The anti-commutation relations for Gamma matrices $\big\{\gamma ^\mu , \gamma ^\nu \big\} = 2g ^{\mu \nu} $ can be used for interchanging positions of the respective matrices in a given expression, for example : $-i\gamma ^\mu \gamma ^2 \gamma ^0 = i\gamma ^2 \gamma ^0 \gamma ^{\mu} $. Question - Do we have any similar prescription for interchanging positions (of objects) that don't belong to the same space, for examlple: can we interchange the positions of $\lambda _A $ with $\tau _2 $ or (say) $\tau _2 $ with $\gamma _5 $ in the following expression (via some rule):
$\bar {\psi} \gamma ^0 \lambda _A \tau _2 \gamma _5 \gamma _0 \psi $ ?
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2 Answers
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Assuming I understand your question:
The Gell-Mann matrices are the generators of the $SU(3)$ Lie algebra - $3 \times 3$ matrices analogous to the $SU(2)$ matrices of dimension $2 \times2$ - the Pauli spin matrices.
The Dirac or gammas matrices as above are of dimensions $4 \times4$.
Given these matrices are all of different dimensions they cannot be multiplied and therefore do not “mix”, so there is no commutation relation that involve these three different matrices.
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$\begingroup$ since the matrices do not “mix”, interchanging their positions has no meaning whatsoever, or, they can be interchanged in an expression without any consequence? Thanks $\endgroup$ Commented Oct 18, 2020 at 7:58
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1$\begingroup$ The point is you cannot put them together in the first place. It would be meaningless. You cannot multiply 3x3 4x4 or 2x2. This is just a consequence of matrix algebra. Hope this helps. Cheers $\endgroup$– joseph hCommented Oct 18, 2020 at 8:03
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$\begingroup$ they can be multiplied - look at equations (4) and (6) here - link $\endgroup$ Commented Oct 20, 2020 at 10:38
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$\begingroup$ These matrices are a representation of quarks in a higher dimensional - flavour- space and not the 2x2 Pauli matrices you stated above. Very interesting paper though. Thanks for the read. Cheers $\endgroup$– joseph hCommented Oct 20, 2020 at 11:31
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$\begingroup$ you're welcome! But my question remains the same - can we/can't we interchange the positons of $\lambda _A$'s, $\gamma $'s and $\tau $'s (strictly with reference to the paper shared above) without any consequence? Thanks $\endgroup$ Commented Oct 20, 2020 at 13:04
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Add indices to all matrices. The only thing you should then worry about is whether they are real or Grassmann numbers.
