How do quarks transform under SO(3)?

[kexue]

Quarks transform in the fundamental representation under SU(3), gluons in the adjoint represention under SU(3), leaving the theory (QCD) invariant.

But how can I find out how quarks transform under SO(3)? I know how to decompose a Lorentz tensor under SO(3), like a four-vector or EM field strength vector. But what about fields in SU(3) representations? How can I see that the quarks are spinors and the glouns are vectors?

hope the question makes sense, thank you

 

[tom.stoer]

I guess by SO(3) you mean the rotation sub-group of the Lorentz group SO(3,1). Quarks transform as Dirac-spinors = in the (1/2, 1/2) representation similar to electrons. The indices look like

$$q^i_a$$

where i=1..3 means the SU(3) color index and a=1..4 means the Dirac spinor index. That means q is a 3*4 matrix with 3 color and 4 Lorentz indices. The two different transformations act on "their" indices. They look like

$$q^i_a \to (q^\prime)^i_a = S(\Lambda)_{ab} q^i_b$$

$$q^i_a \to (q^\prime)^i_a = U_{ik} q^k_a$$

S means a 4*4 Lorentz transformation; U means a 3*3 SU(3) color transformation.

 

[kexue]

Thanks, Tom!

And for gluons similar, I suppose?

 

[tom.stoer]

Gluons look like

$$A_\mu^a$$

where a=1..8 is the color-index in the adjoint rep. Alternatively one can define

$$(A_\mu)_{ik} = A_\mu^a (T^a)_{ik}$$

where you can see the "bi-color" index i,k

 

[kexue]

thanks again, Tom!

I got lost in the indices, I guess..

 

[Parlyne]

It's probably worthwhile to point out that there is no connection whatsoever between the SU(3) representation and the spin of the particles in the representation. For instance, if the SU(3) symmetry were actually just a part of some larger (spontaneously broken) symmetry, such as SU(5), there would be (heavy) spin-1 particles in the fundamental representation SU(3), while if SUSY is correct, there are spin-1/2 particles in the adjoint.

At a more fundamental level, we know that quarks are fermions because proton is a fermion, meaning that something among its constituents must be fermionic, and that gluons are spin-1 because that is what is required of the gauge connection of a simple Lie group.

 

[tom.stoer]

That's a good point; in the standard model space-time symmetries like SO(3,1) and internal symmetries like U(1), SU(2) and SU(3) are unrelated.