theories which are not Yang-Mills.


**Actually gauge invariance is usually considered as something more general. Any transformation which is not a physical symmetry, but rather a redundancy (several labels for the same thing), is called a gauge transformation.

 

[daschaich]

Excellent explanation element4.

I am not sure what you mean with >>How is the existence of quarks expressed in the class. of QCD?<<. Quarks are included, essentially, just by including some fermions, minimally coupled to the gauge field.

I believe the question is just how physicists label the number of fermions transforming under the gauge group. We call each different type of fermion a "flavor", and use $N_f$ to refer to the number of flavors. A slight complication is that we tend to assume all flavors have the same mass unless otherwise indicated. So QCD itself would be called $SU(N_c)$ gauge theory with $N_c=3$ (three "colors") and $N_f=2$ light fermions, since the up and down quarks have roughly the same mass. To include the strange quark, we would specify $N_f=2+1$; to include strange and charm, $N_f=2+1+1$. (Here's an example.)

There's an additional complication that we also have to specify what representation of the gauge group the fermions transform under, but that's a more advanced topic. (If you're interested, the quarks of QCD transform in the fundamental representation.)

QCD cousin: without quarks: YM SU(N=3)

"QCD cousin" is not a term I've ever heard before. To specify a Yang--Mills theory without fermions, I would say "pure gauge" or "pure Yang--Mills".

...include Electroweak theory, Super-Yang-Mills and stuff like that..

The electroweak gauge group is $SU(2)\times U(1)$. Note that this $U(1)$ is not the same one as QED. Supersymmetric Yang--Mills theories are just Yang--Mills theories that are also supersymmetric; supersymmetry itself is a different topic that you can already find discussed in many other threads on this site.

 

[MManuel Abad]

Wow! Thank you very much! Those were actually very good and complete explanations! You both complemented the whole point of my thread! That was very useful! This forum always gives me what I want, so I thank you both again!

Greetings from Mexico!