2 Note added in proof to properly address the question by OP edited Sep 26 '17 at 20:41 DanielC 1,97311 gold badge1010 silver badges2020 bronze badges [quote][...]However, if i consider the following:(1/2,1/2)=(1/2,0)⊗(0,1/2), i don't understand how to decompose it. I know that represents a four vector field, with a temporal scalar component (spin 0) and vector component (spin 1) but i don't really understand why. [/quote] There is nothing to decompose with respect to the full $$\mbox{SL}(2,\mathbb{C})$$ because the spinor tensor with one undotted index and one dotted index is irreducible. Actually, you should write it backwards and apply the same rules as you already did for the 1-form representations: $$\left(\frac{1}{2},0\right)\otimes \left(0,\frac{1}{2}\right) = \left(\frac{1}{2},\frac{1}{2}\right)$$, and this because $$\frac{1}{2}+0 = \frac{1}{2}-0 = \frac{1}{2}$$ for both terms. Now here is something that addresses your question. $$\mbox{SU}(2)$$ is a proper subgroup of $$\mbox{SL}(2,\mathbb{C})$$. So one may ask if the representation $$\left(\frac12,\frac12\right)$$, which as I said is irreducible with respect to $$\mbox{SL}(2,\mathbb{C})$$, is perhaps reducible with respect to $$\mbox{SU}(2)$$. The answer is positive. The general formula below becomes $$\left(\frac12,\frac12\right)|_{\mbox{SU}(2)} \equiv D^0 \oplus D^1$$ [quote][...]However, if i consider the following:(1/2,1/2)=(1/2,0)⊗(0,1/2), i don't understand how to decompose it. I know that represents a four vector field, with a temporal scalar component (spin 0) and vector component (spin 1) but i don't really understand why. [/quote] There is nothing to decompose with respect to $$\mbox{SL}(2,\mathbb{C})$$ because the spinor tensor with one undotted index and one dotted index is irreducible. Actually, you should write it backwards and apply the same rules as you already did for the 1-form representations: $$\left(\frac{1}{2},0\right)\otimes \left(0,\frac{1}{2}\right) = \left(\frac{1}{2},\frac{1}{2}\right)$$, and this because $$\frac{1}{2}+0 = \frac{1}{2}-0 = \frac{1}{2}$$ for both terms. [quote][...]However, if i consider the following:(1/2,1/2)=(1/2,0)⊗(0,1/2), i don't understand how to decompose it. I know that represents a four vector field, with a temporal scalar component (spin 0) and vector component (spin 1) but i don't really understand why. [/quote] There is nothing to decompose with respect to the full $$\mbox{SL}(2,\mathbb{C})$$ because the spinor tensor with one undotted index and one dotted index is irreducible. Actually, you should write it backwards and apply the same rules as you already did for the 1-form representations: $$\left(\frac{1}{2},0\right)\otimes \left(0,\frac{1}{2}\right) = \left(\frac{1}{2},\frac{1}{2}\right)$$, and this because $$\frac{1}{2}+0 = \frac{1}{2}-0 = \frac{1}{2}$$ for both terms. Now here is something that addresses your question. $$\mbox{SU}(2)$$ is a proper subgroup of $$\mbox{SL}(2,\mathbb{C})$$. So one may ask if the representation $$\left(\frac12,\frac12\right)$$, which as I said is irreducible with respect to $$\mbox{SL}(2,\mathbb{C})$$, is perhaps reducible with respect to $$\mbox{SU}(2)$$. The answer is positive. The general formula below becomes $$\left(\frac12,\frac12\right)|_{\mbox{SU}(2)} \equiv D^0 \oplus D^1$$ 1 answered Sep 26 '17 at 20:16 DanielC 1,97311 gold badge1010 silver badges2020 bronze badges [quote][...]However, if i consider the following:(1/2,1/2)=(1/2,0)⊗(0,1/2), i don't understand how to decompose it. I know that represents a four vector field, with a temporal scalar component (spin 0) and vector component (spin 1) but i don't really understand why. [/quote] There is nothing to decompose with respect to $$\mbox{SL}(2,\mathbb{C})$$ because the spinor tensor with one undotted index and one dotted index is irreducible. Actually, you should write it backwards and apply the same rules as you already did for the 1-form representations: $$\left(\frac{1}{2},0\right)\otimes \left(0,\frac{1}{2}\right) = \left(\frac{1}{2},\frac{1}{2}\right)$$, and this because $$\frac{1}{2}+0 = \frac{1}{2}-0 = \frac{1}{2}$$ for both terms.