# Lattice QCD, path integral, single "path", what goes on at a point?

1. Nov 5, 2014

### Spinnor

Say we try and calculate the ground state energy of the bound state of a quark antiquark meson via lattice QCD. Say I look at one space time lattice point of one path. Do the fermi fields "live" on the lattice points? Do the boson fields "live" on the legs between the space time lattice points?

Could it be the other way, fermi fields on the legs and boson fields at the lattice points?

How many numbers do I need to specify the state of the fermi fields at a space time point?

How many numbers nail down the state of the boson fields on one of the legs?

Thanks for any help!

2. Nov 11, 2014

### Greg Bernhardt

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 11, 2014

### Spinnor

I am working on it, further reading and I have learned they are called links, not legs. Also a google of "lattice gauge theory introduction" has lots of info just not very basic.

Thanks!

4. Nov 14, 2014

### The_Duck

This sentence doesn't seem directly related to the rest of your post, but if you are interested in this kind of thing you might look at this tutorial which explains how to do similar calculations.

Yes.

Yes.

The usual formulation with fermion fields at the lattice sites and gauge fields on the links between the lattice sites is specially constructed to ensure that gauge invariance is exactly satisfied even at finite lattice spacing. You will have some trouble getting exact gauge invariance if you switch this around.

In an SU(3) gauge theory like QCD, a Dirac field in the fundamental representation (e.g., a quark field) has four spinor indices and three color indices. So you need 12 complex numbers at each point in spacetime, or, on the lattice, 12 complex numbers at each lattice site.

Eight real numbers, one for each SU(3) generator.

5. Nov 14, 2014

### Spinnor

Thanks for taking time to help!

12 complex numbers, do I need to multiply by 3 if I include all three families of quarks?

The number 8 for the gluons seems low but I don't know. 8 real numbers for the 8 different gluons? Do we need numbers for the phase of the gluons and numbers for their polarization? Gluons can be polarized like photons?

Thanks for the link, I will look for answers there.

Thanks for you help!

6. Nov 15, 2014

### The_Duck

Sure, you need 12 complex numbers for each quark flavor. Most lattice simulations today include just three flavors: up, down, and strange.

Right.

In continuum field theory the gluon field is a real field $A^a_\mu(x)$ where the flavor index $a$ runs from 1 to 8 and the Lorentz index $\mu$ runs from 0 to 3. So there are $8 \times 4$ real numbers at each spacetime point. Similarly on the lattice, there are 8 real numbers per link and four links per spacetime point (one link in each spacetime direction). The factor of four accounts for the polarization degrees of freedom.

7. Nov 16, 2014

### Spinnor

That is right, I forgot there are more links then points. Thanks for clearing that up for me!