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 $SU(2) \times SU(2)$."

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

Minor comment to the post (v3): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. — Qmechanic, Dec 14 '18 at 19:36
“The universe is an enormous direct product of representations of symmetry groups.” – Steven Weinberg . — Cosmas Zachos, Jul 01 '20 at 13:59

Qmechanic · Accepted Answer · 2018-12-14T19:59:40.767

6

On p. 67 Tseng means a tensor product of representations.
On p. 70 Tseng means a direct product of groups.

Note however that the actual statement about the Lorentz group is wrong/imprecise as explained in e.g. this Phys.SE post.

Concerning direct product vs. tensor product of groups, see also my related Phys.SE answer here.

edited Dec 14 '18 at 19:59

answered Dec 14 '18 at 19:49

Qmechanic

201,751

score 1 · Answer 2 · answered Dec 14 '18 at 19:18

In both the cases the author is talking about direct product. Addition of two angular momentums $j_1$ and $j_2$ is represented using the direct product of the two angular momentum spaces. The wave functions or kets in the product space are represented using tensor product notation.

Direct Product vs Tensor Product

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