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In Peskin & Schroeder section $3.2$ they begin by telling us that they want to construct spin $1/2$ representations of the Lorentz Lie algebra. One way to do that, they say, is to first find a representation of the Dirac algebra from which one can construct the Lorentz Lie algebra via commutator. Then they say that the smallest possible representation of the Dirac Algebra are $4 \times 4$ matrices, implying that the smallest possible (complex) dimension of the representation space is $4$. But then in problem $3.1$ when we are asked to find all the representations of the Lorentz Lie algebra we find there are $0$ and $2$ dimensional representations as well (the $(0,1/2)$, and $(1/2,0)$ representations for instance). So how are the above two statements compatible? Aren't the $(0,1/2)$, and $(1/2,0)$ also spin $1/2$ representations of the Lorentz Lie algebra as well?

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    $\begingroup$ The $(1/2,0),(0,1/2)$ representations do not contain Dirac algebra ($\gamma$ matrices) so the statements are compatible. The Dirac algebra is convenient for constructing non trivial representations but it is not necessary. $\endgroup$
    – LPZ
    Commented May 23, 2023 at 18:54
  • $\begingroup$ Ah. I must say that it is very misleading from P&S to leave this out. They literally say and I quote "we can construct such representations from a trick due to Dirac...". Implying heavily that any such representation can be arrived at by this method. In any case, we are interested in the Dirac algebra precisely because it leads to the "square root of KG" correct? Otherwise, there is no reason to single out the $(1/2,1/2)$ representation out of the others. $\endgroup$
    – Leonid
    Commented May 23, 2023 at 19:09
  • $\begingroup$ I think you meant $(1/2,0)\oplus(0,1/2)$ (Dirac spinor like the electron field) and not $(1/2,1/2):=(1/2,0)\otimes(0,1/2)$ (4-vector like the EM potential field). While it is reducible in terms of the restricted Lorentz group, it is irreducible for the entire Lorentz group (including parity). It is therefore the smallest non trivial irreducible representation (which is why KG can be deduced from it), and the basic building block for constructing more elaborate representations by direct sum/tensor product. $\endgroup$
    – LPZ
    Commented May 23, 2023 at 20:19

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