# Direct Product vs Tensor Product

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:

1. On page 67, it mentioned that

"you can take a direct product of two $$j = 1/2$$ representations"

and build representations of higher j.

2. 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

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.