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I am going through the notes on QFT by Srednicki.

When describing fermions, from the very beginning he introduces the Lorentz Group and its algebra, and proves that it is equivalent to two copies of $SU(2)$, so that a representation is specified by two (half) integers, say $n,n'$ (see pages 213-214). He writes such a representation as $(2n+1,2n'+1)$.

For example, some important representations are $(1,1)$: scalar, $(2,1)$: left-handed spinor and so on. Some pages later (p. 217), he writes the relation $2\otimes 2=1\oplus 3$, which is just the usual result from addition of angular momentum. My problem is, some pages later (p. 219) he writes the following "group theoretic relation" $$(2,2)\otimes (2,2)=(1,1)\oplus(1,3)\oplus(3,1)\oplus(3,3).$$ I am having a hard time trying to understand this relation.

At first glance, it seems that we have to add four spin one-half momenta, that is, $$(2,2)\otimes(2,2)=2\otimes2\otimes2\otimes2.$$ If I go through the usual steps to construct such a sum, there is no way I get the expected result.

On the other hand, if I write $(2,2)=1\oplus 3$ and "distribute $\oplus$ over $\otimes$" as if they were actual products and sums, I get the result given by Srednicki, but I feel there is something wrong about that. Maybe I feel it's wrong just because there is something I'm missing or that I don't understand.

If this "distribute $\oplus$ over $\otimes$" is the right thing to do, I would really appreciate someone to explain why is that. If it's not the right thing, then I'd be glad if someone told me how should I deal with "group theoretic relations" like these, or where could I find some literature about this subject.

For example, on page 218, Srednicki writes $$(2,1)\otimes(1,2)\otimes(2,2)=(1,1)\oplus...$$

If my approach is the right one, then, as $$(2,1)\otimes(1,2)=1\oplus3=(2,2),$$ the full answer is $$(2,1)\otimes(1,2)\otimes(2,2)=(2,2)\otimes(2,2)=(1,1)\oplus(1,3)\oplus(3,1)\oplus(3,3).$$ Is this it?

AccidentalFourierTransform
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1 Answers1

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There is a subtle difference between saying $(2,2)$ and $2\otimes 2$. In the latter case we are thinking of both reps as transforming under the same element of the group $SU(2)$. In the former case we are thinking of $(2,2)$ as transforming under the Lorentz group, which contains two distinct copies of $SU(2)$. Call one copy the $L$ copy and the other the $R$ copy. Then the four basis vectors of $(2,2)$ are $0_L 0_R, 0_L 1_R,\dots$ etc. These four basis vectors do not separate out into $1\oplus 3$ since I can choose elements of the Lorentz group that only rotate one of the two representations.

So think of $(2,2)\otimes(2,2) = (2\otimes 2,2\otimes 2),$ which has basis vectors like e.g. $0_{L1} 1_{R1} \otimes 0_{L2} 1_{R2}$, so then I can apply addition of angular momenta between the two Ls and Rs. Then $(1\oplus 3,1\oplus 3)$ means you take all of the basis vectors of $(1\oplus 3)_L$ and tensor product with all the basis vectors of $(1\oplus 3)_R$. So it does distribute.

So when you write $(2,1)\otimes (1,2)$ think of it like $$(2,1)\otimes (1,2)=(2\otimes 1,1\otimes 2)= (2,2)$$

octonion
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