Physical interpretation of charges and casimirs for SU(2) and SU(3)

[Bobhawke]

For SU(2) rotational invariance there is a clear interpretation of what all the generators mean. They correspond to operators whose eigenvalues are some observable. The noether charges correspond to the x,y and z components of angular momentum. The casimir is just the total angular momentum squared.

I want a similar interpretation for SU(2) weak and SU(3) colour in the standard model. For SU(2) weak there is one casimir and 3 generators - What observable quantities do they correspond to? SU(3) has 2 casimirs and 8 generators, again what observables do these correspond to?

 

[AndyW]

Hello, thank you for your post. It is great to see interest in understanding the interpretations of SU(2) weak and SU(3) color in the standard model. Let me provide some clarification and explanations for these groups.

Firstly, for SU(2) rotational invariance, the generators correspond to the x, y, and z components of the angular momentum because they describe the rotation of a system around these axes. The eigenvalues of these generators are observables, meaning they can be measured in experiments.

For SU(2) weak, the group represents the symmetry of the weak nuclear force in the standard model. The generators of this group correspond to the three gauge bosons: W+, W-, and Z. These particles are responsible for the weak interactions between particles and their eigenvalues represent the strength of these interactions.

The casimir in SU(2) weak is related to the Higgs field, which is responsible for giving particles mass. The eigenvalue of the casimir is proportional to the mass of the particles, making it an observable quantity as well.

Moving on to SU(3) color, this group represents the symmetry of the strong nuclear force in the standard model. The generators of this group correspond to the eight gluons, which are responsible for the strong interactions between quarks. The eigenvalues of these generators represent the strength of these interactions.

The two casimirs in SU(3) color correspond to the color charge and the color magnetic moment. The color charge is a property of quarks and the eigenvalue of the casimir is proportional to the strength of this charge. The color magnetic moment is related to the interaction between gluons and the eigenvalue of the casimir represents the strength of this interaction.

In summary, the generators and casimirs of SU(2) weak and SU(3) color correspond to observable quantities related to the weak and strong nuclear forces in the standard model. They provide a mathematical framework for understanding the symmetries and interactions of particles at the subatomic level. I hope this helps to clarify their interpretations.

 

[sir-pinski]

For SU(2) weak in the standard model, the one casimir corresponds to the weak isospin, which is a fundamental property of elementary particles that determines their interactions with the weak force. The three generators correspond to the three components of the weak isospin, just like the x,y, and z components of angular momentum in the rotational invariance interpretation. These generators are responsible for the transformations between different states of the weak isospin, similar to how the angular momentum operators transform between different states of angular momentum.

For SU(3) color in the standard model, the two casimirs correspond to the color charge and the color hypercharge. The color charge is a property of elementary particles that determines their interactions with the strong force, while the color hypercharge is related to the symmetry breaking of the SU(3) gauge group. The eight generators correspond to the eight gluons, which are the force carriers of the strong force. These generators are responsible for the transformations between different states of color charge and color hypercharge, just like how the angular momentum operators transform between different states of angular momentum.

In summary, the generators and casimirs of SU(2) weak and SU(3) color in the standard model have a similar interpretation to those in SU(2) rotational invariance, but with different fundamental properties and force carriers. They all correspond to observable quantities that play a crucial role in understanding the interactions of elementary particles.