# R. Slansky, Group theory for unified model building

1. Jul 14, 2008

### jpg15

I'm a beginning QFT student, trying to slog my way through Slansky's paper...

I have some basic questions, the answers to which everyone around me seems to take for granted. Any help would be appreciated! It would also be great if we could strike up a conversation on group theory as it relates to particle physics, too...

I'll start with page 7, where Slansky discusses "construction of the fermion kinetic energy and fermion mass in a Yang-Mills Lagrangian." He says that the goal is to show that "the kinetic energy couples fR and fL."

First, what are fR and fL? In the paragraph above, he uses them in a sentence like this: "the left-handed fermions transforming as fL" so I take it that he means some sort of representation... but what does this actually mean mathematically? Matrix? Vector?

Second, it seems that people talk about matrices and states interchangeably when they talk about group representations... I completely understand matrix representations. But how can states (like Dirac spinors) represent anything?

As an example, SO(3) refers to a set of matrices used to multiply vectors. The vectors themselves don't tell me anything about SO(3). Or do they?

Third, what exactly does the notation fR $$\times$$ fL mean? Cartesian product? Direct product? Tensor product?

Fourth, what is the conjugate of something like fR? Is this the same as the adjoint representation?

I have so many questions, but I'll leave it here for now.

-j

2. Jul 14, 2008

### Mentz114

If you are a beginner in QFT, starting with Yang-Mills is ambitious. There are free books and lectures available that introduce the subject. These 3 look OK.

arXiv:hep-th/9505152

arXiv:hep-th/0510040

arXiv:physics/0504062

3. Jul 14, 2008

### jpg15

When I say beginner, I mean that I am beginning, while those around me are continuing. Like it or not, this is the level of the discussions at meetings, so this is where I'm jumping in.

Believe me, I will be the first to agree that this material is too advanced. Thanks for the links, though. I have mountains of QFT books and papers, and enough time to look at about 0.001 of it with any sort of rigor.

4. Jul 14, 2008

### Haelfix

Oye...

Ok, you need to put that down for now and go back a few steps, b/c you are headed to war without any ammunition. Start with Peskin and Schroeder and do a thorough word by word review of it for a few weeks. The latter half if you're already comfortable with Feynman diagrams, starting from chapter 15. You can't just jump into Slansky from the getgo like that, it will be completely opaque and you will waste more time by not starting earlier. There is a rather long list of notational changes from what you are probably used too, and many different complexities that you need to disentangle first and foremost.

Once you've mastered the Standard model, or at least up to and past chiral gauge theories (circa chapter 19ish), thats when you pick up a copy.

What you are looking at is known as a helicity basis, and the left and right Dirac fields can be made to transform under different representations of the group G. The idea being to analyze what happen when we restrict say the gauge fields to only couple to the left handed fields (this is the basis and beginning of the theory of the weak interaction). Now be careful, b/c there can be some relabeling at play here as well (I dont have Slansky in front of me).

5. Jul 14, 2008

### jpg15

I am trying to learn QFT concurrently, but at the moment, Slansky is what is being studied in journal club. I don't think that the group-theoretic discussion is unmanageable without a mastery of P&S, which won't come for some time.

Following up on your last paragraph, I interpret your remarks as saying that fR and fL are generic symbols to represent fermions in the helicity basis (is this the Weyl basis?). So the simplest case would be 4 component spinors with the top 2 components representing right-handed fermions and the bottom 2 representing left-handed fermions... is this correct?

6. Jul 14, 2008

### humanino

If I may add, I think you did not pay attention to what Haelfix told you.