# Standard Model Lagrange Density, 2D vectors, Lattice Theory.

1. Oct 23, 2009

### Spinnor

In the article "The Lattice Theory of Quark Confinement", by Claudio Rebbi (Scientific American) there is a graphic representing the chromoelectric field. The caption reads:

"Chromoelectric field is a gauge field similar in principle to the electromagnetic field but more complicated mathematically. At every point on a lattice there are three arrows instead of one; they correspond to the color charges of a quark. Moreover, the color guage field affects not only the direction of each arrow but also its length. ..."

Are such lattice calculations considered classical or does one need to bring the full force of the quantized theory?

On a lattice can I assume we can let 2-dimensional arrows represent every "degree of freedom" of the Standard Model Lagrange Density?

If so, how many arrows at a point do we need to account for the "freedom" of both the matter fields and the force fields?

The color force seems to need three arrows, two arrows for the weak force, one arrow for the electric field, a non-interacting Dirac particle needs 4 arrows at each point in a space-time lattice for each Dirac particle?

Lots of arrows? If string theory turns out to be true how many arrows do we need to represent whats going on at a point in a space-time lattice?

Are there relationships among the various arrows which reduces their "freedom"?

If there is a lattice approximation of classical general relativity? Can we represent the physics of general relativity with some number of 2-dimensional arrows at each point in the space-time lattice? (yikes, now the space-time lattice is not fixed?)

2. Oct 27, 2009

### Spinnor

Doing a Google search of "internal degrees of freedom" I came up with this thread which I think addressed one of my questions above. The post I think has a simple math error and I would appreciate your thoughts on the accuracy of the post. The thread is found at:

The question is:

"How many degrees of freedom exist?

My question is straight forward: how many degrees of freedom exist for any given sub atomic particle? I would say that 3 spatial dimensions as well as time do. Quarks exhibit 3 degrees of freedom in the form of color charge; would another degree be electromagnetic charge? Similarly, I\'ve read that spin offers another 4 degrees of freedom: particle and antiparticle states with two possibilities for handedness. That would bring the total number of degrees of freedom for a particle up to 12, though some particles don\'t experience all these degrees (Neutrinos only see two spin degrees, leptons don\'t seem to carry color charge, etc). Is this all correct? Are there perhaps any more degrees of freedom?"

"This is almost certainly not the right way to think about the problem. In field theory, spatial coordinates are not properties of particles; rather, particles are properties of spatial coordinates. In addition, most of the degrees of freedom you specified should be considered different kinds of particle. So the real question would be, how many different kinds of particle are there?

Quarks
3 generations
3 colors
4 isospins (2 in a doublet + 2 in singlets)
2 spins
2 matter-antimatter
= 144 DoF

Leptons
3 generations
1 color
3 isospins (2 in a doublet + 1 in a singlet) <-- assumes no right-handed neutrino!
2 spins
2 matter-antimatter <-- assumes neutrino not its own antiparticle
= 108 DoF "

Math error? The number 108 should be 48?

"Vector Bosons
8 gluons + 4 W-Z-photons
2 polarizations
= 24 DoF

Higgs
2 isospins
2 complex
= 4 DoF <-- 3 become transverse polarizations of W and Z, 1 remains higgs

Total: 280 DoF?"

For a total of 220 DoF?

"If I got all these right, that is...

Xerxes"

So to answer my question, the Standard Model Lagrange Density has about 220 degrees of freedom?

So do I understand it correctly, string theory has all these degrees of freedom and more?

Does string theory nearly double the number of degrees of freedom of the standard model because of super-symmetry?

Can someone give me a hand waving explanation how in principle we would count the degrees of freedom in string theory?

Thank you for any thoughts!

Last edited by a moderator: Apr 24, 2017
3. Oct 27, 2009

### RUTA

There is a "lattice approximation" to GR, it's called Regge calculus. The basic variable is the square of a link length. The action is varied with respect to this quantity and it is, of course, directly related to the metric. There is a good introduction in Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman, San Francisco (1973), p 1162 (or thereabouts).

4. Nov 3, 2009

### Spinnor

Is there a pattern below?

5 arrows worth of information at a point for General Relativity,
4 arrows worth of information at a point for Dirac field,
3 arrows worth of information at a point for color field,
2 arrows worth of information at a point for weak field,
1 arrows worth of information at a point for electric field?