Direct Product vs Tensor Product

Question

I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product:

On page 67, it mentioned that "you can take a direct product of two j = 1/2 representations" and build representations of higher j.

On page 70, it mentioned "we can think of [the Lorentz Group] as the direct product SU(2) × SU(2)"

In each of the above, does the author mean Direct Product or Tensor Product?

score 1 · Accepted Answer · answered Dec 14 '18 at 19:15

1

On page $70$ the author speaks about the Lie algebras, not of the groups. So he means the direct sum $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$ of Lie algebras. In fact, $(8.14),(8.15),(8.16)$ are Lie brackets. He calls this "$SU(2)$ algebras". On the group level, $SU(2)\times SU(2)$ denotes the direct product in the usual sense. For the "complexities product" he describes, see here:

Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)\times SU(2)$, or their Lie algebras

answered Dec 14 '18 at 19:15

Dietrich Burde

130,978

So in page 70 it's Direct Product; what about page 67? is it tensor or direct product? – The Gypsy King Dec 15 '18 at 06:08
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See wikipedia, in particular the section " the $SU(2)$ case". – Dietrich Burde Dec 15 '18 at 09:12
Ah I see.. so the p67 is Tensor Product? (just to double check) – The Gypsy King Dec 15 '18 at 09:16
Yes, $\rho\otimes \sigma$ denotes tensor product. – Dietrich Burde Dec 15 '18 at 09:17

Direct Product vs Tensor Product

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