- #1

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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

- Thread starter yola
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- #1

- 17

- 0

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

- #2

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- #3

tom.stoer

Science Advisor

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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.

- #4

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

- #5

tom.stoer

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[tex]\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}[/tex]

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

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