Lattice Field Theory - QCD. Trace?

[anony]

Hi guys,

So I'm working on this project to simulate QCD using a computer using (hybrid) monte carlo. I follow the majority of what I've read thus far, though there are a few things I'm uncertain about. Firstly, the Wilson loop is often written as U_{P} and the invariant gauge action is

$$S_{G} = \sum_{P} Tr[ \frac{1}{2} ( U_{P} + U^{\dagger}_{P} ) ]$$

where the sum is over loops (plaquettes) P.

I don't follow what the Trace here means? U_{P} is an su(3) matrix (as opposed to SU(3)) right? But there are 3 colours in QCD. Does that mean there are 3 different loops, one for each colour? and that trace is a sum over colours? And then, doesn't U_{P} have a colour index, dirac index, spatial index and all that? So

$$Tr(U_{P}) = \sum_{c} U_{P}^{c}$$

Help appreciated with this.

Also, if you have any recommended texts for anything related to lattice gauge theories, particularly on a computer, please share :)

Thanks

EDIT: I don't know why the maths isn't showing, if a moderator sees it, please fix my tags :)

 

[eljose79]

Hi there,

Great to hear that you're working on simulating QCD using Monte Carlo methods! The Wilson loop and the gauge action are key concepts in lattice gauge theories, so it's important to have a clear understanding of them. Let me try to address your questions.

Firstly, the Trace in this context refers to the sum over the indices of the matrix. In this case, the matrix is an su(3) matrix, which means it has 8 complex numbers (or 8 generators) since SU(3) is a Lie group with 8 generators. So the sum over the Trace is essentially a sum over these 8 numbers.

U_{P} is indeed an su(3) matrix, and it represents the gauge field configuration at a given point on the lattice. The sum over loops (plaquettes) P is essentially a sum over all possible configurations of the gauge field on the lattice.

Now, for the second part of your question, U_{P} does have a color index, a Dirac index, and a spatial index. But the Trace is taken over the color indices, so it's a sum over the 8 complex numbers in the matrix. This is because the color indices are the only ones that contribute to the gauge action, as the other indices are taken care of by the U_{P} matrix.

As for recommended texts, I would suggest "Lattice Gauge Theories: An Introduction" by Thomas DeGrand and "Monte Carlo Methods in Lattice Field Theory" by I. Montvay and G. Münster. Both are excellent resources for understanding lattice gauge theories and their implementation on a computer.

I hope this helps clarify things for you. Good luck with your project!