I am a newbie in group theory. In Peskin & Schroeder's QFT P.42 (3.29) it says that, since we have
$$\Lambda_{\frac{1}{2}}^{-1}\gamma ^{\mu}\Lambda_{\frac{1}{2}}~=~\Lambda^{\mu}_{~~\nu }\gamma^{\nu }, \tag{3.29}$$
we can say that "$\gamma$ matrices are invariant under simultaneous rotations of their vector and spinor indices (just like the $\sigma$ under spatial rotations)." In other words, "we can take the vector index $\mu$ on $\gamma^{\mu}$ seriously," and dot $\gamma ^{\mu }$ into $\partial _{\mu }$ to form a Lorentz-invariant differential operator.
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I don't understand the phrase that "$\gamma$ matrices are invariant under simultaneous rotations of their vector and spinor indices (just like the $\sigma$ under spatial rotations)." It seems that there are two different things, one is related to spin representation and the other is related to Lorentz transformation.
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What does "we can take the vector index $\mu$ on $\gamma^{\mu}$ seriously," mean?
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What is the difference between spin indices and the space-time indices?
Best Answer
As you probably know, there are different irreducible representations of the different symmetry groups that one gets in relativistic quantum field theory. Here we are dealing with the Lorentz symmetry (part of the Poincare symmetry).
One representation, the spin-half representation, is associated with fermions as one finds in the Dirac equation. The indices of this representations are the spin indices, as denoted by the rows and columns of the Dirac matrices. In this representation the Lorentz transformations are represented by spin transformations.
Another representation, the spin-one representation, is associated with vector fields, such as the gauge bosons. It is also the representation for the coordinate vectors. The indices in this case are the space-time indices that are usually denoted by some Greek letter such as $\mu$. In this representation the Lorentz transformations transform the space-time indices (rotations and boosts).
One can combine a pair of spin-half quantities in a why that they act like spin-one quantities. (This is part of a more general process whereby tensor products of irreducible representations of a symmetry group become direct sums of different irreducible representations of that symmetry group.) This means that a product of two spin-half quantities, each transforming by spin transformations, can act like a single spin-one quantity, transforming as a space-time vector, provided that the two quantities are combined in a particular way. It turns out that the Dirac matrices provide the correct way to combine two spin-half quantities so that they transform as a single spin-one quantity. Equation (3.29) demonstrates this property.
Hope this addresses all your questions.