QCD Lagrangian: Multiplying 4x4 and 1x3 Matrices

[yola]

Hello,
In the lagrangian of QCD, there is q which is the quark field and it is the fundamental representation of SU(3). This q is multiplied by a gamma matrix and a q bar. So, how can we have a 4x4 matrix multiplying 1x3 matrix?
Thanks

 

[chrispb]

q has not only a color index, but also a dirac spinor index. It is the spinor index that the gamma matrices contract. The color indices of q and qbar are contracted with each other for the spatial derivative term, and contracted with the T^a for the interaction with the gluon field term.

 

[tom.stoer]

$$\bar{q} \gamma^\mu A_\mu q = \bar{q}_{i\alpha} \left(\gamma^\mu)^{\alpha\beta} \left(A_\mu\right)_{ik} q_{k\beta} = \bar{q}_{i\alpha} \left(\gamma^\mu)^{\alpha\beta} A^a_\mu \left(T^a\right)_{ik} q_{k\beta}$$

So the q's are 4-spinors (greek indices) with an additional color index i=1..3, the A's are 4-vectors with an additional color index a (in the adjoint rep. i.e. a=1..8) or an additional color-index pair ik=1..3.

 

[dextercioby]

Indices for flavor should be included as well, so that the gauge field has 3 types of indices.

 

[tom.stoer]

OK, let's do that for the interaction term using f as the flavor index ...

$$\bar{q} \gamma^\mu A_\mu q = \bar{q}_{i\alpha f} \left(\gamma^\mu)^{\alpha\beta} \left(A_\mu\right)_{ik} q_{k\beta f} = \bar{q}_{i\alpha f} \left(\gamma^\mu)^{\alpha\beta} A^a_\mu \left(T^a\right)_{ik} q_{k\beta f}$$

... which means that the kinetic energy (which haven't written down here) and the gluon-interaction is flavor-neutral.